Problem set — phase coexistence

Problem set: phase coexistence¶

Exam-style problems. Work them with a pen before touching a computer; problems 4 and 5 are meant to be finished in the laboratory, and problem 6 is a challenge. Solutions and marking rubrics live with the instructor material and are deliberately not on this site.

Take $\kB = 1.381 \times 10^{-23}\ \mathrm{J\,K^{-1}}$, $R = 8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}$ and $N_A = 6.022 \times 10^{23}\ \mathrm{mol^{-1}}$ throughout. Carbon dioxide has $T_c = 304.13\ \mathrm{K}$ and $P_c = 7.3773\ \mathrm{MPa}$; water has $T_c = 647.10\ \mathrm{K}$ and $P_c = 22.064\ \mathrm{MPa}$.

Problem 1 — the spinodal and where it ends¶

The van der Waals isotherm is $P = \kB T/(v - b) - a/v^2$.

(a) Show that $(\partial P/\partial v)_T = 0$ exactly where $\kB T = 2a(v - b)^2/v^3$, and explain, using module 09’s stability criterion, why the states between the two solutions of this equation cannot exist.

(b) Show that the right-hand side has its maximum at $v = 3b$, that the maximum equals $\kB T_c$, and hence that the equation has two solutions for $T < T_c$, one for $T = T_c$ and none above. What are the two solutions called, and what do they become at $T_c$?

(c) Show that the loop’s minimum sits at negative pressure for $T < \tfrac{27}{32}\,T_c$. (Hint: $P = 0$ and $\partial P/\partial v = 0$ together.) What does a negative pressure mean for a liquid, and why is it not absurd?

(d) Using $a$ and $b$ from carbon dioxide’s critical point, find the two spinodal volumes and the corresponding pressures at $280\ \mathrm{K}$. Express the volumes in $\mathrm{L/mol}$ and compare with the critical volume $v_c = 3b$.

Problem 2 — equal areas is equal mu¶

The van der Waals gas has energy per particle $u = \tfrac32\kB T - a/v$ and entropy per particle $s = \kB[\ln((v - b)\,n_Q) + \tfrac52]$, with $n_Q(T)$ the quantum concentration.

(a) From $\mu = f + Pv$ with $f = u - Ts$ the Helmholtz free energy per particle, show that

and give $g(T)$ explicitly.

(b) Verify directly that $(\partial\mu/\partial v)_T = v\,(\partial P/\partial v)_T$, which is Gibbs–Duhem at fixed $T$.

(c) Integrate $\mathrm{d}\mu = v\,\mathrm{d}P$ along the isotherm from $v_\ell$ to $v_g$, both at pressure $P_{\mathrm{sat}}$, and show by parts that $\mu_g = \mu_\ell$ is equivalent to $\int_{v_\ell}^{v_g} P\,\mathrm{d}v = P_{\mathrm{sat}}(v_g - v_\ell)$. Explain in one sentence why the line integral is legitimate even though it passes through states that do not exist.

(d) Define $L_s = T(s_g - s_\ell)$ and $L_u = (u_g - u_\ell) + P_{\mathrm{sat}}(v_g - v_\ell)$, both evaluated at any pressure $P$ for which the isotherm has three crossings, with $v_\ell$ and $v_g$ the outer two. Show that $L_u - L_s = \mu_g - \mu_\ell$, so that the two latent heats agree exactly at $P_{\mathrm{sat}}$ and nowhere else.

(e) Sketch $\mu$ against $P$ along a sub-critical isotherm. Mark the two spinodal points, the crossing, and the two metastable branches, and explain why the middle crossing of the horizontal line is never a state of the fluid.

Problem 3 — Clausius–Clapeyron, and the end of the curve¶

(a) Starting from $\mu_\ell(T, P_{\mathrm{sat}}(T)) = \mu_g(T, P_{\mathrm{sat}}(T))$ and Gibbs–Duhem, derive $\mathrm{d}P_{\mathrm{sat}}/\mathrm{d}T = L/(T\,\Delta v)$ with $\Delta v = v_g - v_\ell$.

(b) Near $T_c$ the van der Waals coexisting volumes are $v_{g,\ell} \approx v_c\,(1 \pm 2\sqrt{t})$ with $t = 1 - T/T_c$. Show that $L \approx 6\,\kB T_c\sqrt{t}$ and $\Delta v \approx 4v_c\sqrt{t}$, so that both vanish at the critical point while the slope of the coexistence curve stays finite.

(c) Show that the limiting slope is $\mathrm{d}P_{\mathrm{sat}}/\mathrm{d}T \to \kB/(2b)$, i.e. $\mathrm{d}P_r/\mathrm{d}T_r \to 4$ in reduced variables. Check this against the values $P_r = 0.9960$ at $T_r = 0.999$ and $P_r = 0.9605$ at $T_r = 0.99$ from the laboratory.

(d) Water boils at $373.15\ \mathrm{K}$ at $1\ \mathrm{atm}$ with $L = 40.7\ \mathrm{kJ/mol}$. Estimate its boiling point at $0.5\ \mathrm{atm}$, stating the approximations you use.

(e) Ice melts at $273.15\ \mathrm{K}$ at $1\ \mathrm{atm}$ with $L_{\mathrm{fus}} = 6.01\ \mathrm{kJ/mol}$, and its molar volume exceeds the liquid’s by $1.63\ \mathrm{cm^3/mol}$. Find the slope of the melting curve in $\mathrm{K/bar}$, and explain its sign. Does a $70\ \mathrm{kg}$ skater on blades of total contact area $2\ \mathrm{cm^2}$ melt ice at $-5\,{}^\circ\mathrm{C}$?

Problem 4 — one curve for every substance, and how far it is from the truth¶

This problem is meant to be finished in the laboratory.

(a) Explain why the van der Waals coexistence curve, expressed in reduced variables $P_{\mathrm{sat}}/P_c$ against $T/T_c$, is the same for every substance.

(b) In the laboratory, compute phases.coexistence_curve for carbon dioxide and for nitrogen ($T_c = 126.19\ \mathrm{K}$, $P_c = 3.3958\ \mathrm{MPa}$) on the same grid of $T/T_c$ from 0.6 to 1, and show that the reduced curves coincide. Report the largest relative difference.

(c) Overlay the measured saturation curve of carbon dioxide, shipped as data/14-co2-saturation.csv, in the same reduced variables. Report the ratio of the model’s $P_{\mathrm{sat}}$ to the measured one at $T/T_c = 0.75$, 0.85 and 0.95, and describe how the discrepancy depends on temperature.

(d) A rigid $1\ \mathrm{L}$ cylinder holds $5\ \mathrm{mol}$ of carbon dioxide at $280\ \mathrm{K}$. Using the model’s coexisting volumes at that temperature, find the fraction of the molecules that is vapour. Repeat using the measured plateau of module 02’s isotherm, $v_\ell = 0.0532$ and $v_g = 0.3834\ \mathrm{L/mol}$. Which number would you trust, and why?

(e) The model’s latent heat at $280\ \mathrm{K}$ is $4.12\ \mathrm{kJ/mol}$. Using the measured slope of the saturation curve, which you can take from the data file by a central difference, and the measured $\Delta v$ of (d), compute the real latent heat by Clausius–Clapeyron. What does the comparison say about the model’s attraction term?

Problem 5 — the latent heat of water, with its error bar audited¶

This problem is meant to be finished in the laboratory.

(a) Show that if the vapour is ideal, $v_\ell \ll v_g$ and $L$ is constant, $\ln P_{\mathrm{sat}} = C - L/(\kB T)$. Which of the three assumptions would you expect to fail first for water between 0 and $150\,{}^\circ\mathrm{C}$, and in which direction does each one bias a fitted $L$?

(b) In the laboratory, fit phases.clausius_clapeyron_fit to the 31 points of data/14-water-saturation.csv and report $L$ with its standard error, in $\mathrm{kJ/mol}$ and in $\mathrm{MJ/kg}$. Compare with the tabulated $40.66\ \mathrm{kJ/mol}$ at $100\,{}^\circ\mathrm{C}$.

(c) Plot the residuals $\ln P_{\mathrm{data}} - \ln P_{\mathrm{fit}}$ against $T$. Are they random? What shape do they have, and what does the shape say about the assumption of constant $L$?

(d) Repeat the fit on 275–$325\ \mathrm{K}$ and on 345–$405\ \mathrm{K}$. Report both values of $L$ with their errors, and the boiling point at $1\ \mathrm{atm}$ each fit predicts. Explain why the two $L$ values differ by far more than their quoted errors, and what the quoted error does and does not measure.

(e) From the two table rows closest to $373\ \mathrm{K}$ alone, estimate $L$ by the two-point formula of worked example 3, and explain why this “measurement” from two numbers is closer to the tabulated value than the fit to all thirty-one.

Problem 6 — how hot can water get without boiling? (challenge)¶

Microwaved water in a smooth cup is regularly observed to reach 105 to $110\,{}^\circ\mathrm{C}$ at atmospheric pressure without boiling, and then to erupt when a spoon or sugar is dropped in. In careful laboratory experiments, clean water at $1\ \mathrm{atm}$ has been superheated to about $300\,{}^\circ\mathrm{C}$.

(a) Where, in the $P$–$v$ plane, does a superheated liquid at $1\ \mathrm{atm}$ sit relative to the binodal and the spinodal? Why is it not forbidden by the stability criterion?

(b) Using $a$ and $b$ for water from its critical point, the van der Waals model’s liquid spinodal reaches $P = 1\ \mathrm{atm}$ at $546\ \mathrm{K}$. Reproduce this number in the laboratory (the spinodal pressure as a function of $T$ is available from phases.spinodal_volumes and the equation of state), and compare it with the observed limit of $\sim 575\ \mathrm{K}$. Given what problem 4 showed about the model, is the agreement meaningful?

(c) Why does microwaved water stop at $110\,{}^\circ\mathrm{C}$, a hundred and sixty degrees short of either number? Estimate the radius of the smallest bubble that can grow at $110\,{}^\circ\mathrm{C}$ and $1\ \mathrm{atm}$, given water’s surface tension $\sigma = 0.057\ \mathrm{N/m}$ and $P_{\mathrm{sat}}(383\ \mathrm{K}) = 143\ \mathrm{kPa}$: a bubble of radius $r$ is in mechanical balance when $P_{\mathrm{inside}} - P_{\mathrm{outside}} = 2\sigma/r$.

(d) The work to form that critical bubble is $W^* = 16\pi\sigma^3/(3\,\Delta P^2)$. Evaluate it in units of $\kB T$ and comment on the probability of its forming by thermal fluctuation alone. What, then, makes the water in the cup erupt when a spoon is dropped in — and what does this say about which of the two numbers in (b) a model of the states can ever be expected to predict?