Problem set — chemical potential

Problem set: chemical potential¶

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 $\kB = 1.381 \times 10^{-23}\ \mathrm{J\,K^{-1}}$, $h = 6.626 \times 10^{-34}\ \mathrm{J\,s}$, $N_A = 6.022 \times 10^{23}\ \mathrm{mol^{-1}}$, $1\ \mathrm{u} = 1.661 \times 10^{-27}\ \mathrm{kg}$ and $g = 9.81\ \mathrm{m\,s^{-2}}$ throughout.

Problem 1 — two exchanges at once¶

Two systems with fundamental relations $S_1(U_1, V_1, N_1)$ and $S_2(U_2, V_2, N_2)$ are separated by a fixed wall that conducts heat and lets particles through. The pair is isolated, with $U_1 + U_2 = U$ and $N_1 + N_2 = N$.

(a) Write $S_{\mathrm{tot}}$ as a function of $U_1$ and $N_1$, and show that its maximum requires $T_1 = T_2$ and $\mu_1 = \mu_2$.

(b) Near equilibrium, with the temperatures already equal, show that $\mathrm{d}S_{\mathrm{tot}} = (\mu_2 - \mu_1)\,\mathrm{d}N_1/T$. Which way do particles flow?

(c) Now suppose the temperatures are not equal and the wall passes particles but no heat. Show that the quantity that equalizes is $\mu/T$, not $\mu$. Why can such a wall not really exist for particles that carry energy?

(d) Explain in two sentences why this argument is “module 09’s argument with $N$ in place of $U$”, and name the corresponding result for a movable wall.

Problem 2 — G = mu N¶

(a) From $\mathrm{d}U = T\,\mathrm{d}S - P\,\mathrm{d}V + \mu\,\mathrm{d}N$ and $G = U - TS + PV$, derive $\mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}P + \mu\,\mathrm{d}N$ and hence $\mu = (\partial G/\partial N)_{T,P}$.

(b) Show that this $\mu$ is the same as $-T(\partial S/\partial N)_{U,V}$.

(c) Using the Euler relation $U = TS - PV + \mu N$, show that $G = \mu N$. Why does it follow that $\mu$ of a pure substance depends on $T$ and $P$ only?

(d) Show that $F = \mu N - PV$, and explain in one sentence why $F$ is not $\mu N$.

(e) For the ideal gas, write $\mu(T, P)$ explicitly, and show that $(\partial\mu/\partial P)_T = v$, the volume per particle.

Problem 3 — a column with a sticky floor¶

A tall isothermal column of nitrogen ($m = 28\ \mathrm{u}$) at $T = 300\ \mathrm{K}$ stands on a floor of area $A = 1\ \mathrm{cm^2}$ that carries $N_s = 10^{15}$ adsorption sites, each binding one molecule with energy $\varepsilon = -0.25\ \mathrm{eV}$.

(a) Using $\mu = \kB T\ln(n/n_Q) + m g z$, show that the density in the column is $n(z) = n(0)\,e^{-z/H}$, and find $H$.

(b) Evaluate $n_Q$ for nitrogen at $300\ \mathrm{K}$, and the chemical potential at the floor when $n(0) = 2.4 \times 10^{25}\ \mathrm{m^{-3}}$.

(c) The floor’s sites are in equilibrium with the gas at $z = 0$. Find the fraction of sites occupied.

(d) Show that the total number of molecules in the gas above the floor is $n(0) A H$, and compare it with the number adsorbed.

(e) At what floor density $n(0)$ is half the surface covered? Express the answer as a pressure.

Problem 4 — how steady is equal mu?¶

This problem is meant to be finished in the laboratory.

Two lattice-gas boxes of $M$ sites each exchange $N$ particles, with box B’s floor $\Delta u$ above box A’s.

(a) For a dilute system, find the mean number $\bar N_A$ in box A in terms of $N$ and $\Delta u$.

(b) Treat each particle as independently in A with probability $p = \bar N_A/N$. Show that the standard deviation of $N_A$ is $\sqrt{N p (1 - p)}$, and that the chemical-potential difference fluctuates by

(c) Evaluate (b) for $\Delta u = 2\,\kB T$ and $N = 100$, 10⁴ and 10²².

(d) In the laboratory, run chemical.particle_exchange_sim for $N = 250$, 1000, 4000 and $16\,000$ with $M = 100 N$ and $\Delta u = 2\,\kB T$, each for at least 20 relaxation times (chemical.relaxation_steps). Measure the standard deviation of $\mu_A - \mu_B$ in the second half of each run, fit a power law in $N$, and compare the exponent and the prefactor with (b).

(e) Why does the simulated standard deviation come out smaller than (b) if the records are taken too close together? What does this say about the error bar on a time average?

Problem 5 — a Langmuir isotherm from a simulation¶

This problem is meant to be finished in the laboratory.

(a) A site that is empty (energy 0) or holds one molecule (energy $\varepsilon$) is in contact with a particle reservoir at $T$ and $\mu$. Using the grand-canonical weight, derive $\langle n \rangle = 1/(e^{(\varepsilon - \mu)/\kB T} + 1)$. List the hypotheses you used.

(b) With the gas as the reservoir, $e^{\mu/\kB T} = P/(n_Q \kB T)$. Derive the Langmuir isotherm $\theta = P/(P + P_0)$ and give $P_0$.

(c) In the laboratory, simulate a surface of 200 sites at $\varepsilon = -2\,\kB T$ exchanging particles with a reservoir of $20\,000$ sites at energy 0, for at least eight total particle numbers between 20 and 4000. For each run, record the surface coverage, the reservoir’s mean occupied fraction $c$, and its chemical potential (chemical.lattice_gas_mu).

(d) With $z = e^{\mu/\kB T}$, fit $\theta = z/(z + z_0)$ to your data, with $z_0$ the fit parameter. Compare $z_0$ with the prediction $e^{\varepsilon/\kB T}$ and quote the fit’s uncertainty. Is the quoted uncertainty trustworthy?

(e) Repeat the fit with the occupied fraction $c$ in place of $z$. Why does $c_0$ come out smaller than $z_0$, and at which of your particle numbers does the difference arise? What does this say about using a gas’s density in the Langmuir isotherm?

Problem 6 — a membrane ladder (challenge)¶

Three compartments, 1, 2 and 3, sit side by side, separated by two membranes that pass water but not sugar. Compartment 1 holds pure water; 2 holds a $0.1\ \mathrm{mol/L}$ sucrose solution; 3 holds $0.3\ \mathrm{mol/L}$. Each compartment is open at the top to a vertical tube of the same cross-section, and the whole system is at $298\ \mathrm{K}$.

(a) Using the ideal-solution model, show that the solvent’s chemical potential in a solution of solute fraction $x$ at pressure $P$ is $\mu_w^{\circ}(T, P) + \kB T\ln(1 - x)$, and derive $\Pi = -(\kB T/v)\ln(1 - x) \approx n_s \kB T$.

(b) At equilibrium, which way does water cross each membrane? Rank the three final water levels.

(c) Neglecting the dilution caused by the water that crosses, find the equilibrium height differences $h_2 - h_1$ and $h_3 - h_2$.

(d) Now include dilution: if each tube has cross-section $0.1\ \mathrm{cm^2}$, each compartment initially holds $1\ \mathrm{L}$, and all three levels start equal, set up the equations for the final levels. Solve them numerically, compare with (c), and find how much water leaves compartment 1.

(e) Explain why the answer to (c) does not depend on the size of the solute molecules, and why a $0.1\ \mathrm{mol/L}$ solution of sodium chloride would give almost twice the pressure.