08.11.03 · stat-mech / real-gases

Real gases — virial expansion and van der Waals

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Anchor (Master): Landau-Lifshitz *Statistical Physics, Part 1* §74-§84 (canonical anchor); Mayer-Mayer *Statistical Mechanics* (Wiley, 1940); Hill *Statistical Mechanics* (McGraw-Hill, 1956); Reichl *A Modern Course in Statistical Physics*, 4th ed. (Wiley-VCH, 2016) Ch. 4; Yang-Lee 1952 *Phys. Rev.* 87 phase-transition zeros; Wilson 1971 *Phys. Rev. B* 4 renormalisation-group view

Intuition Beginner

The ideal-gas law $P V = N k_{B} T$ pretends that gas molecules are points that never touch one another. Real molecules occupy space, and they pull on each other weakly when they pass nearby. As you compress a real gas, these two effects start to matter. The excluded volume of the molecules raises the pressure above the ideal value because the molecules have less room to fly around than the container volume suggests. The weak attraction between molecules lowers the pressure because near the container wall a molecule is pulled back toward the bulk and hits the wall a little less hard.

Van der Waals in 1873 cooked up a two-parameter correction that captures both effects in one tidy formula. Replace $V$ by $V - N b$ to subtract the volume taken up by the molecules themselves, and subtract $N^{2} a / V^{2}$ from the pressure to account for the attractive pull. The two parameters $a$ and $b$ are tuned by experiment, but they have a clean microscopic meaning that the master derivation pins down precisely.

Why does any of this matter? Because the van der Waals equation predicts something the ideal gas cannot: a gas-liquid phase transition. At low enough temperature the equation produces an unstable loop where pressure decreases with decreasing volume, and the system splits into a dense liquid phase and a dilute gas phase. The critical point at which liquid and gas become indistinguishable is predicted from $a$ and $b$ alone. The same prediction holds approximately for every non-polar fluid in nature, and that universality is one of the famous early successes of statistical physics.

Visual Beginner

A pressure-volume diagram with several isotherms (curves at fixed temperature) for the van der Waals equation. At high temperature each isotherm is a smooth decreasing curve close to the ideal-gas hyperbola $P = N k_{B} T / V$ . As the temperature drops toward the critical temperature $T_{c}$ , the isotherms develop a flat inflection at the critical point. Below $T_{c}$ , each isotherm shows an unphysical wiggle with a local maximum and a local minimum. A horizontal line (the Maxwell construction) cuts across the wiggle so that the area above the line equals the area below, and along that horizontal cut the system coexists as a mixture of liquid and gas.

The picture is the whole content of the unit at the informal level. The smooth high-temperature curves correspond to a fluid that behaves like a gas. The wiggly low-temperature curves show where the gas-liquid coexistence begins. The flat inflection at $T_{c}$ is the critical point at which the wiggle first appears.

Worked example Beginner

Use argon gas as the test case. Argon is a noble gas, so its molecules are nearly spherical and weakly interacting — the simplest setting where the van der Waals equation works well.

Step 1. Look up the measured critical-point data for argon: critical temperature $T_{c} = 150.9$ K, critical pressure $P_{c} = 48.9$ atm, critical molar volume $V_{c} = 75$ cm $^{3}$ per mole. These three numbers are the experimental anchors.

Step 2. The van der Waals critical-point relations give $T_{c} = 8 a / (27 b k_{B})$ , $V_{c} = 3 N b$ , and $P_{c} = a / (27 b^{2})$ . Three equations, two unknowns $a, b$ . Use any two to solve and the third becomes a consistency check.

Step 3. Extract $b$ from the volume relation. Per mole, $V_{c} = 3 N_{A} b$ gives $b = V_{c} / (3 N_{A}) = (75 cm^{3}) / (3 \times 6.022 \times 1 0^{23}) = 4.15 \times 1 0^{- 23}$ cm $^{3}$ per molecule. Convert: $b \approx 4.15 \times 1 0^{- 29}$ m $^{3}$ per molecule, the excluded-volume parameter.

Step 4. Extract $a$ from the pressure relation. $a = 27 b^{2} P_{c} = 27 \times (4.15 \times 1 0^{- 29})^{2} \times (48.9 \times 1.013 \times 1 0^{5})$ Pa m $^{6}$ per molecule squared, giving $a \approx 2.3 \times 1 0^{- 49}$ Pa m $^{6}$ per pair. Convert to per-mole units: $a_{molar} \approx 0.136$ Pa m $^{6}$ per mole squared, in good agreement with the tabulated argon value $0.1355$ .

Step 5. Predict the second virial coefficient at room temperature, $T = 300$ K. The van der Waals form is $B_{2} (T) = b - a / (k_{B} T)$ , where here $b$ and $a$ are per-molecule. At $T = 300$ K, $k_{B} T = 4.14 \times 1 0^{- 21}$ J, so $a / (k_{B} T) = 2.3 \times 1 0^{- 49} /4.14 \times 1 0^{- 21} = 5.55 \times 1 0^{- 29}$ m $^{3}$ . Then $B_{2} (300 K) = 4.15 \times 1 0^{- 29} - 5.55 \times 1 0^{- 29} = - 1.40 \times 1 0^{- 29}$ m $^{3}$ per molecule. Multiply by Avogadro: $B_{2}^{molar} (300 K) \approx - 8.4$ cm $^{3}$ per mole.

Step 6. Compare to measurement. The measured second virial of argon at 300 K is about $- 15$ cm $^{3}$ per mole. The van der Waals prediction has the correct sign (attraction dominates at room temperature) and the correct order of magnitude. The factor-of-two discrepancy is the standard accuracy of the van der Waals equation: good qualitatively, off by a tens-of-percent factor in the quantitative tail.

What this tells us: the van der Waals equation is a remarkable two-parameter mean-field model that pins down the right physics of a real gas. It is not exact, but it captures the gas-liquid phase transition, the law of corresponding states, and the basic temperature dependence of the second virial coefficient using two parameters that have direct microscopic meaning.

Check your understanding Beginner

Exercise (easy, true-false).

True or false: the second virial coefficient $B_{2} (T)$ of a real gas is always positive.

Hint

Consider the sign of the Mayer function $e^{- u (r) / k_{B} T} - 1$ in the attractive part of the potential (where $u < 0$ ) versus the repulsive part (where $u > 0$ ). The temperature determines which contribution dominates.

Answer

False. The second virial coefficient depends on temperature. At low temperature the attractive part of the interatomic potential dominates and $B_{2} (T)$ is negative (the gas is more compressible than ideal). At high temperature the hard-core repulsion dominates and $B_{2} (T)$ is positive (the gas is less compressible than ideal). The temperature at which $B_{2} (T) = 0$ is called the Boyle temperature. For argon the Boyle temperature is about 410 K, well above the 300 K computed in the worked example, which is why $B_{2} (300 K)$ comes out negative.

Formal definition Intermediate+

Consider a classical gas of $N$ identical structureless particles in a volume $V$ at temperature $T$ , with $β = 1/ (k_{B} T)$ . The particles interact pairwise through a two-body potential $u (r)$ that depends only on the inter-particle separation $r = ∣ r_{i} - r_{j} ∣$ . The canonical partition function factorises into a kinetic part and a configurational part: $$ Z_N(V, T) = \frac{1}{N!, \lambda_T^{3N}} Q_N(V, T), \qquad Q_N(V, T) = \int_{V^N} \exp!\Bigl(-\beta \sum_{i < j} u(r_{ij})\Bigr) d^{3N} r, $$ where $λ_{T} = (2 π ℏ^{2} / m k_{B} T)^{1/2}$ is the thermal de Broglie wavelength and $Q_{N}$ is the configurational integral. For an ideal gas, $u \equiv 0$ , $Q_{N} = V^{N}$ , and $Z_{N} = V^{N} / (N! λ_{T}^{3 N})$ .

The Mayer function is $$ f(r) = e^{-\beta u(r)} - 1, $$ which vanishes wherever the interaction vanishes and is short-ranged for any physically reasonable $u$ . The Boltzmann factor expands as $$ \exp!\Bigl(-\beta \sum_{i<j} u(r_{ij})\Bigr) = \prod_{i<j} \bigl(1 + f(r_{ij})\bigr) = \sum_{\text{graphs } G} \prod_{(i,j) \in E(G)} f(r_{ij}), $$ where the sum runs over all simple graphs $G$ on the vertex set ${1, \dots, N}$ . This is the Mayer cluster expansion.

The grand-canonical partition function $Ξ (μ, V, T) = \sum_{N \geq 0} e^{β μ N} Z_{N}$ assembles the canonical partition functions weighted by fugacity $z = e^{β μ} / λ_{T}^{3}$ . The Mayer linked-cluster theorem identifies $$ \log \Xi = V \sum_{n \geq 1} b_n z^n, $$ where $b_{n}$ is the cluster integral corresponding to all connected graphs on $n$ labelled vertices, normalised by $1/ (V \cdot n!)$ : $$ b_n = \frac{1}{V \cdot n!} \int_{V^n} \sum_{\substack{G \text{ connected}\\text{on } n \text{ vertices}}} \prod_{(i,j) \in E(G)} f(r_{ij}), d^{3n} r. $$ The density is $n = N / V = z (\partial lo g Ξ/ \partial z) / V = \sum_{m \geq 1} m b_{m} z^{m}$ , and inverting this series defines the virial expansion $$ \frac{P}{k_B T} = \frac{\log \Xi}{V} = \sum_{n \geq 1} B_n(T) , n^n, $$ where $n$ on the right is the number density and the virial coefficients $B_{n} (T)$ are determined by the cluster integrals. The first three are $B_{1} = 1$ , $B_{2} = - b_{2}$ , $B_{3} = 4 b_{2}^{2} - 2 b_{3}$ , with $b_{n}$ given above.

The second virial coefficient is therefore $$ B_2(T) = -\frac{1}{2} \int_{\mathbb{R}^3} f(r), d^3 r = -2\pi \int_0^\infty \bigl(e^{-\beta u(r)} - 1\bigr) r^2, dr, $$ a temperature-dependent integral of the Mayer function over relative position.

The van der Waals equation is the leading-order resummation of the cluster expansion for the model potential $$ u(r) = \begin{cases} +\infty & r < r_0 \ -\epsilon (r_0 / r)^6 & r \geq r_0 \end{cases} $$ with $ϵ ≪ k_{B} T$ . Direct computation (carried out in the key derivation below) gives $$ P = \frac{N k_B T}{V - N b} - \frac{N^2 a}{V^2}, \qquad b = \frac{2\pi r_0^3}{3}, \qquad a = \frac{2\pi \epsilon r_0^3}{3}. $$ The critical point is the unique point at which an isotherm has $\partial P / \partial V = 0$ and $\partial^{2} P / \partial V^{2} = 0$ simultaneously. Solving these conditions yields $$ T_c = \frac{8 a}{27 b k_B}, \qquad V_c = 3 N b, \qquad P_c = \frac{a}{27 b^2}, \qquad Z_c \equiv \frac{P_c V_c}{N k_B T_c} = \frac{3}{8}. $$ The law of corresponding states asserts that in reduced variables $\tilde{P} = P / P_{c}$ , $\tilde{V} = V / V_{c}$ , $\tilde{T} = T / T_{c}$ , every van der Waals fluid satisfies the universal equation $(\tilde{P} + 3/ \tilde{V}^{2}) (\tilde{V} - 1/3) (8/3) = \tilde{T}$ , with no remaining microscopic parameters.

The Maxwell equal-area construction resolves the unstable loop below $T_{c}$ . The actual coexistence pressure $P_{coex} (T)$ at fixed $T$ is the horizontal line in the $P$ - $V$ plane drawn so that the areas above and below the line, between the line and the van der Waals isotherm, are equal. Equivalently, $P_{coex}$ is the Legendre transform of the convex hull of the canonical free energy $F (V, T)$ as a function of $V$ at fixed $T$ .

Counterexamples to common slips

The Mayer function $f (r)$ is not the potential $u (r)$ . Inside the hard core, $u = + \infty$ and $f = - 1$ (not $+ \infty$ ). The expansion in $f$ converges even for hard-core potentials precisely because $f$ is bounded.
The virial expansion is in number density $n = N / V$ , not in fugacity $z$ . Mixing these conventions is a common slip in textbook presentations. The grand-canonical density-fugacity inversion is required to convert between them.
The Maxwell construction does not modify the van der Waals equation. It supplements it with the physical coexistence pressure, replacing the unstable loop by a horizontal line. Below the line lies a metastable region (superheated liquid, supercooled gas) accessible experimentally to those who avoid nucleation.
The compressibility factor $Z_{c} = 3/8 = 0.375$ is a strict van der Waals prediction. Measured values for argon, neon, krypton, methane cluster around $0.29$ to $0.30$ . The discrepancy is a quantitative failure of mean-field theory near $T_{c}$ , not an experimental artefact.

Key derivation Intermediate+

Theorem (second virial coefficient). For a classical gas with pairwise potential $u (r)$ , the leading correction to the ideal-gas equation of state is $$ \frac{P}{k_B T} = n + B_2(T) n^2 + O(n^3), \qquad B_2(T) = -\frac{1}{2} \int_{\mathbb{R}^3} \bigl(e^{-\beta u(r)} - 1\bigr), d^3 r. $$

Derivation. Start from the grand-canonical partition function and the Mayer cluster expansion.

Step 1: cluster expansion of $lo g Ξ$ . Write the configurational integral in the cluster form. For each graph $G$ on ${1, \dots, N}$ , denote by $C (G)$ the set of its connected components. The integral factorises over components: $$ \int_{V^N} \prod_{(i,j) \in E(G)} f(r_{ij}), d^{3N} r = \prod_{C \in \mathcal{C}(G)} \int_{V^{|C|}} \prod_{(i,j) \in E(G|C)} f(r{ij}), d^{3|C|} r. $$ Multiplying by $z^{N} / N!$ and summing over $N$ converts the sum over graphs into an exponential of connected pieces. The standard combinatorial identity (Mayer's linked-cluster theorem; the same combinatorial step that appears again in graph-theoretic generating functions) yields $$ \log \Xi(\mu, V, T) = V \sum_{n \geq 1} b_n z^n, \qquad b_n = \frac{1}{V \cdot n!} \int_{V^n} \mathcal{B}_n(\vec r_1, \ldots, \vec r_n), d^{3n} r, $$ where $B_{n}$ is the sum over connected graphs on $n$ vertices of products of $f$ -bonds. The factor of $V$ comes from translation invariance: in the thermodynamic limit each $b_{n}$ is intensive.

Step 2: pressure and density from $lo g Ξ$ . Thermodynamics identifies $lo g Ξ = β P V$ and $N = z \partial lo g Ξ/ \partial z$ , so $$ \beta P = \sum_{n \geq 1} b_n z^n, \qquad n = \frac{N}{V} = \sum_{n \geq 1} n b_n z^n. $$ Both series are in the fugacity $z$ . Invert the second series to express $z$ in powers of $n$ : $$ z = n - 2 b_2 n^2 + (8 b_2^2 - 3 b_3) n^3 + O(n^4), $$ using $b_{1} = 1$ (the single-vertex connected graph is the one-point graph with integrand $1$ , normalised to $b_{1} = V \cdot 1/ (V \cdot 1!) = 1$ ). Substituting into $β P$ and collecting powers of $n$ yields $$ \beta P = n - b_2 n^2 + (4 b_2^2 - 2 b_3) n^3 + O(n^4). $$ The coefficient of $n^{2}$ is by definition $B_{2} (T) = - b_{2}$ .

Step 3: explicit form of $b_{2}$ . There is only one connected graph on two vertices, namely the single edge. So $$ b_2 = \frac{1}{V \cdot 2!} \int_{V^2} f(r_{12}), d^3 r_1, d^3 r_2 = \frac{1}{2 V} \cdot V \int_{\mathbb{R}^3} f(r), d^3 r = \frac{1}{2} \int_{\mathbb{R}^3} \bigl(e^{-\beta u(r)} - 1\bigr), d^3 r, $$ using translation invariance to do one integral as $V$ , then taking the thermodynamic limit $V \to \infty$ to extend $\int_{V}$ to $\int_{R^{3}}$ . Therefore $$ B_2(T) = -b_2 = -\frac{1}{2} \int_{\mathbb{R}^3} \bigl(e^{-\beta u(r)} - 1\bigr), d^3 r = -2\pi \int_0^\infty \bigl(e^{-\beta u(r)} - 1\bigr) r^2, dr, $$ the second equality by writing $d^{3} r = 4 π r^{2} d r$ for a spherically symmetric integrand. This is the second-virial-coefficient formula derived from first principles. $□$

Theorem (van der Waals equation from the model potential). For the hard-core-plus-attractive-tail potential $u (r) = + \infty$ for $r < r_{0}$ and $u (r) = - ϵ (r_{0} / r)^{6}$ for $r \geq r_{0}$ , in the regime $ϵ ≪ k_{B} T$ , the second virial coefficient is $B_{2} (T) = b - a / (k_{B} T)$ with $b = 2 π r_{0}^{3} /3$ and $a = (2 π /3) ϵ r_{0}^{3}$ . Resummation of the corresponding leading hard-core and attractive contributions yields the van der Waals equation $P = N k_{B} T / (V - N b) - N^{2} a / V^{2}$ .

Derivation. Split the second-virial integral at $r = r_{0}$ : $$ B_2(T) = -2\pi \int_0^{r_0} (e^{-\infty} - 1) r^2, dr - 2\pi \int_{r_0}^\infty (e^{\beta \epsilon (r_0 / r)^6} - 1) r^2, dr. $$ On the first range the integrand is $- 2 π (- 1) r^{2} = 2 π r^{2}$ , giving $$ -2\pi \int_0^{r_0} (-1) r^2, dr = 2\pi \cdot \frac{r_0^3}{3} = \frac{2\pi r_0^3}{3} =: b. $$ On the second range expand the exponential to first order in $β ϵ$ , valid for $ϵ ≪ k_{B} T$ : $$ e^{\beta \epsilon (r_0 / r)^6} - 1 \approx \beta \epsilon (r_0 / r)^6, $$ so the second piece becomes $$ -2\pi \int_{r_0}^\infty \beta \epsilon (r_0 / r)^6 r^2, dr = -2 \pi \beta \epsilon r_0^6 \int_{r_0}^\infty r^{-4}, dr = -\frac{2 \pi \beta \epsilon r_0^6}{3 r_0^3} = -\frac{2\pi \epsilon r_0^3}{3 k_B T} =: -\frac{a}{k_B T}. $$ Combining, $B_{2} (T) = b - a / (k_{B} T)$ .

To produce the van der Waals equation, resum the cluster expansion in two pieces. The hard-core contribution is treated non-perturbatively as an excluded-volume effect: replace the available volume $V$ by $V - N b$ in the ideal-gas pressure, giving the entropic term $N k_{B} T / (V - N b)$ . This is the geometric mean-field statement that each molecule sees a reduced volume because the other $N$ molecules occupy excluded balls of volume $2 b$ around themselves (factor of $2$ from the symmetry of pairs, halved by the per-molecule normalisation). The attractive contribution is treated in a mean-field approximation: each molecule sees an average attractive potential proportional to the density $n$ times the integral of the attractive tail, giving an internal-pressure correction $- N^{2} a / V^{2}$ . Adding these yields $$ P = \frac{N k_B T}{V - N b} - \frac{N^2 a}{V^2}. $$ Expanding in powers of $N / V$ reproduces $β P / n = 1 + (b - a / k_{B} T) n + O (n^{2})$ , confirming consistency with the second-virial calculation above. $□$

Bridge. The virial expansion is the foundational reason that the equation of state of any real gas is computable from microscopic interactions. The cluster expansion builds toward all of equilibrium classical statistical mechanics for interacting systems: the linked-cluster theorem is exactly the combinatorial identity that appears again in 08.10.06 (one-loop renormalisation in $ϕ^{4}$ ), where connected Feynman diagrams play the role of connected Mayer clusters. The central insight is that the Mayer function $f (r) = e^{- β u (r)} - 1$ is small for short-ranged potentials and short-ranged in $r$ , so a power-series expansion in $f$ converges where the original Boltzmann factor $e^{- β u}$ does not. The bridge is the recognition that the van der Waals equation is exactly the leading two-cluster resummation of this expansion for a hard-core-plus-attractive-tail potential, and that this resummation generalises to mean-field theory for every short-ranged interacting system. Putting these together, the same mean-field organisation appears again in 08.02.01 (mean-field theory for the Ising model), where the Weiss approximation plays the role of the van der Waals approximation: each spin sees an effective field generated by its neighbours, exactly as each molecule sees an effective attractive pressure generated by the density of its neighbours. This identifies van der Waals theory with Curie-Weiss mean-field theory and with Landau theory of the liquid-gas phase transition.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Derive the critical-point conditions $\partial P / \partial V = \partial^{2} P / \partial V^{2} = 0$ for the van der Waals equation $P = N k_{B} T / (V - N b) - N^{2} a / V^{2}$ , and solve to obtain $T_{c}$ , $V_{c}$ , $P_{c}$ in terms of $a$ , $b$ .

Hint

Compute the two derivatives, set both to zero, and solve the resulting two equations for $V_{c}$ and $T_{c}$ simultaneously. Then plug back to obtain $P_{c}$ .

Answer

Compute $\partial P / \partial V = - N k_{B} T / (V - N b)^{2} + 2 N^{2} a / V^{3} = 0$ and $\partial^{2} P / \partial V^{2} = 2 N k_{B} T / (V - N b)^{3} - 6 N^{2} a / V^{4} = 0$ . From the first: $N k_{B} T / (V - N b)^{2} = 2 N^{2} a / V^{3}$ , so $k_{B} T = 2 N a (V - N b)^{2} / V^{3}$ . From the second: $2 N k_{B} T / (V - N b)^{3} = 6 N^{2} a / V^{4}$ , so $k_{B} T = 3 N a (V - N b)^{3} / V^{4}$ . Setting these equal: $2 (V - N b)^{2} / V^{3} = 3 (V - N b)^{3} / V^{4}$ , giving $2 V = 3 (V - N b)$ , so $V_{c} = 3 N b$ . Plugging back: $k_{B} T_{c} = 2 N a (2 N b)^{2} / (3 N b)^{3} = 8 N^{3} a b^{2} / (27 N^{3} b^{3}) = 8 a / (27 b)$ , so $T_{c} = 8 a / (27 b k_{B})$ . The pressure $P_{c} = N k_{B} T_{c} / (V_{c} - N b) - N^{2} a / V_{c}^{2} = (8 a /27 b) \cdot N / (2 N b) - N^{2} a / (9 N^{2} b^{2}) = 4 a / (27 b^{2}) - a / (9 b^{2}) = a (4 - 3) / (27 b^{2}) = a / (27 b^{2})$ . Rubric: full credit for all three critical-point values plus the derivation steps.

Exercise 4 (medium, symbolic).

Show that in reduced variables $\tilde{P} = P / P_{c}$ , $\tilde{V} = V / V_{c}$ , $\tilde{T} = T / T_{c}$ the van der Waals equation becomes the parameter-free form $(\tilde{P} + 3/ \tilde{V}^{2}) (\tilde{V} - 1/3) = (8/3) \tilde{T}$ .

Hint

Substitute $P = \tilde{P} P_{c}$ , $V = \tilde{V} V_{c}$ , $T = \tilde{T} T_{c}$ into the van der Waals equation and simplify using $T_{c} = 8 a / (27 b k_{B})$ , $V_{c} = 3 N b$ , $P_{c} = a / (27 b^{2})$ .

Answer

The van der Waals equation $(P + N^{2} a / V^{2}) (V - N b) = N k_{B} T$ . Substitute: $\tilde{P} P_{c} + N^{2} a / (\tilde{V}^{2} V_{c}^{2}) = \tilde{P} (a /27 b^{2}) + N^{2} a / (\tilde{V}^{2} \cdot 9 N^{2} b^{2}) = (a /27 b^{2}) (\tilde{P} + 3/ \tilde{V}^{2})$ . And $\tilde{V} V_{c} - N b = 3 N b \tilde{V} - N b = N b (3 \tilde{V} - 1)$ . Multiplying: $(a /27 b^{2}) (\tilde{P} + 3/ \tilde{V}^{2}) \cdot N b (3 \tilde{V} - 1) = a N (3 \tilde{V} - 1) (\tilde{P} + 3/ \tilde{V}^{2}) / (27 b)$ . Setting equal to $N k_{B} T = N k_{B} \tilde{T} T_{c} = N k_{B} \tilde{T} \cdot 8 a / (27 b k_{B}) = 8 a N \tilde{T} / (27 b)$ : $a N (3 \tilde{V} - 1) (\tilde{P} + 3/ \tilde{V}^{2}) / (27 b) = 8 a N \tilde{T} / (27 b)$ , simplifying to $(\tilde{P} + 3/ \tilde{V}^{2}) (3 \tilde{V} - 1) = 8 \tilde{T}$ , equivalent to $(\tilde{P} + 3/ \tilde{V}^{2}) (\tilde{V} - 1/3) = (8/3) \tilde{T}$ . The microscopic parameters $a$ , $b$ , $N$ , $k_{B}$ have all cancelled. Rubric: full credit for the reduced equation in parameter-free form.

Exercise 5 (medium, symbolic).

State the Maxwell equal-area construction and prove it by demanding equality of the Gibbs free energy of the liquid and gas phases at coexistence.

Hint

At coexistence the chemical potentials of liquid and gas are equal, $μ_{L} = μ_{G}$ . Use $d μ = - s d T + v d P$ at fixed $T$ to relate $μ_{G} - μ_{L}$ to an integral of $v d P$ along the isotherm.

Answer

At coexistence $μ_{L} (T, P_{coex}) = μ_{G} (T, P_{coex})$ . Along an isotherm $d μ = v d P$ where $v = V / N$ is the molar volume. Integrate $μ_{G} - μ_{L} = \int_{L}^{G} v d P$ along the van der Waals isotherm from the liquid branch (small $v$ ) to the gas branch (large $v$ ) through the unstable region. Reparametrise the integral in terms of $v$ : $\int_{L}^{G} v d P = [v P]_{L}^{G} - \int_{v_{L}}^{v_{G}} P d v = P_{coex} (v_{G} - v_{L}) - \int_{v_{L}}^{v_{G}} P d v$ . Setting $μ_{G} - μ_{L} = 0$ at coexistence: $P_{coex} (v_{G} - v_{L}) = \int_{v_{L}}^{v_{G}} P (v) d v$ , where $P (v)$ is the van der Waals isotherm. The right-hand side is the area under the isotherm from $v_{L}$ to $v_{G}$ ; the left-hand side is the area under the horizontal line at $P_{coex}$ over the same range. So the two areas (above the line, below the curve, and below the line, above the curve) are equal: the equal-area construction. Rubric: full credit for the chemical-potential argument and identification of the equal-area condition.

Exercise 6 (medium, numeric).

Compute the second virial coefficient $B_{2} (T)$ for the square-well potential $u (r) = + \infty$ for $r < σ$ , $u (r) = - ϵ$ for $σ < r < λσ$ , and $u (r) = 0$ for $r > λσ$ , where $λ > 1$ .

Hint

Split the integral into three regions and evaluate each. The middle region contributes a $(e^{β ϵ} - 1)$ factor.

Answer

$B_{2} (T) = - 2 π \int_{0}^{\infty} (e^{- β u} - 1) r^{2} d r$ . Split at $σ$ and $λσ$ : $(0, σ)$ contributes $- 2 π (- 1) σ^{3} /3 = 2 π σ^{3} /3$ ; $(σ, λσ)$ contributes $- 2 π (e^{β ϵ} - 1) \int_{σ}^{λσ} r^{2} d r = - 2 π (e^{β ϵ} - 1) σ^{3} (λ^{3} - 1) /3$ ; $(λσ, \infty)$ contributes $0$ . So $B_{2} (T) = (2 π σ^{3} /3) [1 - (λ^{3} - 1) (e^{β ϵ} - 1)]$ . As $T \to \infty$ , $β ϵ \to 0$ and $B_{2} \to 2 π σ^{3} /3$ , recovering the hard-sphere result. As $T \to 0$ , $β ϵ \to \infty$ and $B_{2} \to - \infty$ , signalling the breakdown of the gas phase. Rubric: full credit for the closed-form expression and both limiting behaviours.

Exercise 7 (hard, short-answer).

Derive the third virial coefficient $B_{3} (T) = 4 b_{2}^{2} - 2 b_{3}$ from the cluster expansion, identifying $b_{3}$ as an integral over connected three-vertex graphs.

Hint

The connected graphs on three vertices are the three two-edge "chains" (with one vertex of degree two and two of degree one) and the one triangle (three-cycle). Sum these with the correct combinatorial weights.

Answer

The connected graphs on three labelled vertices are the three "path" graphs (e.g., $1$ - $2$ - $3$ , $1$ - $3$ - $2$ , $2$ - $1$ - $3$ , etc., three in total) and the triangle (one in total). The corresponding integrand is $3 f_{12} f_{23} + f_{12} f_{23} f_{13}$ (after using symmetry). Then $b_{3} = (1/3! V) \int_{V^{3}} [3 f_{12} f_{23} + f_{12} f_{23} f_{13}] d^{9} r$ . By translation invariance one integration gives a factor $V$ , so $b_{3} = (1/6) \int_{R^{6}} [3 f_{12} f_{23} + f_{12} f_{23} f_{13}] d^{3} r_{2} d^{3} r_{3} = (1/2) (\int f d^{3} r)^{2} -$ (combinations involving the triangle). After standard manipulation, $b_{3} = (1/2) (\int f d^{3} r)^{2} -$ triangle correction, and inverting the density-fugacity relation to $O (n^{3})$ yields $B_{3} = 4 b_{2}^{2} - 2 b_{3}$ , with the triangle graph contributing the "irreducible" three-body correlation. For a hard-sphere gas, the triangle integral is finite and computable, giving $B_{3} = 5 b^{2} /8$ in van der Waals units. Rubric: full credit for the connected-graph enumeration plus the relation $B_{3} = 4 b_{2}^{2} - 2 b_{3}$ .

Exercise 8 (hard, short-answer).

Show that on the critical isotherm $T = T_{c}$ the van der Waals equation predicts $∣ P - P_{c} ∣ \sim ∣ V - V_{c} ∣^{δ}$ with critical exponent $δ = 3$ , and explain why the experimental value for non-polar fluids is $δ \approx 4.8$ .

Hint

Expand the reduced van der Waals equation around the critical point $\tilde{V} = \tilde{P} = \tilde{T} = 1$ in small deviations. The leading non-vanishing terms give the critical exponent.

Answer

Set $\tilde{T} = 1$ , $\tilde{V} = 1 + v$ , $\tilde{P} = 1 + p$ in $(\tilde{P} + 3/ \tilde{V}^{2}) (\tilde{V} - 1/3) = 8/3$ . Expand: $3/ \tilde{V}^{2} = 3 (1 - 2 v + 3 v^{2} - 4 v^{3} + \dots)$ , $\tilde{V} - 1/3 = 2/3 + v$ . Multiplying out and keeping terms through $v^{3}$ : $(\tilde{P} + 3 - 6 v + 9 v^{2} - 12 v^{3}) (2/3 + v) = 8/3$ . Expanding: $2 \tilde{P} /3 + 2 - 4 v + 6 v^{2} - 8 v^{3} + \tilde{P} v + 3 v - 6 v^{2} + 9 v^{3} + O (v^{4}) = 8/3$ . Using $\tilde{P} = 1 + p$ : $2/3 + 2 p /3 + 2 - 4 v + 6 v^{2} - 8 v^{3} + v + p v + 3 v - 6 v^{2} + 9 v^{3} + O (v^{4}) = 8/3$ . Simplifying: $2 p /3 + p v + v^{3} /2 + O (v^{4}) = 0$ , so $p \approx - 3 v^{3} /4$ at $\tilde{T} = 1$ . Therefore $∣ P - P_{c} ∣ \sim ∣ V - V_{c} ∣^{3}$ , so $δ = 3$ . The experimental exponent for non-polar fluids is $δ \approx 4.8$ , the value in the three-dimensional Ising universality class. The van der Waals prediction $δ = 3$ fails because mean-field theory ignores critical fluctuations; the renormalisation-group analysis of Wilson (1971) explains the discrepancy. Rubric: full credit for the cubic expansion plus the identification of $δ = 3$ and a note on the experimental value.

Advanced results Master

Theorem (Mayer linked-cluster theorem; Mayer 1937). For a classical gas with pairwise potential $u (r)$ , the logarithm of the grand-canonical partition function admits the expansion $$ \log \Xi(\mu, V, T) = V \sum_{n \geq 1} b_n(T) z^n, $$ where $z = e^{β μ} / λ_{T}^{3}$ is the fugacity and $b_{n} (T)$ is the cluster integral generated by all connected labelled graphs on $n$ vertices weighted by products of Mayer $f$ -bonds, normalised by $1/ (V \cdot n!)$ . Each $b_{n}$ is intensive in the thermodynamic limit and depends on $T$ but not on $V$ .

The theorem is the workhorse of classical many-body physics. Its proof rests on a graph-theoretic identity: the generating function of all labelled graphs is the exponential of the generating function of connected labelled graphs (the "exponential formula" in combinatorics). The diagrammatic structure organises higher-order corrections systematically and is the classical precursor of the linked-cluster theorem in quantum field theory.

Theorem (Yang-Lee circle theorem; Yang-Lee 1952). For a lattice gas with finite hard-core repulsion and arbitrary attractive interactions, the grand-canonical partition function $Ξ (z, V)$ at finite volume $V$ is a polynomial in $z$ whose zeros lie on the unit circle $∣ z ∣ = 1$ . In the thermodynamic limit $V \to \infty$ , these zeros either avoid the positive real axis (no phase transition at any $μ$ ) or accumulate at a point $z_{c}$ on the positive real axis (phase transition at fugacity $z_{c}$ ).

The Yang-Lee theorem connects the analytic structure of $Ξ (z)$ to the existence of phase transitions. A phase transition occurs precisely where the zeros of $Ξ$ , in the thermodynamic limit, pinch the positive real fugacity axis, producing a singularity in $lo g Ξ = β P V$ . For a real gas, the analogous statement is that the van der Waals isotherm below $T_{c}$ corresponds to a coexistence region in fugacity space where the Yang-Lee zeros accumulate. Modern extensions (Fisher zeros in complex temperature, Lee-Yang singularities in spin systems) play a central role in non-perturbative analysis of phase transitions.

Theorem (failure of van der Waals near $T_{c}$ ; mean-field exponents). The van der Waals equation predicts the mean-field critical exponents $$ \alpha = 0, \qquad \beta = 1/2, \qquad \gamma = 1, \qquad \delta = 3, $$ for the specific-heat divergence, the order-parameter scaling, the isothermal compressibility divergence, and the critical-isotherm shape, respectively. Experimentally, non-polar fluids in the three-dimensional Ising universality class exhibit $α \approx 0.110$ , $β \approx 0.326$ , $γ \approx 1.237$ , $δ \approx 4.8$ .

The order-parameter exponent is the most dramatic discrepancy. Near $T_{c}$ from below, the density difference between liquid and gas scales as $ρ_{L} - ρ_{G} \sim (T_{c} - T)^{β}$ , and the van der Waals prediction $β = 1/2$ exceeds the measured Ising value $β = 0.326$ by 50 percent. The failure is not due to error in the model but due to a structural deficiency: mean-field theory ignores correlations between fluctuations on different length scales. Below the upper critical dimension $d_{c} = 4$ , these correlations dominate the long-wavelength physics and force a renormalised value of $β$ .

Theorem (Wilson renormalisation-group view; Wilson 1971). The van der Waals equation is the mean-field limit of the Landau-Ginzburg theory of the liquid-gas transition. Near the critical point the order parameter $ϕ = (ρ - ρ_{c}) / ρ_{c}$ obeys a Landau free-energy functional $F [ϕ] = \int d^{d} x [(1/2) (\nabla ϕ)^{2} + (r /2) ϕ^{2} + (u /4!) ϕ^{4}]$ with $r \propto (T - T_{c})$ . The Wilson-Fisher fixed point analysis shows that for $d < 4$ the renormalisation-group flow drives the system away from the mean-field (Gaussian) fixed point toward an interacting fixed point with non-classical critical exponents in the Ising universality class. The van der Waals equation is the result of evaluating the Landau-Ginzburg free energy at its saddle point, ignoring all fluctuations.

The renormalisation-group viewpoint identifies the van der Waals equation as the mean-field theory of the liquid-gas transition, in the same sense that Curie-Weiss is the mean-field theory of the ferromagnet, and Landau theory is the universal mean-field theory of any second-order phase transition. The upper critical dimension $d_{c} = 4$ is the dimension above which the Gaussian fixed point is stable and mean-field exponents are exact; below $d_{c} = 4$ , fluctuations are relevant and the exponents flow to non-mean-field values. The Ising universality class to which the three-dimensional liquid-gas transition belongs is the same class that contains the three-dimensional Ising magnet 08.01.02, a deep structural identification.

Theorem (modern equations of state). The Carnahan-Starling equation $P V / (N k_{B} T) = (1 + η + η^{2} - η^{3}) / (1 - η)^{3}$ , with packing fraction $η = N \cdot (4 π r_{0}^{3} /3) / V$ , gives an essentially exact description of the hard-sphere fluid up to the freezing transition at $η \approx 0.49$ . The Benedict-Webb-Rubin (BWR) equation, an empirical eight-parameter form, accurately describes light hydrocarbons over a wide range of $(T, P)$ relevant to industrial processes. Modern molecular-dynamics simulations and integral-equation theories (Percus-Yevick, hypernetted-chain) supersede the van der Waals approximation for quantitative predictions.

The Carnahan-Starling formula is a Padé-style resummation of the hard-sphere virial expansion that recovers the exact $B_{2}$ , $B_{3}$ , $B_{4}$ to within a few percent and extrapolates accurately to high density. It is the modern replacement for the van der Waals excluded-volume term. The BWR equation and its descendants (Soave-Redlich-Kwong, Peng-Robinson) are used industrially. The van der Waals equation persists as a pedagogical and conceptual tool: it captures the qualitative physics of the gas-liquid transition with two parameters, and it is the simplest setting in which the law of corresponding states is predicted in a substantive way from microscopic interactions.

Synthesis. The virial expansion is the foundational reason that classical gas thermodynamics admits a systematic perturbation theory in the density. The central insight is that the Mayer cluster expansion organises all interaction effects by connected-graph topology, and the leading two-cluster resummation generates the van der Waals equation with parameters $b$ (excluded volume) and $a$ (attractive integral) that have direct microscopic meaning. The bridge between the microscopic potential and the macroscopic equation of state is exactly the second-virial integral $B_{2} (T) = - (1/2) \int f (r) d^{3} r$ , and this is exactly the same organising principle that appears again in 08.02.01 (mean-field Ising) and in 08.10.06 (one-loop $ϕ^{4}$ theory), where the leading connected-graph contributions resum to give the mean-field equations of state of each system. Putting these together, the van der Waals equation, the Curie-Weiss equation, and the Landau-Ginzburg saddle-point equation are three presentations of the same mean-field structure, identified by the renormalisation-group analysis of Wilson 1971 as the Gaussian fixed point of the Landau theory above the upper critical dimension $d_{c} = 4$ .

The duality between the microscopic and the macroscopic also generalises in a precise sense. The Yang-Lee theorem identifies the analytic structure of $lo g Ξ (z)$ with the locus of phase transitions, and the van der Waals isotherm below $T_{c}$ is the projection of this analytic structure onto the $P$ - $V$ plane. The Maxwell equal-area construction is the Legendre-transform statement that the canonical free energy must be convex in the volume, and the unstable van der Waals loop is replaced by the convex hull, with the horizontal coexistence line equal to the Legendre transform of the slope at the spinodal. The bridge between the cluster expansion (microscopic) and the Maxwell construction (thermodynamic) is the recognition that both are statements about the same partition function, viewed from different sides of the Legendre transform between $F (V, T)$ and $G (P, T)$ . The Carnahan-Starling resummation of the hard-sphere virial series is the modern descendant of the van der Waals approximation, and its accuracy demonstrates that the cluster expansion is quantitatively reliable in regimes where the underlying assumption (short-ranged interactions, density not too close to close-packing) is valid. The breakdown of mean-field exponents below $d_{c} = 4$ is the structural fact that locates van der Waals theory in the Landau universality class and that motivates the Wilson-Fisher renormalisation-group analysis of 08.04.02, where the cluster expansion of $ϕ^{4}$ theory plays the role of the Mayer expansion and the renormalisation flow plays the role of the Maxwell-construction convexification.

Full proof set Master

Proposition (Mayer linked-cluster theorem; combinatorial proof). Let $G_{n}$ denote the set of simple graphs on ${1, \dots, n}$ and $C_{n}$ the subset of connected graphs. For each graph $G$ assign a weight $w (G) = \prod_{(i, j) \in E (G)} f (r_{ij})$ . Then $$ \sum_{n \geq 0} \frac{z^n}{n!} \int_{V^n} \sum_{G \in \mathcal{G}n} w(G), d^{3n} r = \exp!\Bigl( \sum{n \geq 1} \frac{z^n}{n!} \int_{V^n} \sum_{G \in \mathcal{C}_n} w(G), d^{3n} r \Bigr). $$

Proof. The combinatorial input is the exponential formula: if $A (z) = \sum_{n} a_{n} z^{n} / n!$ generates "labelled structures" and $B (z) = \sum_{n} b_{n} z^{n} / n!$ generates the connected components, then $A (z) = exp B (z)$ , where $a_{n}$ counts all structures (a structure is an unordered collection of connected components) and $b_{n}$ counts connected structures. For Mayer graphs, the "structure" is a graph on $n$ vertices, and the "connected components" are the connected subgraphs. Decomposing a graph $G$ on ${1, \dots, n}$ into its connected components $C_{1}, \dots, C_{k}$ partitions the vertex set into disjoint subsets $V_{1}, \dots, V_{k}$ with $G ∣_{V_{i}} = C_{i}$ . The number of ways to choose such a partition and assign vertex labels is $n! / \prod_{i} ∣ V_{i} ∣!$ , and the weight factorises as $w (G) = \prod_{i} w (C_{i})$ . The integral factorises by independence: $\int_{V^{n}} w (G) d^{3 n} r = \prod_{i} \int_{V^{∣ V_{i} ∣}} w (C_{i}) d^{3∣ V_{i} ∣} r$ . Summing over all partitions and using the exponential formula in generating-function form yields the claimed identity. Equivalently, taking logarithms, $$ \log \Xi(z, V) = \sum_{n \geq 1} \frac{z^n}{n!} \int_{V^n} \sum_{G \in \mathcal{C}_n} w(G), d^{3n} r, $$ and translation invariance plus extraction of the volume factor gives $lo g Ξ = V \sum_{n} b_{n} z^{n}$ . $□$

Proposition (van der Waals critical-point values). Setting $\partial P / \partial V = \partial^{2} P / \partial V^{2} = 0$ in the van der Waals equation $P = N k_{B} T / (V - N b) - N^{2} a / V^{2}$ yields $V_{c} = 3 N b$ , $T_{c} = 8 a / (27 b k_{B})$ , $P_{c} = a / (27 b^{2})$ , and the dimensionless ratio $Z_{c} = P_{c} V_{c} / (N k_{B} T_{c}) = 3/8$ .

Proof. Write $W = V - N b$ and $u = N / V$ . The pressure is $P = N k_{B} T / W - N^{2} a / V^{2}$ . Differentiating with respect to $V$ (with $T$ , $N$ held fixed): $$ \frac{\partial P}{\partial V} = -\frac{N k_B T}{W^2} + \frac{2 N^2 a}{V^3}, \qquad \frac{\partial^2 P}{\partial V^2} = \frac{2 N k_B T}{W^3} - \frac{6 N^2 a}{V^4}. $$ Setting both to zero produces two equations. From the first, $k_{B} T = 2 N a W^{2} / V^{3}$ . From the second, $k_{B} T = 3 N a W^{3} / V^{4}$ . Dividing eliminates $T$ : $2/ V^{3} = 3 W / V^{4}$ , so $V_{c} = 3 W = 3 (V_{c} - N b)$ , giving $V_{c} = 3 N b$ . Then $W_{c} = V_{c} - N b = 2 N b$ , and $k_{B} T_{c} = 2 N a (2 N b)^{2} / (3 N b)^{3} = 2 N a \cdot 4 N^{2} b^{2} /27 N^{3} b^{3} = 8 a / (27 b)$ . The critical pressure is $$ P_c = \frac{N k_B T_c}{W_c} - \frac{N^2 a}{V_c^2} = \frac{N \cdot 8 a / 27 b}{2 N b} - \frac{N^2 a}{9 N^2 b^2} = \frac{4 a}{27 b^2} - \frac{a}{9 b^2} = \frac{4 a - 3 a}{27 b^2} = \frac{a}{27 b^2}. $$ The compressibility factor is $Z_{c} = P_{c} V_{c} / (N k_{B} T_{c}) = (a /27 b^{2}) (3 N b) / (N \cdot 8 a /27 b) = 3 a / (8 a) = 3/8$ . $□$

Proposition (law of corresponding states). In reduced variables $\tilde{P} = P / P_{c}$ , $\tilde{V} = V / V_{c}$ , $\tilde{T} = T / T_{c}$ , the van der Waals equation is parameter-free: $(\tilde{P} + 3/ \tilde{V}^{2}) (\tilde{V} - 1/3) = (8/3) \tilde{T}$ .

Proof. Solved in Exercise 4 above. The microscopic parameters $a$ , $b$ , $N$ , $k_{B}$ cancel completely after substitution, leaving a universal equation in dimensionless variables. The corresponding-states law was proposed empirically before its theoretical basis was understood and remains the strongest qualitative success of the van der Waals model. $□$

Proposition (Maxwell construction is convexification). The physical equation of state below $T_{c}$ is obtained from the van der Waals isotherm by replacing the unstable loop with the horizontal segment at coexistence pressure $P_{coex} (T)$ , determined by the equal-area condition. Equivalently, the physical canonical free energy $F_{phys} (V, T)$ is the convex hull of the van der Waals free energy $F_{vdW} (V, T)$ as a function of $V$ .

Proof. The Helmholtz free energy at fixed $N$ , $T$ satisfies $P = - \partial F / \partial V$ . Convexity of $F (V)$ in $V$ corresponds to $\partial^{2} F / \partial V^{2} = - \partial P / \partial V \geq 0$ , equivalently $\partial P / \partial V \leq 0$ (the isothermal compressibility is non-negative). Where the van der Waals isotherm has $\partial P / \partial V > 0$ , the free energy is concave and unstable: phase separation lowers $F$ by exchanging the concave segment for the linear segment connecting the spinodal endpoints. The Maxwell equal-area construction at fixed $T$ is the statement that $P_{coex}$ is the slope of the common tangent to $F (V_{L}, T)$ and $F (V_{G}, T)$ , with the equality of slopes encoding the equal-pressure condition and the equality of intercepts encoding the equal-chemical-potential condition. The construction is the Legendre-transform statement that the physical $G (P, T)$ is the lower envelope of the van der Waals $G$ , and the coexistence pressure is the value at which two branches of $G$ cross. $□$

Proposition (mean-field critical exponents from van der Waals). The van der Waals equation predicts $β = 1/2$ for the order-parameter exponent, defined by $ρ_{L} - ρ_{G} \sim (T_{c} - T)^{β}$ as $T \to T_{c}^{-}$ .

Proof. Set $t = (T_{c} - T) / T_{c}$ , $ϕ = (ρ - ρ_{c}) / ρ_{c}$ . The reduced van der Waals equation $(\tilde{P} + 3/ \tilde{V}^{2}) (\tilde{V} - 1/3) = (8/3) \tilde{T}$ becomes, on expansion about the critical point in small $ϕ$ and small $t$ , the Landau form $$ \tilde P - 1 \approx -4 t - 6 t \phi - (3/2) \phi^3 + O(\phi^4, t \phi^2). $$ The coexistence condition $\partial \tilde{P} / \partial ϕ ∣_{T} = 0$ at fixed $\tilde{P}$ below $T_{c}$ gives $ϕ \neq = 0$ solutions $ϕ^{2} = - 4 t$ (after using the cubic to identify the spinodal). The two non-vanishing roots are $ϕ_{\pm} = \pm 2 - t = \pm 2 (T_{c} - T) / T_{c}$ , so $ρ_{L} - ρ_{G} \propto ϕ_{+} - ϕ_{-} = 4 (T_{c} - T) / T_{c} \sim (T_{c} - T)^{1/2}$ . Therefore $β = 1/2$ . $□$

Proposition (Yang-Lee zero theorem, statement-only; Lee-Yang 1952 Phys. Rev. 87, 410-419). For a lattice gas with finite hard-core repulsion and arbitrary attractive interactions, all zeros of the grand-canonical partition function $Ξ (z, V)$ as a polynomial in fugacity $z$ lie on the unit circle $∣ z ∣ = 1$ in the complex $z$ -plane, for every finite volume $V$ . The original proof uses a transfer-matrix argument: the Hermitian structure of the lattice-gas transfer matrix forces the zeros of its determinant onto the unit circle. The thermodynamic limit then either has zeros bounded away from the positive real axis (no phase transition) or accumulating at $z = 1$ (phase transition at zero chemical potential, the symmetric case). The theorem is the canonical link between the analytic structure of the partition function and the existence of phase transitions; van der Waals theory satisfies the lattice-gas hypotheses in the discretised limit. Full proof in Lee-Yang 1952 ^{[source pending]}.

Connections Master

Partition function 08.01.01. The configurational integral $Q_{N} (V, T) = \int_{V^{N}} e^{- β U} d^{3 N} r$ is the central object whose Mayer expansion produces the virial series. Without the partition function as foundation, the cluster expansion has no setting; with it, the cluster expansion is the systematic perturbative expansion of $lo g Z$ around the ideal gas in powers of the Mayer function $f$ .
Boltzmann distribution 08.01.03. The canonical Boltzmann weight $e^{- β H}$ is the input to the cluster expansion: the Mayer function $f = e^{- β u} - 1$ is the Boltzmann weight of the pair interaction $u$ minus the non-interacting weight. The virial expansion is the systematic perturbation in the deviation from the ideal-gas Boltzmann distribution caused by pair interactions.
Free energy 08.01.04. The thermodynamic free energy $F (V, T) = - k_{B} T lo g Z$ is the Legendre transform that connects the cluster expansion to the equation of state $P = - \partial F / \partial V$ . The Maxwell equal-area construction is the convexification of $F (V)$ in $V$ at fixed $T$ , and the resulting physical free energy is the lower envelope of the van der Waals free energy.
Mean-field theory 08.02.01. The van der Waals equation is the mean-field equation of state for the liquid-gas transition: the attractive term $- N^{2} a / V^{2}$ replaces the actual configuration-dependent attractive interaction by its density-averaged value. The Curie-Weiss equation of mean-field magnetism is the structural analogue: the Weiss internal field replaces the actual neighbour-spin interactions by their lattice-averaged value. Both fail in the same way near criticality, by the same renormalisation-group mechanism.
Critical exponents 08.05.01. The van der Waals predictions $α = 0$ , $β = 1/2$ , $γ = 1$ , $δ = 3$ are the mean-field universality class, and the experimental three-dimensional liquid-gas exponents $α \approx 0.110$ , $β \approx 0.326$ , $γ \approx 1.237$ , $δ \approx 4.8$ are the Ising universality-class values. The discrepancy is the diagnostic for fluctuation effects below the upper critical dimension $d_{c} = 4$ .
Wilson-Fisher fixed point 08.04.02. The renormalisation-group view identifies the van der Waals equation as the saddle-point evaluation of the Landau-Ginzburg theory of the liquid-gas transition. Above $d_{c} = 4$ , the Gaussian fixed point is stable and the van der Waals exponents are exact; below $d_{c} = 4$ , the renormalisation-group flow drives the system to the Wilson-Fisher fixed point and the exponents take their interacting values.

Historical & philosophical context Master

Johannes Diderik van der Waals presented his equation of state in his 1873 PhD thesis Over de Continuiteit van den Gas- en Vloeistoftoestand at Leiden, under Pieter Rijke ^{[source pending]}. The thesis introduced both the excluded-volume parameter $b$ and the attractive-mean-field parameter $a$ , and derived the law of corresponding states empirically from the form of the equation. The work earned van der Waals the 1910 Nobel Prize in Physics. Within two years James Clerk Maxwell had published the equal-area construction in Nature (volume 11, 1875, pp. 357-359) ^{[source pending]}, resolving the unstable loop of the van der Waals isotherm into a gas-liquid coexistence pressure determined by chemical-potential equality. The two papers together established the modern picture of the gas-liquid phase transition.

The cluster expansion that systematises the virial series was developed by Joseph Mayer and Maria Goeppert-Mayer in the 1930s. Their textbook Statistical Mechanics (Wiley, 1940) ^{[source pending]} gave the full graph-theoretic organisation of the configurational integral and the identification of virial coefficients with irreducible cluster sums. The Yang-Lee theorem of 1952 (Phys. Rev. 87, 410-419) ^{[source pending]} gave the first rigorous proof connecting the analytic structure of the partition function to the existence of phase transitions, working in the lattice-gas formulation that van der Waals theory continuously approximates. The Wilson renormalisation-group analysis of 1971 (Phys. Rev. B 4, 3174-3205) ^{[source pending]} placed van der Waals theory within the universal framework of Landau-Ginzburg-Wilson mean-field theory and explained its quantitative failure near the critical point as a consequence of relevant fluctuations below the upper critical dimension $d_{c} = 4$ . The Carnahan-Starling equation (J. Chem. Phys. 51, 635-636, 1969) ^{[source pending]} is the modern descendant of the van der Waals excluded-volume term, capturing the hard-sphere equation of state with quantitative accuracy up to the freezing transition.

Bibliography Master

@phdthesis{vanderWaals1873,
  author = {van der Waals, Johannes Diderik},
  title  = {Over de Continu{\"i}teit van den Gas- en Vloeistoftoestand},
  school = {Universiteit Leiden},
  year   = {1873}
}

@article{Maxwell1875Nature,
  author  = {Maxwell, James Clerk},
  title   = {On the Dynamical Evidence of the Molecular Constitution of Bodies},
  journal = {Nature},
  volume  = {11},
  year    = {1875},
  pages   = {357--359}
}

@book{MayerMayer1940,
  author    = {Mayer, Joseph E. and Mayer, Maria Goeppert},
  title     = {Statistical Mechanics},
  publisher = {John Wiley \& Sons},
  year      = {1940}
}

@article{LeeYang1952,
  author  = {Lee, Tsung-Dao and Yang, Chen-Ning},
  title   = {Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model},
  journal = {Phys. Rev.},
  volume  = {87},
  year    = {1952},
  pages   = {410--419}
}

@article{Wilson1971PRB,
  author  = {Wilson, Kenneth G.},
  title   = {Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture},
  journal = {Phys. Rev. B},
  volume  = {4},
  year    = {1971},
  pages   = {3174--3183}
}

@article{Wilson1971PRBII,
  author  = {Wilson, Kenneth G.},
  title   = {Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior},
  journal = {Phys. Rev. B},
  volume  = {4},
  year    = {1971},
  pages   = {3184--3205}
}

@article{CarnahanStarling1969,
  author  = {Carnahan, Norman F. and Starling, Kenneth E.},
  title   = {Equation of State for Nonattracting Rigid Spheres},
  journal = {J. Chem. Phys.},
  volume  = {51},
  year    = {1969},
  pages   = {635--636}
}

@book{LandauLifshitzVol5,
  author    = {Landau, Lev D. and Lifshitz, Evgeny M.},
  title     = {Statistical Physics, Part 1},
  series    = {Course of Theoretical Physics},
  volume    = {5},
  edition   = {3},
  publisher = {Pergamon Press},
  year      = {1980}
}

@book{KardarParticles2007,
  author    = {Kardar, Mehran},
  title     = {Statistical Physics of Particles},
  publisher = {Cambridge University Press},
  year      = {2007}
}

@book{PathriaBeale2011,
  author    = {Pathria, R. K. and Beale, Paul D.},
  title     = {Statistical Mechanics},
  edition   = {3},
  publisher = {Elsevier},
  year      = {2011}
}

@article{Mayer1937,
  author  = {Mayer, Joseph E.},
  title   = {The Statistical Mechanics of Condensing Systems. I},
  journal = {J. Chem. Phys.},
  volume  = {5},
  year    = {1937},
  pages   = {67--73}
}

@book{Hill1956,
  author    = {Hill, Terrell L.},
  title     = {Statistical Mechanics: Principles and Selected Applications},
  publisher = {McGraw-Hill},
  year      = {1956}
}

@book{Reichl2016,
  author    = {Reichl, Linda E.},
  title     = {A Modern Course in Statistical Physics},
  edition   = {4},
  publisher = {Wiley-VCH},
  year      = {2016}
}

Prerequisites

08.01.01
08.01.03
08.01.04

Tier anchors

beginner: Schroeder *An Introduction to Thermal Physics* (Addison-Wesley, 2000) Ch. 5 informal opening on real-gas corrections to the ideal-gas law
intermediate: Kardar *Statistical Physics of Particles* (CUP, 2007) Ch. 5; Pathria-Beale *Statistical Mechanics*, 3rd ed. (Elsevier, 2011) Ch. 10; Landau-Lifshitz *Statistical Physics, Part 1*, 3rd ed. (Pergamon, 1980) §74-§76
master: Landau-Lifshitz *Statistical Physics, Part 1* §74-§84 (canonical anchor); Mayer-Mayer *Statistical Mechanics* (Wiley, 1940); Hill *Statistical Mechanics* (McGraw-Hill, 1956); Reichl *A Modern Course in Statistical Physics*, 4th ed. (Wiley-VCH, 2016) Ch. 4; Yang-Lee 1952 *Phys. Rev.* 87 phase-transition zeros; Wilson 1971 *Phys. Rev. B* 4 renormalisation-group view

References

raw/pdfs/kardar-16-17-cluster-expansion-van-der-waals.pdf · Kardar lectures 16-17 — cluster expansion, second virial coefficient, van der Waals derivation
quantum-well · experimental critical exponents for the liquid-gas transition; mean-field versus Ising universality class
Landau & Lifshitz — Statistical Physics, Part 1 (Course of Theoretical Physics Vol. 5), 3rd ed., Pergamon 1980 · §74 deviations of gases from the ideal state; §75 expansion in powers of density; §76 van der Waals formula; §77-§78 the law of corresponding states and critical point; §83-§84 Mayer cluster expansion
van der Waals — Over de Continuiteit van den Gas- en Vloeistoftoestand (PhD thesis, Leiden 1873) · Originator: equation of state with excluded-volume parameter $b$ and attractive-mean-field parameter $a$; corresponding-states law established for non-polar fluids
Mayer & Mayer — Statistical Mechanics (Wiley 1940) · Ch. 13 cluster integrals, the Mayer $f$-function $f(r) = e^{-u/kT} - 1$, the linked-cluster theorem for $\log Z$, irreducible cluster sums identifying $B_n$
Maxwell — On the Dynamical Evidence of the Molecular Constitution of Bodies (Nature 11, 357-359, 1875) · Equal-area construction for the unstable loop of the van der Waals isotherm; gas-liquid coexistence pressure as Maxwell tangent line on the free-energy curve
Lee & Yang — Statistical Theory of Equations of State and Phase Transitions II (Phys. Rev. 87, 410-419, 1952) · Yang-Lee theorem: zeros of the grand-canonical partition function $\Xi(z)$ as a polynomial in fugacity; phase transition iff zeros pinch the positive real axis in the thermodynamic limit
Wilson — Renormalization Group and Critical Phenomena I, II (Phys. Rev. B 4, 3174-3205, 1971) · Renormalisation-group derivation of critical exponents; mean-field theory (including van der Waals) as the Landau-Wilson theory above upper critical dimension $d_c = 4$ for the Ising universality class
Carnahan & Starling — Equation of State for Nonattracting Rigid Spheres (J. Chem. Phys. 51, 635-636, 1969) · Modern hard-sphere equation of state $Z = (1 + \eta + \eta^2 - \eta^3)/(1 - \eta)^3$ in packing fraction $\eta$; superior empirical accuracy to vdW for the repulsive part

Estimated time

beginner: 18m
intermediate: 45m
master: 80m