Physics 4311: Thermal Physics – Homework 8

Problem 1: Entropy of the ideal gas (12 points)

The equation of state of an ideal gas is $\mathit{pV}=N{k}_{B}T$ with $p$ being pressure, $V$ volume, $N$ the number of particles, $k_B$ the Boltzmann constant, and $T$ the temperature. The internal energy is given by $U=(3∕2)N{k}_{B}T$.

a)

Start from the first law, $\mathit{dU}=T\mathit{dS}-p\mathit{dV}$, and derive an expression for the entropy of the ideal gas as a function of $T$ and $V$.

b)

Determine the behavior of $S$ for $T\to 0$. What does the result mean?

Problem 2: Maxima of entropy (16 points)

As system can be in $N$ different states with probabilities $p_i$ ($i=1\dots N$). Determine which $p_i$ lead to the maximum (Gibbs) entropy under the following constraints:

a)

Fixed normalization ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_i{p}_i=1$

b)

fixed normalization ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_i{p}_i=1$ and fixed average energy $⟨E⟩={\sum <!--\; FUNCTION\; APPLICATION\; -->}_i{p}_i{E}_i$.

Hint: Use Lagrange multipliers to enforce the constraints.

Problem 3: Additivity of the entropy (12 points)

A system consists of two subsystems A and B. Subsystem A can be in states $i=1\dots n$ and subsystem B can be states $j=1\dots m$. The Gibbs entropy of this system reads

where $p(i,j)$ are the joint probabilities for the states of the subsystems.

Show that if the two subsystems A and B are statistically independent, then the entropy $S$ of the total system is the sum of the entropies of the two subsystems.