Thermodynamics: Gas: Problems

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 = τlog. Remember that an ideal gas must have nn_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 6.02×10²³. Room temperature is 25^oC, which is 298K. Therefore τ = 298k_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 = Nlog + . Now = 100. Remembering that log refers to ln, we solve to find that σ = 4.28×10²⁴. 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.