Chapter 5, Entropy & the Boltzmann Law Video Solutions, Molecular Driving Forces

Problem 1

One-dimensional lattice. You have a one-dimensional lattice that contains $N_{A}$ particles of type $A$ and $N_{A}$ particles of type $B$. They completely fill the lattice, so the number of sites is $N_{A}+N_{B}$.
(a) Express the entropy $S\left(N_{A}, N_{B}\right)$ as a function of $N_{A}$ and $N_{B-}$
(b) Give the relationship between the chemical potential $\mu_{A}$ and the quantity $\left(\partial S / \partial N_{A}\right)_{N_{B^{*}}}$
(c) Express $\mu_{A}\left(N_{A}, N_{B}\right)$ as a function of $N_{A}$ and $N_{D}$.

Sana Riaz

Numerade Educator

01:53

Problem 2

The entropy of an ideal gas. Show that the entropy of an ideal gas is
$$
S(V, N)=N k \ln V .
$$

Aspen Fenzl

Numerade Educator

02:10

Problem 3

Entropy changes don't depend on a process pathway. For an ideal gas, the entropy is $S=N k \ln V$ (see above).
(a) Express $\Delta S_{V}=S_{2}\left(V_{2}\right)-S_{1}\left(V_{1}\right)$, the entropy change upon changing the volume from $V_{1}$ to $V_{2}$, at fixed particle number $N$.
(b) Express $\Delta S_{N}=S_{2}\left(N_{2}\right)-S_{1}\left(N_{1}\right)$, the entropy change upon changing the particle number from $N_{1}$ to $N_{2}$, at fixed volume $V$.
(c) Write an expression for the entropy change, $\Delta S$, for a two-step process $\left(V_{1}, N_{1}\right) \rightarrow\left(V_{2}, N_{1}\right) \rightarrow\left(V_{2}, N_{2}\right)$ in which the volume changes first at fixed particle number, then the particle number changes at fixed volume.
(d) Show that the entropy change $\Delta S$ above is exactly the same as for the two-step process in reverse order: changing the particle number first, then the volume.

Ajay Singhal

Numerade Educator

01:50

Problem 4

Compute $\Delta S(V)$ for an ideal gas. What is the entropy change if you double the volume from $V$ to $2 V$ in a quasistatic isothermal process at temperature $T$ ?

Averell Hause

Carnegie Mellon University