**Problem : **

We can define the spin of any collection of particles to be the sum of the spins of the individual particles that comprise it. Given that protons and electrons are of spin 1/2, specify whether the hydrogen atom is a fermion or a boson.

The Hydrogen atom consists of an electron and a proton, so the total spin is 1. Therefore, the Hydrogen atom is a boson.

**Problem : **

What is the sign of the chemical potential for an ideal gas, and when does our expression for it break down?

Recall that the chemical potential for an ideal gas is
*μ* = *τ*log. Remember that an ideal gas must have
*n**n*_{Q}. Therefore, 1. The log of a number between
0 and 1 is negative, and the temperature for any ideal gas must be
positive. Therefore, the chemical potential *μ* is negative for an
ideal gas. The equation breaks down as *n*→*n*_{Q}, for we are leaving
the classical regime and *μ*→ 0.

**Problem : **

What is the energy of one mole of an ideal gas at room temperature?

This problem tests whether you remember all of the conversions between
fundamental and conventional units, and tests whether you can recall the
equation we derived for the energy of an ideal gas. Recall that
*U* = *Nτ*. *N* here will be Avogadro's Number, which is
6.02×10^{23}. Room temperature is 25^{o}*C*, which is
298*K*. Therefore *τ* = 298*k*_{B}. The final result gives us *U* = 2477
Joules.

**Problem : **

What is the entropy of one mole of an ideal gas whose concentration *n*
is one-hundreth of the quantum concentration *n*_{Q}?

Recall that *σ* = *N*log + .
Now = 100. Remembering that log refers to ln, we solve to find
that *σ* = 4.28×10^{24}. Notice that as *n* gets smaller, the
entropy gets bigger. You can imagine that a gas with more room to move
around per particle would have more randomness then one in which the
particles were forced together in a small space.

**Problem : **

Give the two heat capacities for a mole of ideal gas.

We recall that *C*_{V} = *N* and *C*_{p} = *N*. Therefore,
*C*_{V} = 9.03×10^{23} and *C*_{p} = 1.51×10^{24}.

Take a Study Break!