For an atom or ion with \(Z\) protons and \(n\) electrons, each proton contributes \(+e\) and each electron \(-e\) to the total charge:

\[ q_{net} = Z(+e) + n(-e) = (Z-n)e \]

For a neutral atom, \(q_{net}=0\), which forces \(n=Z\): the number of electrons must exactly equal the number of protons. Removing \(k\) electrons (\(n=Z-k\)) gives a cation of charge \(+ke\); adding \(k\) electrons (\(n=Z+k\)) gives an anion of charge \(-ke\) — e.g. \(Na\) (\(Z=11\)) losing one electron gives \(Na^+\), and \(Cl\) (\(Z=17\)) gaining one gives \(Cl^-\).

With both an electric field \(E\) and a magnetic field \(B\) applied (perpendicular to each other and to the beam), the fields are tuned until the beam is undeflected: the electric force balances the magnetic force,

\[ eE = evB \quad\Longrightarrow\quad v = \frac{E}{B} \]

giving the electron's speed directly. Next, the magnetic field is switched off, leaving only \(E\), which deflects the beam by \(y\) over a plate length \(L\) (time in the field \(t=L/v\)) via acceleration \(a=eE/m\):

\[ y = \tfrac{1}{2}at^2 = \tfrac{1}{2}\frac{eE}{m}\left(\frac{L}{v}\right)^2 \]

Solving for \(e/m\) and substituting \(v=E/B\):

\[ \frac{e}{m} = \frac{2yv^2}{EL^2} = \frac{2yE}{B^2L^2} \]

Every quantity on the right (\(y\), \(E\), \(B\), \(L\)) is directly measurable, giving \(e/m \approx 1.76\times10^{11}\ \text{C kg}^{-1}\) — about 1800 times larger than any known ion's ratio, hinting the electron is far lighter than any atom.

Model the unknown neutral radiation as a particle of mass \(m\) and speed \(v\) making an elastic, head-on collision with a stationary target nucleus of mass \(M\). Conservation of momentum and kinetic energy for this 1-D elastic collision gives the target's recoil speed:

\[ V = \frac{2mv}{m+M} \]

Chadwick measured the maximum recoil speed for two different targets: protons (\(M=m_p\), giving recoil \(V_p\)) and nitrogen nuclei (\(M=14m_p\), giving recoil \(V_N\)). Taking the ratio eliminates the unknown incoming speed \(v\):

\[ \frac{V_p}{V_N} = \frac{m+14m_p}{m+m_p} \]

With \(V_p\) and \(V_N\) both measured experimentally, this is one equation in one unknown, \(m\). Solving gives \(m \approx m_p\) — the new particle's mass is approximately equal to the proton's, far too heavy to be a gamma-ray photon (which is massless), confirming a new, roughly proton-mass, neutral particle.

For a head-on collision, an alpha particle (charge \(+2e\)) approaching a gold nucleus (charge \(+79e\)) slows down as Coulomb repulsion converts its kinetic energy into potential energy, until, at the distance of closest approach \(r_{min}\), all of its kinetic energy has been converted:

\[ KE_{\alpha} = \frac{1}{4\pi\varepsilon_0}\cdot\frac{(2e)(79e)}{r_{min}} \]

Solving for \(r_{min}\):

\[ r_{min} = \frac{1}{4\pi\varepsilon_0}\cdot\frac{158e^2}{KE_{\alpha}} \]

For alpha particles of a few MeV, this gives \(r_{min}\) on the order of \(10^{-14}\ \text{m}\) — roughly 10,000 times smaller than the atom's overall radius (\(\sim 10^{-10}\ \text{m}\)), directly showing the nucleus must be extremely small and dense to repel the alpha particle back the way it came.

The average atomic mass is a weighted mean over all naturally occurring isotopes, weighted by their fractional abundance \(f_i\) (which sum to 1):

\[ M_{avg} = \sum_i f_i\,M_i \]

Worked example (chlorine): naturally occurring chlorine is 75.77% \({}^{35}Cl\) (mass 34.969 u) and 24.23% \({}^{37}Cl\) (mass 36.966 u):

\[ M_{avg} = (0.7577)(34.969) + (0.2423)(36.966) \]

\[ M_{avg} = 26.494 + 8.960 = 35.45\ \text{u} \]

This matches the non-integer value, 35.45, listed for chlorine on the periodic table — a direct signature that the tabulated mass is an isotopic average, not the mass of any single atom.

In one period \(T=1/\nu\), a wave advances exactly one wavelength \(\lambda\), so its speed is distance over time:

\[ c = \frac{\lambda}{T} = \lambda\nu \]

Now check the empirical Rydberg formula against the first line of the Balmer series (\(n_1=2\), \(n_2=3\)):

\[ \frac{1}{\lambda} = R_H\left(\frac{1}{4}-\frac{1}{9}\right) = R_H\times 0.1389 \]

\[ \lambda = \frac{1}{(1.097\times10^7)(0.1389)} \approx 6.56\times10^{-7}\ \text{m} = 656\ \text{nm} \]

This matches the observed red hydrogen line (Hα) almost exactly — striking empirical success for a formula with no theoretical basis at the time it was proposed.

The average energy of a quantised oscillator at temperature \(T\) (from Boltzmann statistics over the allowed levels \(E=nh\nu\)) works out to:

\[ \langle E \rangle = \frac{h\nu}{e^{h\nu/k_BT}-1} \]

In the low-frequency limit, \(h\nu \ll k_BT\), expand the exponential using \(e^x \approx 1+x\) for small \(x\), with \(x=h\nu/k_BT\):

\[ e^{h\nu/k_BT}-1 \approx \frac{h\nu}{k_BT} \]

Substituting back:

\[ \langle E \rangle \approx \frac{h\nu}{h\nu/k_BT} = k_BT \]

This recovers the classical equipartition result, \(\langle E\rangle = k_BT\), independent of frequency — showing quantisation reduces smoothly to classical behaviour at low frequency, while at high frequency (\(h\nu \gg k_BT\)) the exponential in the denominator forces \(\langle E\rangle \to 0\), which is exactly what avoids the ultraviolet catastrophe.

Rearranging Einstein's equation with \(\phi = h\nu_0\) (the threshold condition, where \(KE_{max}=0\)):

\[ KE_{max} = h\nu - h\nu_0 = h(\nu-\nu_0) \]

Experimentally, \(KE_{max}\) is measured via the stopping potential \(V_0\) — the retarding voltage just sufficient to stop even the fastest photoelectrons, where \(eV_0 = KE_{max}\):

\[ eV_0 = h\nu - h\nu_0 \quad\Longrightarrow\quad V_0 = \frac{h}{e}\nu - \frac{h}{e}\nu_0 \]

Plotting \(V_0\) against \(\nu\) gives a straight line of slope \(h/e\) and \(\nu\)-intercept \(\nu_0\) — exactly the graphical method (used by Millikan) that provided an independent, precise experimental measurement of Planck's constant, strongly confirming Einstein's photon model.

Coulomb attraction supplies the centripetal force for a circular orbit:

\[ \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2} = \frac{mv^2}{r} \quad\Longrightarrow\quad v^2 = \frac{e^2}{4\pi\varepsilon_0 mr} \]

Combine with the quantisation condition \(v = nh/(2\pi mr)\), squared:

\[ \frac{n^2h^2}{4\pi^2m^2r^2} = \frac{e^2}{4\pi\varepsilon_0 mr} \quad\Longrightarrow\quad r_n = \frac{n^2h^2\varepsilon_0}{\pi m e^2} \]

giving allowed radii proportional to \(n^2\) (for \(n=1\), this is the Bohr radius, \(a_0 \approx 0.529\ \text{Å}\)). Total energy is kinetic plus Coulomb potential energy, \(E = \tfrac{1}{2}mv^2 - \dfrac{e^2}{4\pi\varepsilon_0 r}\); using \(v^2\) from above and substituting \(r_n\):

\[ E_n = -\frac{me^4}{8\varepsilon_0^2h^2}\cdot\frac{1}{n^2} \]

Numerically, \(E_n = -13.6\ \text{eV}/n^2\) for hydrogen — the negative sign shows the electron is bound, with energy rising (becoming less negative) toward zero as \(n\to\infty\) (ionisation).

The photon energy for a transition from level \(n_2\) to \(n_1\) (\(n_2>n_1\)), using Ch. 9's \(E_n\) formula, is:

\[ h\nu = E_{n_2}-E_{n_1} = \frac{me^4}{8\varepsilon_0^2h^2}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) \]

Dividing by \(hc\) to convert to \(1/\lambda\) reproduces exactly the empirical Rydberg formula from Ch. 6, \(1/\lambda = R_H(1/n_1^2-1/n_2^2)\), identifying:

\[ R_H = \frac{me^4}{8\varepsilon_0^2h^3c} \]

Plugging in fundamental constants gives \(R_H \approx 1.097\times10^7\ \text{m}^{-1}\) — matching the purely empirical value from decades earlier almost exactly. This agreement was Bohr's great triumph for hydrogen. But for helium (\(Z=2\), two electrons), the same style of calculation, which ignores electron–electron repulsion entirely, predicts energy levels that disagree noticeably with experiment — the model's fundamental limitation.

For a photon, combine \(E=h\nu=hc/\lambda\) with the relativistic energy–momentum relation for a massless particle, \(E=pc\):

\[ pc = \frac{hc}{\lambda} \quad\Longrightarrow\quad \lambda = \frac{h}{p} \]

De Broglie proposed this relation holds for any particle, not just photons: \(\lambda = h/(mv)\). This has a striking consequence for Bohr's model (Ch. 9): for an electron's orbit to persist as a stable standing wave, the orbit's circumference must fit a whole number of wavelengths — otherwise the wave would destructively interfere with itself on each pass:

\[ 2\pi r = n\lambda = \frac{nh}{mv} \quad\Longrightarrow\quad mvr = \frac{nh}{2\pi} \]

This is exactly Bohr's angular-momentum quantisation condition — but now it emerges naturally from the wave nature of matter, rather than being assumed as an ad hoc postulate.

To locate an electron's position to within \(\Delta x\), you must illuminate it with light whose wavelength is at least as small as \(\Delta x\) (diffraction limits resolution to roughly \(\lambda\)), so \(\lambda \lesssim \Delta x\). But this photon carries momentum \(p_{photon}=h/\lambda\); when it scatters off the electron to be detected, it transfers an uncontrollable momentum kick of similar size to the electron:

\[ \Delta p \sim \frac{h}{\lambda} \sim \frac{h}{\Delta x} \]

Multiplying the two uncertainties together:

\[ \Delta x \cdot \Delta p \sim h \]

This heuristic argument gives the right order of magnitude; a full wave-packet (Fourier) analysis sharpens it to the precise bound \(\Delta x \cdot \Delta p \ge h/4\pi\). Either way, the key point survives: trying to measure position more precisely (shorter \(\lambda\), higher-energy photon) necessarily disturbs momentum more.

The simplest exactly solvable case: a particle confined to \(0\lt x\lt L\), with \(V=0\) inside and \(V=\infty\) outside (impenetrable walls). Inside the box:

\[ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} = E\psi \quad\Longrightarrow\quad \frac{d^2\psi}{dx^2} = -k^2\psi,\quad k^2=\frac{2mE}{\hbar^2} \]

The general solution is \(\psi(x) = A\sin(kx) + B\cos(kx)\). The walls force \(\psi=0\) there (the particle cannot exist where \(V=\infty\)):

\[ \psi(0)=0 \Rightarrow B=0; \qquad \psi(L)=0 \Rightarrow A\sin(kL)=0 \Rightarrow kL=n\pi,\ n=1,2,3,\dots \]

So \(k=n\pi/L\), and substituting back into \(k^2=2mE/\hbar^2\):

\[ E_n = \frac{n^2h^2}{8mL^2} \]

Quantisation of energy falls directly out of the boundary conditions — exactly the kind of "why" that Bohr's model had to simply assume, and precisely the same mechanism (in three dimensions, with a Coulomb potential rather than a box) that quantises the hydrogen atom's energy levels.

For a given \(n\), \(l\) ranges over \(n\) values (\(0,1,\dots,n-1\)), and for each \(l\), \(m_l\) takes \(2l+1\) values. The total number of orbitals for that \(n\) is:

\[ \sum_{l=0}^{n-1}(2l+1) \]

Using \(\sum_{l=0}^{n-1} l = \dfrac{n(n-1)}{2}\):

\[ \sum_{l=0}^{n-1}(2l+1) = 2\cdot\frac{n(n-1)}{2} + n = n(n-1)+n = n^2 - n + n = n^2 \]

So there are exactly \(n^2\) orbitals for a given \(n\): 1 for \(n=1\) (just \(1s\)), 4 for \(n=2\) (\(2s\) and three \(2p\)), 9 for \(n=3\), and so on — a result used constantly when building up electron configurations (Ch. 16).

Solving the radial Schrödinger equation for hydrogen shows the total number of nodes (surfaces where \(\psi=0\), excluding the origin and infinity) in any orbital is exactly \(n-1\). This total splits between angular and radial nodes as:

\[ \text{angular nodes} = l, \qquad \text{radial nodes} = (n-1) - l = n-l-1 \]

Check, 2p orbital (\(n=2,l=1\)): angular nodes \(=1\), radial nodes \(=2-1-1=0\), total \(=1=n-1\). ✓

Check, 3d orbital (\(n=3,l=2\)): angular nodes \(=2\), radial nodes \(=3-2-1=0\), total \(=2=n-1\). ✓

Check, 3s orbital (\(n=3,l=0\)): angular nodes \(=0\), radial nodes \(=3-0-1=2\), total \(=2=n-1\). ✓

This simple rule lets you predict the node structure of any orbital directly from \(n\) and \(l\), without solving the wave equation each time.

Tabulate \(n+l\) for the orbitals in question:

\(4s\): \(n=4,\ l=0 \Rightarrow n+l=4\).
\(3d\): \(n=3,\ l=2 \Rightarrow n+l=5\).

Since \(4 < 5\), \(4s\) fills first — matching experiment. For a case where \(n+l\) ties:

\(4f\): \(n=4,\ l=3 \Rightarrow n+l=7\).
\(5d\): \(n=5,\ l=2 \Rightarrow n+l=7\).

Tied at \(n+l=7\), so the tie-break rule applies: smaller \(n\) fills first, so \(4f\) fills before \(5d\) — again matching the experimentally observed order (\(\dots 6s, 4f, 5d, 6p\dots\)). Applying this rule systematically reproduces the entire standard filling sequence:

\[ 1s,2s,2p,3s,3p,4s,3d,4p,5s,4d,5p,6s,4f,5d,6p,7s,5f,6d,\dots \]

From Ch. 14, a shell with principal quantum number \(n\) contains \(n^2\) orbitals. Since Pauli's principle allows at most 2 electrons per orbital:

\[ \text{max. electrons in shell } n = 2n^2 \]

Similarly, a subshell of azimuthal number \(l\) contains \(2l+1\) orbitals (Ch. 14), so:

\[ \text{max. electrons in subshell } l = 2(2l+1) = 4l+2 \]

This reproduces the familiar capacities: \(s\ (l=0)\): 2 electrons; \(p\ (l=1)\): 6; \(d\ (l=2)\): 10; \(f\ (l=3)\): 14 — numbers used constantly when building electron configurations.

Compare two ways of placing two electrons into a pair of degenerate \(p\) orbitals:

Paired (both electrons in the same orbital, e.g. \(p_x\)): both electrons occupy the same spatial region, so their average separation \(r_{12}\) is small, giving a large Coulomb repulsion energy (\(\propto 1/r_{12}\)).

Unpaired (one electron in \(p_x\), one in \(p_y\), parallel spins): the electrons occupy different spatial regions, so their average \(r_{12}\) is larger, lowering the classical repulsion — and the additional quantum exchange interaction between same-spin electrons lowers the energy further still.

Since total energy includes this repulsion term, the doubly-occupied configuration costs more energy than the singly-occupied, parallel-spin one — so the ground state of the atom is the Hund's-rule configuration, not the naively "compact" paired one.

Exchange stabilisation scales with the number of unique same-spin electron pairs within a subshell, \(\binom{k}{2}=\dfrac{k(k-1)}{2}\), where \(k\) is the number of parallel-spin electrons.

Expected configuration \(4s^23d^4\): the \(3d^4\) electrons are all parallel-spin (Hund's rule), giving \(\binom{4}{2}=6\) exchange pairs; the paired \(4s^2\) electrons contribute none (opposite spins).

Actual configuration \(4s^13d^5\): the \(3d^5\) electrons are all parallel-spin (half-filled), giving \(\binom{5}{2}=10\) exchange pairs; the single \(4s^1\) electron is unpaired, contributing no repulsion penalty either.

The actual configuration gains 4 additional exchange pairs (10 vs. 6) while also relieving the extra Coulomb repulsion that came from pairing two electrons together in \(4s^2\). Both effects favour \(4s^13d^5\) over \(4s^23d^4\), which is exactly what's observed.

Treat the outermost electron using a hydrogen-like extension of Bohr's energy formula (Ch. 9), with the actual nuclear charge \(Z\) replaced by the effective nuclear charge it experiences:

\[ E_n = -13.6\left(\frac{Z_{eff}}{n}\right)^2\ \text{eV} \]

Ionization energy is the energy needed to remove this electron to \(n=\infty\), i.e. \(IE = |E_n| = 13.6(Z_{eff}/n)^2\ \text{eV}\). This single expression reproduces both trends:

Across a period (n roughly constant, \(Z_{eff}\) increasing): \(IE\) increases, since it scales with \(Z_{eff}^2\).

Down a group (\(Z_{eff}\) roughly constant, \(n\) increasing): \(IE\) decreases, since it scales with \(1/n^2\).

This closes the loop on the whole topic: the very first quantitative atomic model (Ch. 9), extended with the shielding concept, is enough to explain the periodic table's most basic trends.

As two atoms approach from far apart, attractive forces (each nucleus attracting the other's electrons) dominate first, and the system's potential energy falls. At very short range, repulsive forces (nucleus–nucleus, electron–electron) grow rapidly and dominate, sending the energy sharply upward. Between these two regimes lies an energy minimum, at the equilibrium bond length \(r_e\):

\[ \left(\frac{dE}{dr}\right)_{r=r_e} = 0 \]

The depth of this energy well below the separated-atoms baseline is the bond dissociation energy — the same balance of attraction and repulsion that sets both a molecule's bond length and its bond strength (developed further in Ch. 17).

Model the lattice's energy as a balance between long-range Coulombic attraction and short-range repulsion from electron-cloud overlap:

\[ U(r) = -\frac{A}{r} + \frac{B}{r^n}, \qquad A = \frac{N_A M z_+z_-e^2}{4\pi\varepsilon_0} \]

where \(N_A\) is Avogadro's number, \(M\) the Madelung constant (accounting for every ion–ion interaction throughout the lattice, not just nearest neighbours), and \(n\) the Born exponent. At the equilibrium separation \(r_0\), \(dU/dr=0\):

\[ \frac{A}{r_0^2} - \frac{nB}{r_0^{n+1}} = 0 \quad\Longrightarrow\quad B = \frac{Ar_0^{n-1}}{n} \]

Substituting this back into \(U(r_0)\):

\[ U(r_0) = -\frac{A}{r_0} + \frac{A}{nr_0} = -\frac{A}{r_0}\left(1-\frac{1}{n}\right) \]

giving the Born–Landé equation:

\[ U_0 = -\frac{N_AMz_+z_-e^2}{4\pi\varepsilon_0 r_0}\left(1-\frac{1}{n}\right) \]

Let \(N\) be the electrons every atom would need if none were shared (8 per atom, 2 for H), \(A\) the valence electrons actually available, and \(S=N-A\) the electrons that must be shared to make up the difference. The number of bonding pairs is then \(S/2\), and the remaining \(A-S\) electrons are lone pairs.

For \(CO_2\): \(N = 8+8+8=24\) (three atoms, each needing an octet); \(A = 4+6+6=16\) (C contributes 4, each O contributes 6). So:

\[ S = N-A = 24-16=8 \quad\Longrightarrow\quad \text{bonding pairs} = \frac{S}{2}=4 \]

Four bonding pairs across two C–O connections means two double bonds (\(O=C=O\)), matching the known structure. Remaining lone-pair electrons: \(A-S = 16-8=8\), i.e. 4 lone pairs, distributed as two lone pairs on each oxygen — exactly CO₂'s actual Lewis structure.

Ozone has two equivalent resonance structures: \(O=O{-}\ddot{O}{}^-\) and its mirror image, \({}^-\ddot{O}{-}O=O\). In each structure, one O–O bond is a double bond (bond order 2) and the other is a single bond (bond order 1); which bond is which simply swaps between the two structures.

Averaging over both equally weighted resonance structures, each O–O bond's effective bond order is:

\[ \text{average bond order} = \frac{2+1}{2} = 1.5 \]

This matches experiment directly: both O–O bonds in ozone are observed to have identical, intermediate lengths — shorter than a typical O–O single bond, longer than a typical O=O double bond — exactly what a bond order of 1.5 predicts.

Four points maximally spread apart on a sphere sit at the vertices of a regular tetrahedron. Using the four alternating vertices of a cube as coordinates,

\[ \vec{v}_1=(1,1,1), \quad \vec{v}_2=(1,-1,-1) \]

the angle between any two of these position vectors (from the centre) is found from the dot product:

\[ \cos\theta = \frac{\vec{v}_1\cdot\vec{v}_2}{|\vec{v}_1||\vec{v}_2|} = \frac{(1)(1)+(1)(-1)+(1)(-1)}{\sqrt{3}\sqrt{3}} = \frac{1-1-1}{3} = -\frac{1}{3} \]

\[ \theta = \cos^{-1}\!\left(-\frac{1}{3}\right) \approx 109.47^{\circ} \]

This confirms, from pure geometry (not chemistry), that four points repelling each other equally on a sphere naturally settle at the tetrahedral angle — exactly what four identical bonding pairs (as in \(CH_4\)) are observed to adopt.

For two bond dipoles of equal magnitude \(\mu\) separated by angle \(\theta\), resolving both along their common bisector (their perpendicular components cancel by symmetry) gives a resultant magnitude:

\[ \mu_{net} = 2\mu\cos\!\left(\frac{\theta}{2}\right) \]

CO₂ (linear, \(\theta=180^{\circ}\), from Ch. 5's VSEPR result):

\[ \mu_{net} = 2\mu\cos(90^{\circ}) = 0 \]

The two C=O dipoles point in exactly opposite directions and cancel completely — CO₂ is nonpolar.

H₂O (bent, \(\theta\approx104.5^{\circ}\), also from Ch. 5):

\[ \mu_{net} = 2\mu\cos(52.25^{\circ}) \approx 1.23\mu \ne 0 \]

The O–H dipoles do not cancel, leaving H₂O with a significant net dipole — exactly the geometry-dependence that makes bond polarity alone insufficient to predict molecular polarity.

Solving the Schrödinger equation for two approaching H atoms (Heitler and London, 1927) shows the total energy splits into two distinct cases depending on the two electrons' relative spin:

Opposite (paired) spins: the quantum mechanical exchange contribution to the energy is attractive, producing a genuine energy minimum at a finite internuclear distance — exactly the bond-formation curve from Ch. 1. A stable \(H_2\) molecule results.

Same (parallel) spins: the exchange contribution instead is repulsive at every distance — the energy curve simply rises as the atoms approach, with no minimum at all. No bond forms; the atoms simply repel.

This is the same style of quantum exchange effect encountered for electrons within one atom (Hund's rule, Atomic Structure Ch. 18) — here it determines whether two atoms bond at all, not just how electrons arrange within a single atom.

The four \(sp^3\) hybrid orbitals are standard linear combinations of one \(s\) and three \(p\) orbitals:

\[ \psi_1=\tfrac12(s+p_x+p_y+p_z),\quad \psi_2=\tfrac12(s+p_x-p_y-p_z) \]

\[ \psi_3=\tfrac12(s-p_x+p_y-p_z),\quad \psi_4=\tfrac12(s-p_x-p_y+p_z) \]

Each hybrid's \(s\)-coefficient is \(\tfrac12\), so its squared contribution is \(\tfrac14\): each \(sp^3\) hybrid is 25% \(s\) character and 75% \(p\) character — consistent with the 1-part-s-to-3-parts-p naming. The direction each hybrid points is set by the signs of its \(p_x,p_y,p_z\) coefficients: \((+,+,+)\), \((+,-,-)\), \((-,+,-)\), \((-,-,+)\) — exactly the four tetrahedral vertex directions, \((1,1,1)\), \((1,-1,-1)\), \((-1,1,-1)\), \((-1,-1,1)\), used to derive the \(109.5^{\circ}\) angle in Ch. 5. Hybridisation and VSEPR are two routes to the same geometric answer.

Following the same orbital-counting logic as Ch. 8:

\(PCl_5\) (5 electron domains — trigonal bipyramidal by VSEPR, Ch. 5): \(sp^3d\) combines \(1+3+1=5\) atomic orbitals, giving exactly 5 hybrid orbitals — matching the 5 bonding positions required.

\(SF_6\) (6 electron domains — octahedral by VSEPR): \(sp^3d^2\) combines \(1+3+2=6\) atomic orbitals, giving exactly 6 hybrid orbitals — matching the 6 bonding positions required.

In both cases, the hybridisation scheme is chosen precisely so its orbital count matches the number of electron domains VSEPR independently predicts — the two theories are cross-checked against each other by design.

Consider two atomic orbitals of equal energy \(\alpha\) (the Coulomb integral), coupled by an interaction energy \(\beta\) (the resonance/exchange integral, with \(\beta<0\) for a stabilising interaction). Ignoring orbital overlap for simplicity (a standard first approximation), the secular determinant is:

\[ \begin{vmatrix} \alpha-E & \beta \\ \beta & \alpha-E \end{vmatrix} = 0 \]

\[ (\alpha-E)^2 - \beta^2 = 0 \quad\Longrightarrow\quad E = \alpha \mp \beta \]

This gives two solutions:

\[ E_{bonding} = \alpha+\beta \quad (\text{lower, since }\beta<0) \]

\[ E_{antibonding} = \alpha-\beta \quad (\text{higher}) \]

Two atomic orbitals always combine into exactly two MOs — the same orbital-conservation principle seen in hybridisation (Ch. 8): orbitals are recombined, never created or destroyed.

Each atom contributes 4 valence atomic orbitals (\(2s\) plus three \(2p\)); two atoms therefore contribute \(4\times2=8\) atomic orbitals in total. By the orbital-conservation principle from Ch. 10 (LCAO neither creates nor destroys orbitals), these must combine into exactly 8 molecular orbitals:

\[ \sigma_{2s},\ \sigma^*_{2s},\ \sigma_{2p},\ \pi_{2p}(\times2),\ \pi^*_{2p}(\times2),\ \sigma^*_{2p} \]

\[ 1+1+1+2+2+1 = 8\ \checkmark \]

The count checks out regardless of which specific energy ordering (normal or \(s\)–\(p\)-mixed) applies — only the relative energies of \(\sigma_{2p}\) and \(\pi_{2p}\) swap between the two cases, not the total number or identity of the orbitals formed.

\(O_2\) has 12 valence electrons (6 from each O atom: \(2s^22p^4\)). Filling the "normal"-order MO diagram (O₂ is a "later" diatomic, Ch. 11):

\[ \sigma_{2s}^2\,\sigma^{*2}_{2s}\,\sigma_{2p}^2\,\pi_{2p}^4\,\pi^{*2}_{2p} \]

Total: \(2+2+2+4+2=12\ \checkmark\). Bonding electrons: \(\sigma_{2s}^2+\sigma_{2p}^2+\pi_{2p}^4=8\). Antibonding electrons: \(\sigma^{*2}_{2s}+\pi^{*2}_{2p}=4\). Bond order:

\[ \text{bond order} = \tfrac12(8-4) = 2 \]

— matching the O=O double bond from the Lewis structure. But the crucial detail is how the final two electrons fill the two degenerate \(\pi^*_{2p}\) orbitals: by Hund's rule, one electron goes into each, with parallel spins, rather than pairing up in just one. That leaves two unpaired electrons — correctly predicting \(O_2\)'s paramagnetism, something the simple Lewis structure could never capture.

From Ch. 10's LCAO result, a bonding MO's stabilisation scales with the magnitude of the resonance integral, \(|\beta|\), which itself scales with the degree of orbital overlap, \(S\). Head-on (\(\sigma\)) overlap between two \(p\) orbitals pointed directly at each other along the bond axis achieves close to the maximum possible lobe-to-lobe overlap at a given internuclear distance. Side-on (\(\pi\)) overlap between two parallel \(p\) orbitals, by contrast, is inherently less complete geometrically, even at the same distance:

\[ S_{\sigma} > S_{\pi} \quad\Longrightarrow\quad |\beta_{\sigma}| > |\beta_{\pi}| \]

So, for a given pair of atoms, the \(\sigma\) framework is consistently more stabilised than any accompanying \(\pi\) bond — which is why, in a triple bond like \(N_2\)'s, the initial \(\sigma\) bond is the hardest component to break, while the two \(\pi\) bonds are comparatively easier to break first.

Following Ch. 10's principle directly: \(N\) metal atoms, each contributing one valence atomic orbital (e.g. \(3s\) for sodium), combine via LCAO into exactly \(N\) molecular orbitals — the same orbital-conservation rule, just scaled up. For a macroscopic sample, \(N \sim 10^{23}\), so these \(N\) discrete levels, spread across a finite energy range, become effectively continuous: a band.

Roughly half these orbitals are bonding-like (lower energy) and half antibonding-like (higher energy) — the many-orbital analogue of Ch. 10's \(\alpha\pm\beta\) split. Each orbital holds 2 electrons (Pauli, Atomic Structure Ch. 17), but each Na atom supplies only 1 valence electron — so the band ends up only half-filled. Electrons near the top of the filled portion sit immediately below a huge number of essentially degenerate empty states, so they can move into them with negligible energy cost — this mobility is exactly what makes metals such good electrical conductors.

Boiling points of the group 16 hydrides, ignoring \(H_2O\): \(H_2Te \approx -2^{\circ}\text{C}\), \(H_2Se \approx -41^{\circ}\text{C}\), \(H_2S \approx -60^{\circ}\text{C}\) — a smooth trend, decreasing with decreasing molecular weight, consistent with weakening van der Waals forces (Ch. 16) alone.

Extrapolating that smooth trend down to \(H_2O\)'s molecular weight predicts a boiling point of roughly \(-90^{\circ}\text{C}\). The actual boiling point of water is \(+100^{\circ}\text{C}\) — a gap of about \(190^{\circ}\text{C}\) that van der Waals forces alone cannot explain. This large, specific deviation is direct empirical evidence for an additional, much stronger attractive force — hydrogen bonding — present in \(H_2O\) but not in the heavier hydrides (whose central atom isn't electronegative or small enough to hydrogen bond effectively).

An instantaneous dipole \(\mu_A\) on atom A produces an electric field at distance \(r\) roughly \(E \sim \mu_A/(4\pi\varepsilon_0 r^3)\) (the standard dipole field). This field induces a dipole on a neighbouring, polarisable atom B with polarisability \(\alpha\):

\[ \mu_{B,induced} \sim \alpha E \sim \frac{\alpha\mu_A}{4\pi\varepsilon_0 r^3} \]

The interaction energy between the two aligned dipoles scales as \(U \sim -\mu_A\mu_{B,induced}/r^3\); substituting \(\mu_{B,induced}\):

\[ U \sim -\frac{\alpha\mu_A^2}{(4\pi\varepsilon_0)^2}\cdot\frac{1}{r^6} \]

The characteristic \(1/r^6\) dependence falls off far more steeply than Coulomb's \(1/r\) or dipole–dipole's \(1/r^3\) — which is why van der Waals forces only matter when molecules are already very close together.

Naively, tripling the bond order might suggest tripling the bond energy: \(3\times347=1041\ \text{kJ mol}^{-1}\). The actual C≡C energy, 839 kJ mol⁻¹, falls well short of that. Ch. 13 explains why: the first bond formed is a \(\sigma\) bond (strong, high-overlap), while the second and third are \(\pi\) bonds (weaker, lower-overlap, since \(S_{\pi}\lt S_{\sigma}\)). Each additional \(\pi\) bond contributes real, but proportionally smaller, extra energy than the initial \(\sigma\) bond did:

\[ E(\sigma) \approx 347,\quad E(\sigma+\pi_1)-E(\sigma)\approx 267,\quad E(\sigma+\pi_1+\pi_2)-E(\sigma+\pi_1)\approx 225\ \ (\text{kJ mol}^{-1}) \]

Each successive bond adds less energy than the one before — a direct, quantitative echo of Ch. 13's overlap argument (\(S_\sigma>S_\pi\)) showing up in real thermochemical data.

Using Ch. 4's formal charge formula, \(FC = V - N - \tfrac12 B\) (valence electrons minus nonbonding electrons minus half the bonding electrons), for nitrogen in \(NH_4^+\) (4 bonds, 0 lone pairs, \(V=5\)):

\[ FC(N) = 5 - 0 - \tfrac12(8) = 5-4 = +1 \]

For each hydrogen (1 bond, \(V=1\)):

\[ FC(H) = 1-0-\tfrac12(2) = 1-1 = 0 \]

Summing formal charges: \(+1 + 4(0) = +1\), matching \(NH_4^+\)'s overall charge exactly. The formalism correctly places the "extra" positive charge on nitrogen — the atom that donated both electrons of the new bond — even though, once formed, that fourth N–H bond is chemically identical to the other three.

A simple VBT/Lewis treatment of \(O_2\) gives the structure \(O=O\): a double bond (bond order 2), with every electron paired — predicting diamagnetism.

The full MOT treatment (Ch. 12) gives the same bond order, 2, from \(\tfrac12(8\ \text{bonding}-4\ \text{antibonding})\) — so both theories agree on bond order and, roughly, bond strength. But MOT additionally shows the last two electrons must occupy the degenerate \(\pi^*_{2p}\) orbitals singly (Hund's rule), giving two unpaired electrons and correctly predicting paramagnetism — the experimentally observed behaviour that VBT's simple picture misses entirely.

This doesn't mean VBT is "wrong" — with resonance added, it can be patched to explain delocalised systems like benzene reasonably well too. Each theory has weak points that can be patched with extensions; MOT simply builds delocalisation and magnetism in from the start.

From Atomic Structure Ch. 20, silicon's larger atomic radius (relative to carbon) comes from its valence electrons occupying a higher principal shell (\(n=3\) vs. \(n=2\>), with roughly similar effective nuclear charge \(Z_{eff}\) within the group. A larger atomic orbital is necessarily more diffuse — its electron density is spread over a larger volume, with lower peak density at any given point.

From Ch. 13's overlap argument, bond strength scales with the overlap integral \(S\) between the two bonding orbitals. Even though two \(3p\) orbitals on silicon can approach to their own (longer) equilibrium distance, their diffuseness means the overlap achieved there is smaller than the overlap two compact \(2p\) orbitals on carbon achieve at their (shorter) equilibrium distance:

\[ S_{Si-Si} < S_{C-C} \quad\Longrightarrow\quad |\beta_{Si-Si}| < |\beta_{C-C}| \]

via Ch. 10's \(E=\alpha\pm\beta\), a smaller \(|\beta|\) means less bonding stabilisation — a weaker bond. This closes the loop across both topics: atomic radius (Atomic Structure) → orbital diffuseness → overlap (Ch. 13) → LCAO bond strength (Ch. 10) together explain why bonds systematically weaken down a group.

Consider one particle of mass \(m\), moving with speed \(v_x\) along the x-axis, bouncing elastically between two walls separated by distance \(L\). Each collision reverses its momentum:

\[ \Delta p = 2mv_x \]

The particle returns to hit the same wall again after travelling a round trip of \(2L\), taking time \(\Delta t = 2L/v_x\). By Newton's second law, the average force this one particle exerts on the wall is:

\[ F = \frac{\Delta p}{\Delta t} = \frac{2mv_x}{2L/v_x} = \frac{mv_x^2}{L} \]

This single-particle result is the building block for the full pressure derivation: Ch. 7 extends it to \(N\) particles moving in three dimensions to derive \(PV=nRT\) itself.

From Ch. 1, one particle contributes an average force \(F=mv_x^2/L\) to a wall of area \(A\), where the box volume is \(V=LA\). Summing over \(N\) particles and dividing by area to get pressure:

\[ P = \frac{F_{total}}{A} = \frac{Nm\langle v_x^2\rangle}{LA} = \frac{Nm\langle v_x^2\rangle}{V} \]

At constant temperature, average kinetic energy (and hence \(\langle v_x^2\rangle\)) is constant (Ch. 1's KMT postulate), so \(Nm\langle v_x^2\rangle\) is just a fixed number for a given gas sample at that temperature:

\[ P = \frac{\text{constant}}{V} \quad\Longrightarrow\quad PV = \text{constant} \]

Boyle's Law falls directly out of the same single-particle momentum-transfer picture introduced in Ch. 1, simply extended from one particle to \(N\).

From Ch. 2, \(P = Nm\langle v_x^2\rangle/V\). The KMT postulate that average kinetic energy is proportional to \(T\) means \(\langle v_x^2\rangle \propto T\) (made rigorous in Ch. 7–9). Writing \(\langle v_x^2\rangle = cT\) for some constant \(c\):

\[ P = \frac{Nmc}{V}T \quad\Longrightarrow\quad PV = (Nmc)\,T \]

This single relation, \(PV \propto T\) at fixed \(N\), contains both named laws as special cases:

Constant \(P\): \(V = \dfrac{Nmc}{P}T \Rightarrow V\propto T\) (Charles's Law).

Constant \(V\): \(P = \dfrac{Nmc}{V}T \Rightarrow P\propto T\) (Gay-Lussac's Law).

Both familiar laws are just two different constant-variable slices through the same underlying \(PV\propto T\) relationship — which Ch. 4 will formalise fully as the ideal gas equation.

Start from Boyle's Law at fixed \(n,T\): \(V \propto 1/P\). Now let temperature vary (Ch. 3): \(V \propto T\). Now let the amount vary (Avogadro's Law): \(V \propto n\). Combining all three proportionalities into one:

\[ V \propto \frac{nT}{P} \quad\Longrightarrow\quad PV \propto nT \quad\Longrightarrow\quad PV = nRT \]

for some proportionality constant \(R\), determined empirically. Using the molar volume of an ideal gas at STP (0°C, 1 atm): 1 mole occupies 22.4 L:

\[ R = \frac{PV}{nT} = \frac{(1\ \text{atm})(22.4\ \text{L})}{(1\ \text{mol})(273.15\ \text{K})} \approx 0.0821\ \text{L atm mol}^{-1}\text{K}^{-1} \]

For component \(i\) alone, occupying the full mixture volume \(V\) at temperature \(T\) with \(n_i\) moles, the ideal gas equation (Ch. 4) gives its partial pressure:

\[ P_i = \frac{n_iRT}{V} \]

Summing over every component in the mixture:

\[ P_{total} = \sum_i P_i = \frac{RT}{V}\sum_i n_i = \frac{n_{total}RT}{V} \]

— showing the whole mixture obeys the ideal gas equation too, with \(n_{total}=\sum_i n_i\). Dividing \(P_i\) by \(P_{total}\):

\[ \frac{P_i}{P_{total}} = \frac{n_iRT/V}{n_{total}RT/V} = \frac{n_i}{n_{total}} = x_i \quad\Longrightarrow\quad P_i = x_iP_{total} \]

Equal average kinetic energy for two gases at the same temperature:

\[ \tfrac12 m_1\langle v_1^2\rangle = \tfrac12 m_2\langle v_2^2\rangle \]

Rearranging for the speed ratio:

\[ \frac{\langle v_1^2\rangle}{\langle v_2^2\rangle} = \frac{m_2}{m_1} \quad\Longrightarrow\quad \frac{v_1}{v_2} = \sqrt{\frac{m_2}{m_1}} = \sqrt{\frac{M_2}{M_1}} \]

(using \(m \propto M\), since molar mass is just molecular mass scaled by Avogadro's number). Since effusion rate tracks average molecular speed directly:

\[ \frac{\text{rate}_1}{\text{rate}_2} = \frac{v_1}{v_2} = \sqrt{\frac{M_2}{M_1}} \]

Graham's Law follows directly from the single assumption of equal average kinetic energy at a shared temperature.

From Ch. 1–2, summing the single-particle wall force over \(N\) particles gives \(P = Nm\langle v_x^2\rangle/V\). Using isotropy, \(\langle v_x^2\rangle=\langle v^2\rangle/3\):

\[ P = \frac{Nm\langle v^2\rangle}{3V} \quad\Longrightarrow\quad PV = \tfrac13 Nm\langle v^2\rangle = \tfrac23 N\left(\tfrac12 m\langle v^2\rangle\right) = \tfrac23 N\langle KE\rangle \]

where \(\langle KE\rangle\) is the average translational kinetic energy per molecule. A foundational result of kinetic theory (from the equipartition of energy across 3 translational degrees of freedom) is \(\langle KE\rangle = \tfrac32 k_BT\), where \(k_B\) is Boltzmann's constant. Substituting:

\[ PV = \tfrac23 N \cdot \tfrac32 k_BT = Nk_BT \]

Writing \(N=nN_A\) (moles times Avogadro's number) and defining \(R=N_Ak_B\) (the molar gas constant):

\[ PV = nN_Ak_BT = nRT \]

— the ideal gas equation, derived entirely from Newtonian mechanics applied to colliding particles, with no assumption about intermolecular forces at all.

Rather than deriving the full normalised formula (which requires statistical mechanics beyond this scope), the distribution's characteristic shape comes from multiplying two physically distinct factors:

1. The Boltzmann factor: the relative probability of any single velocity state falls off with its kinetic energy as \(e^{-KE/k_BT}=e^{-mv^2/2k_BT}\) — higher-speed (higher-energy) states are exponentially less likely.

2. Velocity-space geometry: the "amount of room" available at speed \(v\), summed over every possible direction, is the surface area of a sphere of radius \(v\) in velocity space: \(4\pi v^2\).

Multiplying these two competing factors together:

\[ f(v) \propto \underbrace{4\pi v^2}_{\text{geometric factor (rises with }v)} \times \underbrace{e^{-mv^2/2k_BT}}_{\text{Boltzmann factor (falls with }v)} \]

At low \(v\), the geometric \(v^2\) factor dominates and \(f(v)\) rises; at high \(v\), the exponential dominates and \(f(v)\) falls — producing the characteristic peaked, asymmetric curve from these two competing effects.

Ch. 7 established \(PV = \tfrac13 Nm\langle v^2\rangle = nRT\). Solving for \(\langle v^2\rangle\), with \(m N = nM\) (total mass equals moles times molar mass):

\[ \langle v^2 \rangle = \frac{3nRT}{Nm} = \frac{3RT}{M} \]

\[ v_{rms} = \sqrt{\langle v^2\rangle} = \sqrt{\frac{3RT}{M}} \]

— derived directly and rigorously from Ch. 7's kinetic theory result, with no new assumptions. The other two speeds require integrating the full Maxwell–Boltzmann distribution (Ch. 8) using calculus, beyond this scope, but their standard results can be checked for consistency: dividing all three formulas by \(\sqrt{2RT/M}\) gives the ratio

\[ v_{mp}:v_{avg}:v_{rms} = \sqrt{2}:\sqrt{8/\pi}:\sqrt{3} \approx 1.000:1.128:1.225 \]

confirming \(v_{mp}\lt v_{avg}\lt v_{rms}\), consistent with the distribution's asymmetric, right-skewed shape.

A general, rigorous way to express real-gas deviation is the virial equation of state, a power series in \(1/V_m\) (molar volume):

\[ Z = \frac{PV_m}{RT} = 1 + \frac{B}{V_m} + \frac{C}{V_m^2} + \dots \]

where \(B\), \(C\), etc. (the virial coefficients) capture successive corrections from intermolecular interactions, and depend on temperature and the specific gas. As pressure drops toward zero, \(V_m\to\infty\), and every correction term vanishes:

\[ \lim_{V_m\to\infty} Z = 1 \]

This confirms, rigorously, that every real gas approaches ideal behaviour in the low-pressure (low-density) limit, regardless of its particular attractive or repulsive character — because at large enough separation, intermolecular forces and molecular volume both become irrelevant. Ch. 11's van der Waals equation is, in effect, one specific, physically motivated model for generating these virial coefficients.

Volume correction: if each mole of molecules excludes volume \(b\) (accounting for the fact that two molecules can't occupy the same space), then the free volume available for motion in \(n\) moles is simply the container volume minus the total excluded volume: \(V_{free} = V - nb\).

Pressure correction: the inward pull felt by a molecule near the wall is proportional to the local concentration of attracting neighbours, \(n/V\); and the number of molecules experiencing this pull (per unit wall area) is also proportional to \(n/V\). The total pressure reduction is therefore proportional to the product of these two factors:

\[ \Delta P \propto \frac{n}{V}\times\frac{n}{V} = \frac{n^2}{V^2} \]

Writing \(\Delta P = an^2/V^2\) (with \(a\) as the gas-specific proportionality constant), the "ideal-equivalent" pressure is \(P_{ideal}=P_{measured}+an^2/V^2\), giving the full corrected equation. At low pressure (large \(V\)), both correction terms vanish and the equation reduces exactly to \(PV=nRT\) — consistent with Ch. 10's virial-limit result.

The critical point is the isotherm's inflection point: both the first and second derivatives of \(P\) with respect to \(V\) vanish there. For 1 mole, \(P = RT/(V-b) - a/V^2\):

\[ \left(\frac{\partial P}{\partial V}\right)_T = -\frac{RT}{(V-b)^2}+\frac{2a}{V^3}=0, \qquad \left(\frac{\partial^2 P}{\partial V^2}\right)_T = \frac{2RT}{(V-b)^3}-\frac{6a}{V^4}=0 \]

Solving the first for \(RT\): \(RT = 2a(V-b)^2/V^3\). Solving the second: \(RT = 3a(V-b)^3/V^4\). Setting these equal and simplifying:

\[ \frac{2(V-b)^2}{V^3} = \frac{3(V-b)^3}{V^4} \quad\Longrightarrow\quad 2V=3(V-b) \quad\Longrightarrow\quad V_c = 3b \]

Substituting \(V_c=3b\) back into \(RT=2a(V-b)^2/V^3\):

\[ T_c = \frac{8a}{27Rb} \]

And substituting both into the original equation of state gives:

\[ P_c = \frac{a}{27b^2} \]

All three critical constants are determined entirely by a gas's own \(a\) and \(b\) — setting up Ch. 14's law of corresponding states.

Along the coexistence plateau, liquid and gas phases coexist in equilibrium. From Chemical Thermodynamics Ch. 18, phase equilibrium requires equal chemical potential in both phases, \(\mu_{liq}=\mu_{vap}\), which holds at exactly one specific pressure for a given temperature — the vapour pressure.

As the volume is reduced along the plateau, more gas condenses into liquid, but as long as both phases remain present, the system's pressure is locked at that single vapour pressure value — it cannot rise or fall while both phases coexist. Only once all the gas has condensed (or all the liquid has vaporised) can pressure begin to change again, producing the sharp corners at each end of the flat plateau.

Substitute \(P=P_rP_c\), \(V=V_rV_c\), \(T=T_rT_c\), using Ch. 12's results \(P_c=a/27b^2\), \(V_c=3b\), \(T_c=8a/27Rb\), into the van der Waals equation \((P+a/V^2)(V-b)=RT\):

\[ \left(P_r\frac{a}{27b^2}+\frac{a}{9V_r^2b^2}\right)(3bV_r-b) = R\cdot T_r\cdot\frac{8a}{27Rb} \]

Factoring \(a/(27b^2)\) from the left bracket and \(b\) from the right one, then simplifying (the full algebra is a satisfying exercise — try Q1 below):

\[ \left(P_r + \frac{3}{V_r^2}\right)(3V_r-1) = 8T_r \]

Every gas-specific constant — \(a\), \(b\), even \(R\) — has cancelled out completely. This single, universal equation, containing only reduced variables, applies (approximately) to any gas obeying van der Waals-type behaviour.

Model a moving molecule of diameter \(d\) sweeping through a sea of (first, for simplicity) stationary molecules. A collision occurs whenever another molecule's centre comes within distance \(d\) of the moving one's centre, so as it travels a distance \(L\), it sweeps out an effective cylindrical volume of cross-section \(\sigma=\pi d^2\) (the collision cross-section) and length \(L\):

\[ \text{swept volume} = \sigma L = \pi d^2 L \]

With number density \(N^*\) (molecules per unit volume), the number of collisions in this swept volume is \(N^*\sigma L\), so the distance travelled per collision (first approximation, stationary targets) is:

\[ \lambda_{approx} = \frac{L}{N^*\sigma L} = \frac{1}{N^*\pi d^2} \]

Correcting for the fact that target molecules are also moving requires using their relative speed rather than the moving molecule's own speed; combining two independent Maxwell–Boltzmann velocity distributions (Ch. 8) shows the average relative speed between two molecules is \(\sqrt{2}\) times a single molecule's average speed. This increases the effective collision rate by \(\sqrt{2}\), giving the corrected result:

\[ \lambda = \frac{1}{\sqrt{2}\,N^*\pi d^2} \]

Substituting the ideal gas relation \(N^* = P/k_BT\) (from Ch. 7) gives the final, pressure-and-temperature form:

\[ \lambda = \frac{k_BT}{\sqrt{2}\,\pi d^2P} \]

Consider flow velocity \(u(z)\) varying with height \(z\) (gradient \(du/dz\)). Using the standard simplified kinetic-theory estimate that, on average, \(1/6\) of molecules move in each of the six \(\pm x,\pm y,\pm z\) directions, molecules cross a horizontal plane at rate \(\tfrac16N\langle v\rangle\) per unit area from each side, carrying momentum characteristic of their origin layer — one mean free path \(\lambda\) away, i.e. offset by \(\lambda(du/dz)\) in flow velocity. Combining the momentum carried by molecules crossing from both directions gives a net shear stress:

\[ \tau = \tfrac13 Nm\langle v\rangle\lambda \left(\frac{du}{dz}\right) \]

Comparing with Newton's law of viscosity, \(\tau = \eta(du/dz)\), and using \(Nm=\rho\):

\[ \eta = \tfrac13\rho\langle v\rangle\lambda \]

A striking consequence: since \(\rho \propto P\) but \(\lambda \propto 1/P\) (Ch. 15), their product is independent of pressure — gas viscosity is predicted (and experimentally confirmed) to be nearly pressure-independent over a wide range.

Using the identical flux argument as Ch. 16 (\(\tfrac16N\langle v\rangle\) molecules crossing a boundary per unit area from each direction, each carrying the energy characteristic of a layer \(\lambda\) away, offset by \(\lambda(dT/dz)\) in temperature), the net heat flux works out to:

\[ q = -\tfrac13N\langle v\rangle\lambda c_v\left(\frac{dT}{dz}\right) \]

Comparing with Fourier's law of heat conduction, \(q=-\kappa(dT/dz)\):

\[ \kappa = \tfrac13N\langle v\rangle\lambda c_v \]

The mathematical form is identical to Ch. 16's viscosity result, with momentum-per-molecule simply replaced by energy-per-molecule — both are examples of the same underlying transport phenomenon: random thermal motion carrying a conserved property (momentum, or energy) from a region where it's abundant to one where it's scarce. Like \(\eta\), \(\kappa\) is therefore also predicted to be nearly pressure-independent, by the same \(\rho\propto P\), \(\lambda\propto1/P\) cancellation.

Total mass of a mixture is \(\sum_i n_iM_i\); total moles is \(n_{total}=\sum_i n_i\). By definition:

\[ M_{avg} = \frac{\text{total mass}}{\text{total moles}} = \frac{\sum_i n_iM_i}{n_{total}} = \sum_i\left(\frac{n_i}{n_{total}}\right)M_i = \sum_i x_iM_i \]

using Ch. 5's mole fraction definition. For density: since the mixture obeys \(PV=n_{total}RT\) (Ch. 5), \(n_{total}=PV/RT\), so:

\[ \rho = \frac{\text{mass}}{V} = \frac{n_{total}M_{avg}}{V} = \frac{(PV/RT)M_{avg}}{V} = \frac{PM_{avg}}{RT} \]

Applying this to humid vs. dry air (same \(T,P\)): replacing some \(N_2\)/\(O_2\) molecules with lighter \(H_2O\) molecules (Avogadro's Law, Ch. 4, means equal moles occupy equal volume at fixed \(T,P\)) lowers \(M_{avg}\), and hence lowers \(\rho\) — humid air really is less dense, and more buoyant, than dry air.

As with van der Waals (Ch. 11) and the virial equation (Ch. 10), any valid real-gas equation must reduce to \(PV=nRT\) at low pressure. As \(V\to\infty\) (low pressure), in the Redlich–Kwong equation:

\[ \frac{RT}{V-b} \to \frac{RT}{V}, \qquad \frac{a}{\sqrt{T}\,V(V+b)} \to \frac{a}{\sqrt{T}\,V^2} \to 0 \]

(the second term vanishes faster, as \(1/V^2\), than the first falls, as \(1/V\)), so:

\[ P \to \frac{RT}{V} \quad\Longrightarrow\quad PV \to RT \]

confirming Redlich–Kwong is a legitimate refinement that still respects the fundamental low-pressure limit shared by every real-gas equation of state. The improvement over van der Waals shows up specifically at higher pressure and away from \(T_c\), where the temperature-dependent attraction term matters most — verified by comparing each equation's predicted \(Z\) against real experimental compressibility data.

Combine hydrostatic equilibrium (pressure decreases with height due to the weight of the air above) with the ideal gas density relation from Ch. 18. Hydrostatic balance for a thin slice of air:

\[ dP = -\rho g\,dh \]

Using \(\rho = PM/RT\) (Ch. 18, with \(M\) the average molar mass of air, and assuming roughly constant temperature — the "isothermal atmosphere" approximation):

\[ dP = -\frac{PM}{RT}g\,dh \quad\Longrightarrow\quad \frac{dP}{P} = -\frac{Mg}{RT}\,dh \]

Integrating from sea level (\(h=0\), \(P=P_0\)) to height \(h\):

\[ \ln\!\frac{P}{P_0} = -\frac{Mgh}{RT} \quad\Longrightarrow\quad P(h) = P_0\,e^{-Mgh/RT} \]

the barometric formula: atmospheric pressure decreases exponentially, not linearly, with altitude — because the air itself becomes less dense at higher altitude too, so each additional metre of height "weighs" progressively less than the one below it.

Whether a substance is solid, liquid, or gas at a given temperature comes down to a competition between average thermal kinetic energy, \(k_BT\) (Gaseous State Ch. 7), and the depth of the intermolecular potential energy well, \(\varepsilon\) (set by the strength of the attractive forces involved, Chemical Bonding Ch. 16):

• \(k_BT \ll \varepsilon\): molecules are trapped in their potential wells with too little energy to escape — a solid.
• \(k_BT \sim \varepsilon\): comparable energies — molecules can move and exchange neighbours, but remain loosely bound overall — a liquid.
• \(k_BT \gg \varepsilon\): kinetic energy dominates entirely, and molecules escape the attraction altogether — a gas.

This directly explains why substances with stronger intermolecular forces (larger \(\varepsilon\), such as hydrogen-bonded liquids, Ch. 14) require higher temperatures — larger \(k_BT\) — before the balance tips toward boiling.

Compare three molecules of similar molar mass, isolating the effect of force type from simple size/dispersion effects:

Despite having similar or even lower molar mass than propane and acetone, ethanol's hydrogen bonding gives it by far the highest boiling point — direct confirmation of Ch. 1's \(k_BT \sim \varepsilon\) framing: it is intermolecular force type and strength (\(\varepsilon\)), not molecular size alone, that primarily governs boiling point.

Only molecules in the high-energy tail of the Maxwell–Boltzmann distribution (Gaseous State Ch. 8) — those with kinetic energy exceeding the intermolecular binding energy \(\varepsilon\) (Ch. 1–2) — can escape the liquid surface. The fraction of molecules meeting this threshold is set by the Boltzmann factor:

\[ f \propto e^{-\varepsilon/k_BT} \]

Since evaporation rate is proportional to this escaping fraction, and equilibrium vapour pressure is set by balancing evaporation against condensation, the equilibrium vapour pressure must follow the same exponential temperature dependence:

\[ P_{vap} \propto e^{-\varepsilon/k_BT} \quad\text{or, in molar terms,}\quad P_{vap} \propto e^{-\Delta H_{vap}/RT} \]

This is exactly the Clausius–Clapeyron relationship, derived rigorously and independently from thermodynamics (equal chemical potentials) in Chemical Thermodynamics Ch. 19. Here, the same exponential form emerges instead from kinetic theory and Boltzmann statistics — two completely independent routes converging on the same law.

Water's vapour pressure is 23.8 torr at 298 K and 760 torr at 373 K (its normal boiling point). Using the two-point Clausius–Clapeyron formula:

\[ \ln\!\frac{760}{23.8} = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{373}-\frac{1}{298}\right) \]

\[ \ln(31.93) = 3.464, \qquad \frac{1}{373}-\frac{1}{298} = -6.747\times10^{-4}\ \text{K}^{-1} \]

\[ 3.464 = \Delta H_{vap}\times\frac{6.747\times10^{-4}}{8.314} \quad\Longrightarrow\quad \Delta H_{vap} \approx 42{,}700\ \text{J mol}^{-1} \approx 42.7\ \text{kJ mol}^{-1} \]

This is close to water's accepted literature value, \(\Delta H_{vap}\approx40.7\ \text{kJ mol}^{-1}\) — the small discrepancy comes from \(\Delta H_{vap}\) not being exactly constant across such a wide temperature span, exactly the caveat noted above.

Combine Gaseous State Ch. 20's barometric formula with this topic's Clausius–Clapeyron application (Ch. 4). First, estimate atmospheric pressure at Everest's summit (\(h\approx8848\ \text{m}\)), using \(M\approx0.029\ \text{kg mol}^{-1}\), \(T\approx250\ \text{K}\):

\[ \frac{Mgh}{RT} = \frac{(0.029)(9.8)(8848)}{(8.314)(250)} \approx 1.21 \quad\Longrightarrow\quad P(h) \approx (1\ \text{atm})e^{-1.21} \approx 0.30\ \text{atm} \]

Now use the Clausius–Clapeyron equation (Ch. 4) to find the temperature \(T_2\) at which water's vapour pressure drops to \(0.30\ \text{atm}\), starting from \(P_1=1\ \text{atm}\) at \(T_1=373\ \text{K}\), with \(\Delta H_{vap}\approx40.7\ \text{kJ mol}^{-1}\):

\[ \ln(0.30) = -\frac{40{,}700}{8.314}\left(\frac{1}{T_2}-\frac{1}{373}\right) \quad\Longrightarrow\quad T_2 \approx 342\ \text{K} \approx 69^{\circ}\text{C} \]

This matches real-world reports of water boiling around 70°C or lower at Everest's summit — a satisfying, fully-worked synthesis connecting two different topics' results into one concrete, verifiable prediction.

In a simple close-packed picture, a bulk molecule has \(z\) nearest neighbours, each pair contributing binding energy \(\varepsilon/2\) to the total energy (shared between the pair, Ch. 1–2). A surface molecule, missing roughly half its neighbours (nothing above it), is missing about:

\[ \Delta E_{surface} \approx \frac{z}{2}\times\frac{\varepsilon}{2} = \frac{z\varepsilon}{4} \]

of stabilisation compared to a bulk molecule. Surface tension is this missing energy per molecule, spread over the area each surface molecule occupies (number density at the surface, \(n_s\)):

\[ \gamma \approx n_s\cdot\frac{z\varepsilon}{4} \]

This is a simplified estimate (the precise numerical prefactor depends on packing geometry), but it correctly captures the key physical result: surface tension scales directly with intermolecular bond strength \(\varepsilon\) — stronger attractions (e.g. hydrogen bonding, Chemical Bonding Ch. 15) give higher surface tension.

An established empirical relationship (Eötvös, 1886) captures this temperature dependence quantitatively:

\[ \gamma V^{2/3} = k(T_c - T) \]

where \(V\) is molar volume and \(k\) is roughly similar across many liquids (\(\approx 2.1\times10^{-7}\ \text{J K}^{-1}\text{mol}^{-2/3}\)). Rearranged:

\[ \gamma = \frac{k(T_c-T)}{V^{2/3}} \]

As \(T\to T_c\), \(\gamma\to0\) exactly — confirming, with a real named empirical law, the qualitative reasoning above and tying surface tension's temperature behaviour directly to the critical temperature concept from Gaseous State Ch. 12.

Balance the upward force from surface tension, acting around the tube's circumference at contact angle \(\theta\), against the downward weight of the raised liquid column:

\[ \underbrace{\gamma(2\pi r)\cos\theta}_{\text{upward, surface tension}} = \underbrace{\rho g h(\pi r^2)}_{\text{downward, weight}} \]

Solving for the rise height \(h\):

\[ h = \frac{2\gamma\cos\theta}{\rho g r} \]

Rise height is inversely proportional to tube radius: narrower tubes show dramatically greater capillary rise — directly relevant to how water ascends through the narrow vessels of plant xylem.

For a liquid molecule to move relative to its neighbours, it must transiently acquire enough energy to "squeeze past" the attractive forces holding it in its local cage of neighbours — conceptually similar to Ch. 3's evaporation, where a molecule needed kinetic energy exceeding \(\varepsilon\) to escape the surface. By the same Boltzmann-factor logic (Gaseous State Ch. 8):

\[ \eta \propto e^{+E_a/k_BT} \]

Note the positive exponent here, opposite to evaporation's \(e^{-\varepsilon/k_BT}\): higher temperature makes it easier to escape the local energy barrier \(E_a\), which lowers viscosity, rather than raising a rate as in evaporation. This qualitative, activation-energy-based reasoning previews the precise, quantitative treatment in Ch. 10.

Taking the logarithm of the Andrade equation linearises it, exactly paralleling Ch. 4's method:

\[ \ln\eta = \ln A + \frac{E_a}{RT} \]

— a plot of \(\ln\eta\) vs. \(1/T\) gives a straight line of slope \(+E_a/R\) (positive, in contrast to Ch. 4's negative slope for vapour pressure, since \(\eta\) increases with \(1/T\), i.e. decreases with \(T\)). Water's viscosity is about 1.002 mPa·s at 293 K and 0.548 mPa·s at 323 K:

\[ \ln\!\frac{0.548}{1.002} = \frac{E_a}{R}\left(\frac{1}{323}-\frac{1}{293}\right) \]

\[ -0.6035 = \frac{E_a}{8.314}\times(-3.17\times10^{-4}) \quad\Longrightarrow\quad E_a \approx 15{,}800\ \text{J mol}^{-1} \approx 15.8\ \text{kJ mol}^{-1} \]

This matches water's accepted activation energy for viscous flow (typically quoted around 15–20 kJ mol⁻¹) — a concrete, real-world-consistent worked example structurally identical to Ch. 4's vapour pressure calculation.

A standard result from the electromagnetic theory of dielectrics (its full derivation requires Maxwell's equations applied to a polarisable medium, beyond this chapter's chemistry focus, but the result itself is central to physical chemistry) relates refractive index directly to molecular polarisability \(\alpha\) and number density \(N\):

\[ \frac{n^2-1}{n^2+2} = \frac{4\pi}{3}N\alpha \]

In molar form, using molar refractivity \(R_m\):

\[ R_m = \frac{n^2-1}{n^2+2}\cdot\frac{M}{\rho} = \frac{4\pi}{3}N_A\alpha \]

This lets chemists determine a molecule's polarisability directly from two easily measured bulk quantities — refractive index and density — a genuinely useful experimental technique, and a direct link between an optical property and the same polarisability parameter that governs London dispersion forces (Chemical Bonding Ch. 16).

The average number of molecules in a thin spherical shell between \(r\) and \(r+dr\) around a reference molecule is:

\[ dN(r) = \rho\, g(r)\, 4\pi r^2\, dr \]

using the same spherical-shell geometry (\(4\pi r^2 dr\)) that appeared in the Maxwell–Boltzmann distribution's velocity-space argument (Gaseous State Ch. 8) and the mean-free-path collision-cylinder derivation (Gaseous State Ch. 15). Integrating from \(r=0\) to \(r_{min}\), the position of \(g(r)\)'s first minimum (the edge of the first coordination shell), gives the coordination number:

\[ \text{coordination number} = \int_0^{r_{min}} \rho\, g(r)\, 4\pi r^2\, dr \]

This connects the abstract statistical function \(g(r)\) to a concrete, physically meaningful quantity — the typical number of nearest neighbours a molecule has, directly measurable via X-ray or neutron diffraction on liquids.

Alignment quality is quantified by the order parameter:

\[ S = \left\langle \frac{3\cos^2\theta - 1}{2} \right\rangle \]

where \(\theta\) is the angle between a molecule's long axis and the average alignment direction (the "director"). Check the limiting cases:

Perfect alignment (\(\theta=0\) for every molecule): \(S=(3(1)-1)/2=1\) — maximum order.

Perfectly random orientation (an ordinary isotropic liquid): averaging \(\cos^2\theta\) uniformly over a sphere gives \(\langle\cos^2\theta\rangle=1/3\) (the same isotropic-averaging result used for \(\langle v_x^2\rangle=\langle v^2\rangle/3\) in Gaseous State Ch. 7), so \(S=(3(1/3)-1)/2=0\) — no order, as expected for a normal liquid.

These two checks confirm the order parameter behaves sensibly across its full range, \(0\le S\le1\), from a completely disordered liquid to perfect molecular alignment.

Compare water's measured enthalpy of vaporisation (Ch. 3–4, \(\Delta H_{vap}\approx40.7\ \text{kJ mol}^{-1}\)) against \(H_2S\), water's heavier group 16 analogue, which does not hydrogen-bond effectively (sulfur is larger and less electronegative, Chemical Bonding Ch. 15): \(H_2S\) has \(\Delta H_{vap}\approx18.7\ \text{kJ mol}^{-1}\), despite being the heavier molecule (\(M=34\) vs. water's \(M=18\)).

If water followed the "normal" size-driven trend (dispersion forces alone), its \(\Delta H_{vap}\) should be lower than \(H_2S\)'s, not more than double it. This large "excess" — roughly \(40.7-18.7\approx22\ \text{kJ mol}^{-1}\) beyond what size alone would predict — is direct energetic evidence of the strength of water's hydrogen-bonded network, extending Chemical Bonding Ch. 15's boiling-point-based argument with a more fundamental energetic quantity.

Treat \(\rho(T)\) as governed by two competing contributions:

Normal thermal contraction: density increases as \(T\) falls (ordinary behaviour, dominant in virtually all liquids at all temperatures).

Structural network expansion (unique to water and a few other network-forming liquids): as \(T\) falls further toward 0°C, increasing tetrahedral ordering pushes density back down.

At \(T=4^{\circ}\text{C}\), these two competing contributions to \(d\rho/dT\) exactly cancel: \(d\rho/dT=0\), the elementary calculus condition defining a maximum — the same "competing effects crossing at an extremum" logic behind Gaseous State Ch. 12's critical point (\(dP/dV=0\)). Above \(4^{\circ}\text{C}\), thermal contraction dominates (\(d\rho/dT<0\) overall, normal behaviour); below \(4^{\circ}\text{C}\), the network effect dominates (density falls as \(T\) falls further) — together fully explaining the observed density maximum.

The heat removed when mass \(\Delta m\) evaporates is \(\Delta H_{vap}\) per mole, drawn directly from the liquid's own thermal energy. This lowers the remaining mass \(m_{remaining}\) by \(\Delta T\), via its specific heat capacity \(c\):

\[ \Delta H_{vap}\cdot\frac{\Delta m}{M} = m_{remaining}\cdot c\cdot\Delta T \quad\Longrightarrow\quad \Delta T = \frac{\Delta H_{vap}\,\Delta m}{M\,m_{remaining}\,c} \]

For 100 g of water losing just 1 g to evaporation (\(\Delta H_{vap}\approx2261\ \text{J g}^{-1}\), \(c=4.18\ \text{J g}^{-1}\text{K}^{-1}\)):

\[ \Delta T = \frac{(2261\ \text{J g}^{-1})(1\ \text{g})}{(99\ \text{g})(4.18\ \text{J g}^{-1}\text{K}^{-1})} \approx 5.5\ \text{K} \]

Evaporating just 1% of the sample's mass cools the rest by roughly 5.5°C — a striking, real-world-scale result that directly explains why sweating is such an effective cooling mechanism: water's \(\Delta H_{vap}\) per gram is large relative to its specific heat.

At equilibrium, the horizontal components of the three interfacial tensions acting on the three-phase contact line must balance: \(\gamma_{SV}\) pulls the contact line outward along the solid, \(\gamma_{SL}\) pulls it inward, and \(\gamma_{LV}\)'s horizontal component, \(\gamma_{LV}\cos\theta\), also pulls inward:

\[ \gamma_{SV} = \gamma_{SL} + \gamma_{LV}\cos\theta \]

Rearranged, this is Young's equation:

\[ \cos\theta = \frac{\gamma_{SV}-\gamma_{SL}}{\gamma_{LV}} \]

This connects directly back to Ch. 6 (via \(\gamma_{LV}\)) and Ch. 8's capillary rise formula, which used exactly this same \(\cos\theta\) — closing the loop between surface tension, capillary action, and wetting.

A diffusing molecule experiences viscous drag from its surroundings, described by Stokes' law for a sphere of radius \(r\) moving through a fluid of viscosity \(\eta\): \(F_{drag}=6\pi\eta rv\). Balancing this drag against the random thermal driving force (related to \(k_BT\) via fluctuation–dissipation reasoning, a result from statistical mechanics beyond this chapter's scope to derive in full) yields the Stokes–Einstein equation:

\[ D = \frac{k_BT}{6\pi\eta r} \]

Larger \(\eta\) (Ch. 9–10) or larger molecular radius \(r\) both mean more drag, and hence slower diffusion — directly connecting this chapter to liquid viscosity. Numeric check: for a small molecule (\(r\approx2\times10^{-10}\ \text{m}\)) in water (\(\eta\approx0.001\ \text{Pa s}\)) at 298 K:

\[ D = \frac{(1.38\times10^{-23})(298)}{6\pi(0.001)(2\times10^{-10})} \approx 1.1\times10^{-9}\ \text{m}^2\text{s}^{-1} \]

— a realistic value, matching typical measured diffusion coefficients for small molecules in water.

Every comparison above follows from where a substance sits on Ch. 1's \(k_BT/\varepsilon\) ratio:

\(k_BT \gg \varepsilon\) (gas): molecules move nearly independently — highly compressible, fast diffusion (long mean free path, Gaseous State Ch. 15), negligible IMFs (justifying the ideal gas model), and viscosity from momentum transfer, which increases with \(T\) (Gaseous State Ch. 16).

\(k_BT \sim \varepsilon\) (liquid): molecules are loosely bound but mobile — nearly incompressible (already touching), slow diffusion (squeezing past close neighbours, Ch. 18), IMFs dominant (this topic's central theme), and viscosity from IMF-resistance, which decreases with \(T\) (Ch. 9–10).

\(k_BT \ll \varepsilon\) (solid, briefly anticipated): molecules essentially fixed — zero compressibility, negligible diffusion, IMFs completely dominant, and no meaningful flow at all.

A single simple ratio, introduced in Ch. 1, qualitatively explains the entire pattern of differences across all three states of matter covered in these two topics.

Raoult's Law extends vapour pressure to mixtures: each component's partial vapour pressure above an ideal solution is proportional to its liquid-phase mole fraction times its pure vapour pressure, \(P_i = x_iP_i^{\circ}\). Total vapour pressure sums exactly as in Dalton's Law (Gaseous State Ch. 5):

\[ P_{total} = x_AP_A^{\circ} + x_BP_B^{\circ} \]

and the vapour-phase mole fraction follows from Dalton's Law applied to the vapour: \(y_A = P_A/P_{total}\). For a 50:50 liquid mixture with \(P_A^{\circ}=200\ \text{torr}\), \(P_B^{\circ}=100\ \text{torr}\):

\[ P_A = 0.5(200)=100,\ P_B=0.5(100)=50,\ P_{total}=150\ \text{torr} \]

\[ y_A = \frac{100}{150} = 0.667 \]

The vapour is 66.7% A, enriched well above the liquid's 50%. Repeatedly condensing and re-vaporising this vapour enriches it further each time — the working principle of fractional distillation columns.

Near its lattice site, a particle's potential energy (the same bond-energy-vs-distance well from Chemical Bonding Ch. 1) is approximately parabolic for small displacements: \(U(x)\approx\tfrac12k_{spring}x^2\), where \(k_{spring}\) characterises the well's steepness. By the equipartition theorem (Gaseous State Ch. 7), average vibrational energy in one dimension is \(\tfrac12k_BT\):

\[ \tfrac12k_BT = \tfrac12k_{spring}\langle x^2\rangle \quad\Longrightarrow\quad \langle x^2\rangle = \frac{k_BT}{k_{spring}} \]

giving RMS vibrational amplitude:

\[ x_{rms} = \sqrt{\frac{k_BT}{k_{spring}}} \]

Amplitude grows modestly with \(\sqrt{T}\) and shrinks with stiffer bonds (larger \(k_{spring}\)) — explaining why solids hold their shape while vibrations stay small relative to interatomic spacing, and foreshadowing melting: the Lindemann criterion states melting occurs roughly when \(x_{rms}\) reaches about 10–15% of the interatomic spacing, at which point the harmonic approximation breaks down entirely.

In a crystalline solid, every particle occupies an equivalent lattice environment (by definition of the repeating structure), so every particle has the same \(k_{spring}\) (Ch. 1) and reaches the Lindemann instability threshold at exactly the same temperature — a sharp, simultaneous, first-order phase transition (Chemical Thermodynamics Ch. 16).

In an amorphous solid, particles occupy a wide distribution of different local environments — structural disorder means a range of different \(k_{spring}\) values are present. Different particles reach their local escape threshold at different temperatures, producing gradual softening over a range rather than one sharp transition — a direct, quantitative extension of Ch. 1's harmonic-well model explaining exactly why the crystalline/amorphous distinction shows up as sharp vs. gradual melting.

Melting occurs roughly when \(k_BT_m\) reaches a characteristic fraction of the lattice/bond energy per particle (Liquid State Ch. 1's \(k_BT\sim\varepsilon\) criterion), so \(T_m \propto \varepsilon/k_B\). For ionic solids, lattice energy \(U\propto z_+z_-/r_0\) (Born–Landé, Chemical Bonding Ch. 2), so \(T_m\) should track \(z_+z_-/r_0\).

Check: NaCl (\(z_+z_-=1\), \(r_0\approx2.8\ \text{Å}\)), \(T_m=801^{\circ}\text{C}\), vs. MgO (\(z_+z_-=4\), \(r_0\approx2.1\ \text{Å}\)), \(T_m=2852^{\circ}\text{C}\). MgO's quadrupled charge product and smaller ionic separation both predict a substantially higher lattice energy — confirmed dramatically by its melting point, more than triple NaCl's. This directly validates Chemical Bonding Ch. 2's lattice energy framework as a predictor of bulk solid-state melting behaviour.

A particle at a cubic cell's corner is shared among 8 adjacent cells (8 cubes meet at each corner), contributing \(1/8\); a particle at a face centre is shared between 2 cells, contributing \(1/2\); a particle at the body centre belongs entirely to one cell, contributing 1. Applying this to the three common cubic cell types:

Simple cubic (SC): \(8\times\tfrac18 = 1\) particle per cell.

Body-centred cubic (BCC): \(8\times\tfrac18 + 1\times1 = 1+1 = 2\) particles per cell.

Face-centred cubic (FCC): \(8\times\tfrac18 + 6\times\tfrac12 = 1+3 = 4\) particles per cell.

This fractional-sharing method is the essential computational tool used throughout the rest of this topic — density calculations (Ch. 8), packing efficiency (Ch. 6–7), and beyond.

The conventional FCC cubic unit cell (edge \(a\)) contains 4 lattice points (Ch. 4's counting). Its primitive (smallest possible) cell, by definition, contains exactly 1 lattice point, so its volume must be exactly \(1/4\) of the conventional cell's:

\[ V_{primitive} = \frac{a^3}{4} \]

Working through the geometry, this primitive cell turns out to be a rhombohedron (not a cube) with edge length \(a/\sqrt2\) and angle \(60^{\circ}\) between edges — a genuinely different, lower-symmetry shape than the cubic description. This confirms FCC is not redundant with some simpler lattice: it requires its own distinct classification, unlike face-centred tetragonal, which is redescribable more simply.

SC (edge touching, \(a=2r\), \(Z=1\) from Ch. 4):

\[ \text{eff.} = \frac{1\times\tfrac43\pi r^3}{(2r)^3} = \frac{\pi}{6} \approx 52.4\% \]

BCC (body-diagonal touching, spanning \(4r=\sqrt3\,a\), so \(a=4r/\sqrt3\), \(Z=2\)):

\[ \text{eff.} = \frac{2\times\tfrac43\pi r^3}{(4r/\sqrt3)^3} = \frac{\pi\sqrt3}{8} \approx 68.0\% \]

FCC (face-diagonal touching, spanning \(4r=\sqrt2\,a\), so \(a=2\sqrt2\,r\), \(Z=4\)):

\[ \text{eff.} = \frac{4\times\tfrac43\pi r^3}{(2\sqrt2\,r)^3} = \frac{\pi}{3\sqrt2} \approx 74.0\% \]

FCC is the most efficiently packed of the three — setting up Ch. 7, where FCC turns out to be one of the two structures achieving the theoretical maximum packing efficiency for spheres.

Any sphere in a close-packed layer touches 6 neighbours within its own layer (the hexagonal 2D arrangement). It also touches 3 spheres in the layer above (nestled in 3 of the depressions directly above it) and 3 spheres in the layer below:

\[ \text{coordination number} = 6\ (\text{same layer}) + 3\ (\text{above}) + 3\ (\text{below}) = 12 \]

This local 3-layer "sandwich" around any given sphere is identical whether the stacking is ABAB (HCP) or ABCABC (CCP) — the difference between the two only shows up at the fourth layer (whether it returns to A or moves on to C), which is too far away to affect the immediate coordination count. Both structures therefore share the same coordination number and packing efficiency, differing only in overall symmetry.

Silver crystallises in FCC (\(Z=4\)), with edge length \(a=408.6\ \text{pm}\) and \(M=107.9\ \text{g mol}^{-1}\). Converting \(a=4.086\times10^{-8}\ \text{cm}\), so \(a^3=6.822\times10^{-23}\ \text{cm}^3\):

\[ \rho = \frac{(4)(107.9)}{(6.022\times10^{23})(6.822\times10^{-23})} \approx \frac{431.6}{41.08} \approx 10.51\ \text{g cm}^{-3} \]

This matches silver's known experimental density (\(\approx10.49\ \text{g cm}^{-3}\)) extremely well — validating the formula and demonstrating the connection between microscopic crystal structure (unit cell dimensions from X-ray diffraction, Ch. 11) and macroscopic, easily measured bulk density.

Applying Ch. 4's fractional-sharing technique to voids rather than spheres, in the FCC cell (\(Z=4\)):

Octahedral voids: one at the body centre (fully inside, contributes 1) plus one at the midpoint of each of the 12 edges (each shared among 4 adjacent cells, contributing \(1/4\) each: \(12\times\tfrac14=3\)):

\[ 1 + 3 = 4\ \text{octahedral voids per cell} \]

— exactly equal to \(Z=4\), confirming the \(N\) octahedral voids per \(N\) spheres relationship (4:4 = 1:1 ✓).

Tetrahedral voids: 8 positions entirely inside the cell, each belonging fully to that cell (contributing 1 each):

\[ 8\times1 = 8\ \text{tetrahedral voids per cell} \]

— exactly double the 4 octahedral voids (8:4 = 2:1 ✓), confirming the \(2N\) tetrahedral voids per \(N\) spheres relationship.

A cation of radius \(r_+\) sits at the centre of an octahedral void, touching 6 anions of radius \(r_-\). Two adjacent anions in octahedral geometry sit \(90^{\circ}\) apart. At the critical ratio, these two anions just touch each other while both touch the central cation, forming a right isosceles triangle: two legs of length \((r_++r_-)\) (cation-to-anion), and a hypotenuse of length \(2r_-\) (anion-to-anion, touching):

\[ (r_++r_-)^2 + (r_++r_-)^2 = (2r_-)^2 \quad\Longrightarrow\quad 2(r_++r_-)^2 = 4r_-^2 \]

\[ (r_++r_-)^2 = 2r_-^2 \quad\Longrightarrow\quad r_++r_- = r_-\sqrt2 \]

\[ r_+ = r_-(\sqrt2-1) \quad\Longrightarrow\quad \frac{r_+}{r_-} = \sqrt2-1 \approx 0.414 \]

This exactly matches the quoted lower boundary for octahedral coordination — a complete, rigorous derivation confirming why 0.414 specifically marks the threshold for a cation to fit an octahedral void (Ch. 9) snugly rather than rattle within it.

Consider two parallel X-ray beams hitting two adjacent parallel crystal planes separated by distance \(d\), both at incidence angle \(\theta\) from the plane. The beam hitting the deeper plane travels extra distance on both the way in and the way out; by simple right-triangle trigonometry (plane spacing \(d\) as hypotenuse, angle \(\theta\) from the plane), each leg contributes \(d\sin\theta\) extra path length:

\[ \text{extra path} = 2d\sin\theta \]

For constructive interference (a detected diffraction peak), this extra path must equal a whole number of wavelengths:

\[ 2d\sin\theta = n\lambda \]

— Bragg's Law, a complete geometric derivation from the path-difference condition.

NaCl: FCC anion lattice, \(Z=4\) anions/cell (Ch. 4, 6); all 4 octahedral voids filled (Ch. 9) \(=4\) cations/cell. Ratio \(4:4=1:1\) ✓.

ZnS: FCC anion lattice, \(Z=4\) anions/cell; half of the 8 tetrahedral voids filled (alternating ones) \(=4\) cations/cell. Ratio \(4:4=1:1\) ✓ (filling all 8 would incorrectly give a 2:1 ratio).

CsCl: simple cubic anion lattice, \(Z=1\) anion/cell (Ch. 4, 6); one cation fully inside at the body centre \(=1\) cation/cell. Ratio \(1:1\) ✓.

All three structures verify correctly using the same fractional-counting toolkit from Ch. 4 and Ch. 9.

For \(n\) defects among \(N\) sites, costing \(\Delta H_{defect}\) each, the configurational entropy uses Boltzmann's formula (Atomic Structure Ch. 11), \(S=k_B\ln W\), with \(W=N!/(n!(N-n)!)\). Using Stirling's approximation (\(\ln N!\approx N\ln N-N\)) for the entropy of mixing, and minimising \(\Delta G=n\Delta H_{defect}-T\Delta S\) with respect to \(n\) — the same "derivative equals zero at equilibrium" logic behind Gaseous State Ch. 12's critical point and Liquid State Ch. 15's density maximum:

\[ \frac{d(\Delta G)}{dn} = \Delta H_{defect} - k_BT\ln\!\frac{N-n}{n} = 0 \]

Solving for \(n/N\) (assuming \(n\ll N\), valid for the small defect concentrations typically observed):

\[ \frac{n}{N} \approx e^{-\Delta H_{defect}/2k_BT} \]

Defect concentration follows the same Boltzmann-factor exponential form that has recurred throughout (Liquid State's vapour pressure and viscosity relations) — increasing with temperature, but never reaching zero at any \(T>0\), confirming that perfectly defect-free crystals are only a theoretical idealisation.

The theoretical shear strength of a perfect, defect-free crystal (estimated from interatomic bond-breaking energetics) is typically around \(G/10\) to \(G/30\), where \(G\) is the material's shear modulus. The observed shear strength of real crystalline metals is typically only \(G/1000\) to \(G/10{,}000\) — a discrepancy of roughly 100–1000× below theoretical prediction.

This dramatic, well-documented gap was historically one of the key pieces of evidence that led Taylor, Orowan, and Polanyi (independently, 1934) to theoretically predict dislocations as the resolution — decades before dislocations were directly observed via electron microscopy. A striking case of theory predicting an unseen structural feature to explain a puzzling quantitative discrepancy, later confirmed by direct observation.

The number of electrons thermally excited across a band gap \(E_g\) follows the same Boltzmann-factor logic recurring throughout this material (Liquid State's vapour pressure and viscosity; Ch. 13's defect concentration):

\[ n_{excited} \propto e^{-E_g/2k_BT} \]

(the factor of 2 reflects the symmetric process of exciting an electron into the conduction band while simultaneously leaving a corresponding "hole" in the valence band). Since conductivity \(\sigma\) is directly proportional to the number of available charge carriers:

\[ \sigma \propto e^{-E_g/2k_BT} \]

This explains why semiconductor conductivity increases sharply with temperature — the opposite of metallic conductors, whose conductivity decreases with temperature as increased lattice vibrations (Ch. 1's growing vibrational amplitude) scatter and impede electron flow. The same exponential Boltzmann-factor structure appears here in yet another physical context.

For a simple reversible elementary reaction \(A \rightleftharpoons B\), the forward and reverse rates (mass-action rate laws for an elementary step) are \(\text{rate}_f=k_f[A]\) and \(\text{rate}_r=k_r[B]\). At equilibrium, by definition, these rates are equal:

\[ k_f[A]_{eq} = k_r[B]_{eq} \]

Rearranging:

\[ \frac{[B]_{eq}}{[A]_{eq}} = \frac{k_f}{k_r} = \text{a constant} \]

since \(k_f\) and \(k_r\) are both fixed at a given temperature. This constant ratio is exactly the equilibrium constant \(K=k_f/k_r\) for this simple case — derived here from first principles (kinetics) rather than just defined empirically, and generalised to arbitrary stoichiometry in Ch. 2.

Generalising Ch. 1's elementary-step rate law form to arbitrary stoichiometry: \(\text{rate}_f=k_f[A]^a[B]^b\), \(\text{rate}_r=k_r[C]^c[D]^d\). At equilibrium these balance:

\[ k_f[A]^a[B]^b = k_r[C]^c[D]^d \]

Rearranging:

\[ \frac{[C]^c[D]^d}{[A]^a[B]^b} = \frac{k_f}{k_r} = K_c \]

the general Law of Mass Action, derived directly from kinetic first principles. (This simple derivation strictly applies to elementary reactions, where the rate law mirrors stoichiometry directly; for complex, multi-step reactions the overall \(K_c\) expression still takes this same form, but \(k_f,k_r\) don't correspond to a single elementary step's rate constants — a nuance worth flagging honestly.)

Substitute \(P_i=[i]RT\) for every species in \(K_p\)'s definition:

\[ K_p = \frac{([C]RT)^c([D]RT)^d}{([A]RT)^a([B]RT)^b} = \frac{[C]^c[D]^d}{[A]^a[B]^b}\cdot(RT)^{(c+d)-(a+b)} \]

\[ K_p = K_c\cdot(RT)^{\Delta n} \]

where \(\Delta n=(c+d)-(a+b)\), the difference in gas-phase stoichiometric coefficients — a clean, complete algebraic derivation reusing Gaseous State Ch. 4's ideal gas law directly.

Substituting \(\Delta G^{\circ}=-RT\ln K\) (Chemical Thermodynamics Ch. 13) into the general relation \(\Delta G=\Delta G^{\circ}+RT\ln Q\):

\[ \Delta G = -RT\ln K + RT\ln Q = RT\ln\!\frac{Q}{K} \]

Applying the spontaneity criterion \(\Delta G\lt0\) (Chemical Thermodynamics Ch. 16):

\(Q\lt K\): \(Q/K\lt1\), so \(\ln(Q/K)\lt0\), so \(\Delta G\lt0\) — forward reaction spontaneous, \(Q\) rises toward \(K\).

\(Q\gt K\): \(\Delta G\gt0\) — forward reaction non-spontaneous, reverse is spontaneous, \(Q\) falls toward \(K\).

\(Q=K\): \(\Delta G=0\) — at equilibrium, no net driving force.

A complete derivation grounding the empirical Q-vs-K rule directly in Chemical Thermodynamics' Gibbs free energy framework.

Consider \(aA+bB\rightleftharpoons cC+dD\) at equilibrium (\(Q=K\)). Suddenly add more \(A\). Since \([A]\) appears in \(Q\)'s denominator, this increase makes \(Q\) momentarily smaller than \(K\) (\(Q\lt K\), right after the addition, before any shift occurs).

By Ch. 4's rule, \(Q\lt K\) means the reaction shifts forward (toward products) to increase \(Q\) back toward \(K\). Shifting forward consumes some of the added \(A\) (and \(B\)), partially reducing \([A]\) back down — partially counteracting the stress of having added \(A\) — while producing more \(C\) and \(D\). This is exactly Le Chatelier's qualitative prediction ("adding a reactant shifts equilibrium toward products"), now derived rigorously from the quantitative Q-vs-K framework.

Take two identical systems, each with volume \(V\) and mass \(m\), and combine them into one system.

Mass and volume both double: \(m \to 2m\), \(V \to 2V\) — confirming both are extensive. Now consider density:

\[ \rho = \frac{m}{V} \quad\longrightarrow\quad \rho' = \frac{2m}{2V} = \frac{m}{V} = \rho \]

Density is unchanged by combining identical systems, so it is intensive. The same argument applies to any ratio of two extensive quantities (e.g. molar volume \(V/n\), specific heat capacity per gram).

Write \(dU = M\,dT + N\,dV\), where \(M=(\partial U/\partial T)_V\) and \(N=(\partial U/\partial V)_T\). If \(U\) is a well-behaved state function, its mixed partial derivatives must be equal regardless of the order of differentiation (Euler's reciprocity relation):

\[ \frac{\partial^2 U}{\partial V\,\partial T} = \frac{\partial^2 U}{\partial T\,\partial V} \quad\Longrightarrow\quad \left(\frac{\partial M}{\partial V}\right)_T = \left(\frac{\partial N}{\partial T}\right)_V \]

This equality is both necessary and sufficient for \(M\,dT + N\,dV\) to be an exact differential — i.e. for \(\int dU\) between two states to be the same along every path. It is exactly this property that fails for \(dq\) and \(dw\): no function \(q(T,V)\) exists whose exact differential reproduces the heat exchanged, which is why heat is a path function and cannot be assigned a single value "possessed" by a system.

For a fixed amount of gas at constant volume, pressure varies linearly with Celsius temperature (Gay-Lussac's Law):

\[ P = P_0(1 + \alpha\,t) \]

where \(t\) is temperature in \(^{\circ}\text{C}\), \(P_0\) is the pressure at \(0^{\circ}\text{C}\), and \(\alpha\) is found experimentally to be very close to \(1/273.15\ ^{\circ}\text{C}^{-1}\) for any dilute gas. Setting \(P=0\) and solving for \(t\):

\[ 0 = P_0\left(1 + \frac{t}{273.15}\right) \quad\Longrightarrow\quad t = -273.15\ ^{\circ}\text{C} \]

Every dilute gas extrapolates to zero pressure at the same temperature, independent of which gas is used. This universal intercept is defined as absolute zero, 0 K — the basis of the Kelvin scale.

Compare two ways of expanding 1 mol of ideal gas isothermally at temperature \(T\) from \(V_1\) to \(V_2\):

(a) Single-step irreversible expansion against a constant external pressure equal to the final pressure, \(P_{ext}=P_2=RT/V_2\):

\[ w_{by,\,irr} = P_{ext}(V_2-V_1) = \frac{RT}{V_2}(V_2-V_1) \]

(b) Reversible expansion, with \(P_{ext}\) matching the gas pressure at every instant (Ch. 5 result):

\[ w_{by,\,rev} = RT\ln\!\frac{V_2}{V_1} \]

For any \(V_2>V_1\), it can be shown (e.g. by plotting or by the inequality \(\ln x > 1 - 1/x\) for \(x>1\), with \(x=V_2/V_1\)) that

\[ RT\ln\!\frac{V_2}{V_1} \;>\; \frac{RT}{V_2}(V_2-V_1) \]

so \(w_{by,\,rev} > w_{by,\,irr}\): the reversible path always extracts more work from the same expansion. This is a general result, not special to ideal gases — a reversible process always represents the maximum work obtainable (or minimum work required for compression) between two given states.

For a reversible expansion, the external pressure equals the gas pressure at every instant: \(P_{ext}=P=nRT/V\). Work done by the gas in an infinitesimal expansion is \(dw_{by} = P\,dV\). Integrating from \(V_1\) to \(V_2\) at constant \(T\):

\[ w_{by} = \int_{V_1}^{V_2} \frac{nRT}{V}\,dV = nRT \ln\!\frac{V_2}{V_1} \]

Using the IUPAC convention (\(w\) = work done on the system, \(w=-w_{by}\)):

\[ w = -nRT \ln\!\frac{V_2}{V_1} \]

Since \(T\) is constant, \(\Delta U = 0\) for an ideal gas (internal energy of an ideal gas depends only on \(T\)). By the First Law, \(q = -w = nRT\ln(V_2/V_1)\): all the heat absorbed is converted directly into work of expansion.

Bonus result, quoted for reference (full derivation in Ch. 6): for an ideal gas, \(C_P - C_V = R\).

For one mole of an ideal gas, \(H = U + RT\). Differentiating with respect to temperature at constant pressure:

\[ \left(\frac{\partial H}{\partial T}\right)_P = \left(\frac{\partial U}{\partial T}\right)_P + R \]

For an ideal gas, \(U\) depends only on \(T\) (not on \(V\) or \(P\)), so \((\partial U/\partial T)_P = (\partial U/\partial T)_V = C_V\). The left-hand side is \(C_P\) by definition, giving:

\[ C_P = C_V + R \quad\Longrightarrow\quad C_P - C_V = R \]

Per mole, \(C_P\) exceeds \(C_V\) by exactly the gas constant \(R \approx 8.314\ \text{J K}^{-1}\text{mol}^{-1}\), regardless of which ideal gas is involved.

Because \(H\) is a state function, \(\Delta H\) for reactants → products is the same by any path. Consider the two-step path: first decompose the reactants entirely into their constituent elements (in standard states), then reassemble those elements into the products.

Step 1 (reactants → elements) is the reverse of forming the reactants, so its enthalpy is \(-\sum \Delta H^{\circ}_f(\text{reactants})\). Step 2 (elements → products) is exactly \(\sum \Delta H^{\circ}_f(\text{products})\). Adding the two steps (path independence of a state function):

\[ \Delta H^{\circ}_{rxn} = -\sum \Delta H^{\circ}_f(\text{reactants}) + \sum \Delta H^{\circ}_f(\text{products}) \]

which rearranges to the formula above. The elements act as a common reference point that every substance's formation enthalpy is measured against, making the whole table self-consistent.

This reaction is hard to isolate experimentally because C tends to burn straight to \(CO_2\). But two related combustions are easy to measure calorimetrically:

(1) \(C(s) + O_2(g) \rightarrow CO_2(g)\), \(\Delta H_1 = -393.5\ \text{kJ}\)

(2) \(CO(g) + \tfrac{1}{2}O_2(g) \rightarrow CO_2(g)\), \(\Delta H_2 = -283.0\ \text{kJ}\)

Target: \(C(s) + \tfrac{1}{2}O_2(g) \rightarrow CO(g)\), \(\Delta H_3 = \,?\)

Reverse equation (2) (flip its sign) so \(CO_2\) appears as a reactant, then add it to equation (1):

\[ C(s)+O_2(g) \rightarrow CO_2(g) \quad(\Delta H_1) \]

\[ CO_2(g) \rightarrow CO(g) + \tfrac{1}{2}O_2(g) \quad(-\Delta H_2) \]

Adding and cancelling the \(CO_2\) and one \(O_2\) that appear on both sides leaves exactly the target equation, so:

\[ \Delta H_3 = \Delta H_1 - \Delta H_2 = (-393.5) - (-283.0) = -110.5\ \text{kJ} \]

For a reversible adiabatic change, \(dq=0\), so the First Law gives \(dU = -P\,dV\). Using \(dU = nC_V\,dT\) and \(P = nRT/V\):

\[ nC_V\,dT = -\frac{nRT}{V}\,dV \quad\Longrightarrow\quad \frac{C_V}{T}\,dT = -\frac{R}{V}\,dV \]

Integrating between states 1 and 2:

\[ C_V \ln\!\frac{T_2}{T_1} = -R \ln\!\frac{V_2}{V_1} \]

Using \(R = C_V(\gamma - 1)\) (from \(C_P - C_V = R\) and \(\gamma = C_P/C_V\)) and dividing through by \(C_V\):

\[ \ln\!\frac{T_2}{T_1} = -(\gamma-1)\ln\!\frac{V_2}{V_1} = (\gamma-1)\ln\!\frac{V_1}{V_2} \]

which exponentiates to \(T_2 V_2^{\gamma-1} = T_1 V_1^{\gamma-1}\), i.e. \(TV^{\gamma-1}=\text{constant}\). Substituting the ideal gas law \(T = PV/nR\) converts this to the equivalent, more commonly quoted form:

\[ PV^{\gamma} = \text{constant} \]

Assume the Clausius statement is false: a "super-refrigerator" transfers heat \(Q\) from a cold reservoir to a hot one with no external work. Pair it with an ordinary heat engine operating between the same two reservoirs, which absorbs heat \(Q_h\) from the hot reservoir, delivers work \(W\), and rejects exactly \(Q\) to the cold reservoir (choose the engine's size so its rejected heat matches the refrigerator's transferred heat).

Combine the two devices. The cold reservoir gains \(Q\) from the engine and loses \(Q\) to the refrigerator — no net effect there. The hot reservoir loses \(Q_h\) to the engine and gains \(Q\) from the refrigerator: a net loss of \(Q_h - Q\). Overall, the combined device extracts heat \(Q_h - Q\) from a single (hot) reservoir and converts all of it into work \(W\), with no other effect — exactly what the Kelvin–Planck statement forbids.

So a violation of Clausius implies a violation of Kelvin–Planck. An analogous construction shows the reverse implication, establishing that the two statements are equivalent: either both hold, or neither does.

From the First Law, \(dU = dq_{rev} - P\,dV\), so \(dq_{rev} = dU + P\,dV\). Using \(dU = nC_V dT\) and the ideal gas law \(P = nRT/V\):

\[ dq_{rev} = nC_V\,dT + \frac{nRT}{V}\,dV \]

Divide through by \(T\) to get \(dS = dq_{rev}/T\):

\[ dS = \frac{nC_V}{T}\,dT + \frac{nR}{V}\,dV \]

Integrating from state 1 to state 2, treating \(C_V\) as constant over the range:

\[ \Delta S = nC_V \ln\!\frac{T_2}{T_1} + nR \ln\!\frac{V_2}{V_1} \]

Because \(S\) is a state function, this result holds even if the actual process was irreversible (e.g. free expansion into a vacuum) — the same formula gives the correct \(\Delta S\), so long as the initial and final states match.

Using the isothermal work result (Ch. 5) and the adiabatic relation \(TV^{\gamma-1}=\text{const}\) (Ch. 9) for the four steps, with volumes \(V_1\rightarrow V_2\rightarrow V_3\rightarrow V_4\rightarrow V_1\):

Isothermal expansion at \(T_h\) (\(V_1\rightarrow V_2\)): \(q_h = nRT_h\ln(V_2/V_1)\).
Isothermal compression at \(T_c\) (\(V_3\rightarrow V_4\)): heat released, magnitude \(q_c = nRT_c\ln(V_3/V_4)\).

The two adiabatic steps give \(T_hV_2^{\gamma-1}=T_cV_3^{\gamma-1}\) and \(T_cV_4^{\gamma-1}=T_hV_1^{\gamma-1}\). Dividing these two equations cancels \(T_h,T_c\):

\[ \left(\frac{V_2}{V_1}\right)^{\gamma-1} = \left(\frac{V_3}{V_4}\right)^{\gamma-1} \quad\Longrightarrow\quad \frac{V_2}{V_1}=\frac{V_3}{V_4} \]

So the logarithms in \(q_h\) and \(q_c\) are equal, giving the clean ratio \(q_c/q_h = T_c/T_h\). Since \(w = q_h - q_c\) over the full cycle:

\[ \eta = \frac{w}{q_h} = \frac{q_h-q_c}{q_h} = 1 - \frac{q_c}{q_h} = 1 - \frac{T_c}{T_h} \]

The Second Law states that for any spontaneous process, the entropy of the universe increases:

\[ \Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} \geq 0 \]

At constant pressure, heat released by the system flows into the surroundings: \(q_{surr} = -\Delta H_{sys}\). If the surroundings are large enough to stay at constant \(T\), \(\Delta S_{surr} = q_{surr}/T = -\Delta H_{sys}/T\). Substituting:

\[ \Delta S_{univ} = \Delta S_{sys} - \frac{\Delta H_{sys}}{T} \geq 0 \]

Multiplying through by \(-T\) (which flips the inequality, since \(T>0\)):

\[ \Delta H_{sys} - T\Delta S_{sys} \leq 0 \]

The left-hand side is defined as \(\Delta G_{sys}\), giving the spontaneity criterion \(\Delta G \leq 0\), with equality at equilibrium. A further standard result, quoted here and derived in Ch. 16, links \(\Delta G\) to the equilibrium constant:

\[ \Delta G^{\circ} = -RT\ln K \]

The Clausius inequality (a generalisation of the Second Law, Ch. 10–11) states that for any process exchanging heat \(q\) with surroundings at temperature \(T\), \(\Delta S \ge q/T\). Combined with the First Law, \(q = \Delta U - w\) (\(w\) = work done on the system):

\[ \Delta S \ge \frac{\Delta U - w}{T} \quad\Longrightarrow\quad w \ge \Delta U - T\Delta S \]

At constant \(T\), the right-hand side is exactly \(\Delta A = \Delta U - T\Delta S\), so:

\[ w \ge \Delta A \]

The work done on the system is always at least \(\Delta A\); equivalently, the work done by the system, \(w_{by}=-w\), satisfies \(w_{by}\le -\Delta A\). So \(-\Delta A\) is the maximum work extractable, achieved only in the reversible limit. Setting \(w=0\) (constant volume, no non-expansion work) recovers the equilibrium criterion \(\Delta A \le 0\).

Differentiate \(\Delta G/T\) with respect to \(T\) at constant \(P\) using the quotient rule:

\[ \frac{\partial}{\partial T}\!\left(\frac{\Delta G}{T}\right) = \frac{1}{T}\left(\frac{\partial \Delta G}{\partial T}\right)_P - \frac{\Delta G}{T^2} \]

Using \((\partial \Delta G/\partial T)_P = -\Delta S\):

\[ = -\frac{\Delta S}{T} - \frac{\Delta G}{T^2} \]

Now substitute \(\Delta S = (\Delta H - \Delta G)/T\), from rearranging \(\Delta G = \Delta H - T\Delta S\) (Ch. 13):

\[ = -\frac{\Delta H - \Delta G}{T^2} - \frac{\Delta G}{T^2} = \frac{-\Delta H + \Delta G - \Delta G}{T^2} = -\frac{\Delta H}{T^2} \]

giving \(\left[\partial(\Delta G/T)/\partial T\right]_P = -\Delta H/T^2\), as stated. Integrating between \(T_1\) and \(T_2\) (treating \(\Delta H\) as constant) gives the practical working form:

\[ \frac{\Delta G_2}{T_2} - \frac{\Delta G_1}{T_1} = -\Delta H\left(\frac{1}{T_2}-\frac{1}{T_1}\right) \]

Start from the Clausius inequality, \(T\,dS \ge dq\), and the First Law for \(PV\)-work only, \(dq = dU + P\,dV\):

\[ T\,dS \ge dU + P\,dV \quad\Longrightarrow\quad dU + P\,dV - T\,dS \le 0 \]

This single master inequality specialises to each named criterion:

Constant \(S,V\) (\(dS=0,\ dV=0\)): directly gives \(dU \le 0\).

Constant \(S,P\): since \(dH = dU + P\,dV\) at constant \(P\), the master inequality reads \(dH - T\,dS \le 0\); with \(dS=0\), \(dH \le 0\).

Constant \(T,V\) (\(dV=0\)): the master inequality reduces to \(dU - T\,dS \le 0\), which at constant \(T\) is exactly \(dA \le 0\) (since \(dA=dU-T\,dS-S\,dT\) and \(dT=0\)).

Constant \(T,P\): the master inequality as written, \(dU+P\,dV-T\,dS\le 0\), is exactly \(dG \le 0\) at constant \(T,P\) (since \(dG=dU+P\,dV+V\,dP-T\,dS-S\,dT\), and the last two terms vanish).

From \(dS = dq_{rev}/T\) and, at constant pressure, \(dq_{rev}=C_P\,dT\):

\[ dS = \frac{C_P}{T}\,dT \]

Integrating from 0 K to temperature \(T\), and using the Third Law boundary condition \(S(0)=0\) for a perfect crystal:

\[ S^{\circ}(T) = \int_0^T \frac{C_P}{T}\,dT \]

In practice this integral is evaluated from measured \(C_P\) data at many temperatures (often plotted as \(C_P/T\) vs. \(T\)), adding a \(\Delta H_{trs}/T_{trs}\) jump term at each phase transition passed through on the way from 0 K to \(T\) (fusion, vaporisation, solid–solid transitions). This calorimetric method is how tables of \(S^{\circ}\) values are actually built.

At constant \(T,P\), a small change in Gibbs energy from changing the amounts of each species is \(dG = \sum_i \mu_i\,dn_i\). For a reaction with extent of reaction \(\xi\), each \(dn_i = \nu_i\,d\xi\) (\(\nu_i>0\) for products, \(<0\) for reactants), so:

\[ \left(\frac{\partial G}{\partial \xi}\right)_{T,P} = \sum_i \nu_i\mu_i \]

From Ch. 16, equilibrium at constant \(T,P\) occurs where \(G\) is at a minimum with respect to any change the system can still make — here, with respect to the extent of reaction, \((\partial G/\partial\xi)_{T,P}=0\). Therefore, at equilibrium:

\[ \sum_i \nu_i \mu_i = 0 \]

This single condition is the chemical-potential form of "reaction has reached equilibrium", and (with \(\mu_i=\mu_i^{\circ}+RT\ln a_i\) for each species) it is exactly what expands into the familiar \(\Delta G^{\circ}=-RT\ln K\) relation from Ch. 13.

Along a two-phase coexistence curve, the chemical potentials of the two phases stay equal (Ch. 18): \(\mu_{liq}(T,P)=\mu_{vap}(T,P)\). Using \(d\mu = -S\,dT + V\,dP\) for each phase and equating the changes along the curve:

\[ -S_{liq}\,dT + V_{liq}\,dP = -S_{vap}\,dT + V_{vap}\,dP \]

Rearranging gives the general Clapeyron equation:

\[ \frac{dP}{dT} = \frac{S_{vap}-S_{liq}}{V_{vap}-V_{liq}} = \frac{\Delta S_{vap}}{\Delta V_{vap}} = \frac{\Delta H_{vap}}{T\Delta V_{vap}} \]

(using \(\Delta S_{vap}=\Delta H_{vap}/T\) for a reversible phase change at equilibrium, Ch. 11). For vaporisation, \(V_{vap}\gg V_{liq}\), so \(\Delta V_{vap}\approx V_{vap}=RT/P\) (ideal gas). Substituting:

\[ \frac{dP}{dT} = \frac{\Delta H_{vap}\,P}{RT^2} \quad\Longrightarrow\quad \frac{d\ln P}{dT} = \frac{\Delta H_{vap}}{RT^2} \]

which integrates to the working form quoted above.

For a real gas at constant \(T\), \(d\mu = V\,dP\); by the definition of fugacity, \(d\mu = RT\,d(\ln f)\), so:

\[ d(\ln f) = \frac{V}{RT}\,dP \]

For an ideal gas at the same \(T\), \(V_{ideal}/RT = 1/P\), so \(d(\ln P) = dP/P\). Subtracting the ideal-gas relation from the real-gas one, and using \(\ln\varphi = \ln f - \ln P\):

\[ d(\ln\varphi) = \left(\frac{V}{RT}-\frac{1}{P}\right)dP = \frac{1}{P}\left(\frac{PV}{RT}-1\right)dP = \frac{Z-1}{P}\,dP \]

Integrating from \(P=0\) (where \(\varphi\to 1\), so \(\ln\varphi\to 0\)) up to pressure \(P\):

\[ \ln\varphi = \int_0^P \frac{Z-1}{P}\,dP \]

Given experimental \(Z\) vs. \(P\) data (or an equation of state), this integral gives the fugacity coefficient, and hence \(f=\varphi P\), directly.