Consider a container like that shown in Figure, with $n_1$ moles of a monatomic gas on one side and $n_2$ moles of a diatomic gas on the other. The monatomic gas has initial temperature $T_{1i}$. The diatomic gas has initial temperature $T_{2i}$. Show that the equilibrium thermal energies are
$\(\begin{aligned}\) E_{1f} &= \(\frac{3n_1}{3n_1 + 5n_2}\) (E_{1i} + E_{2i}) \\ E_{2f} &= \(\frac{5n_2}{3n_1 + 5n_2}\) (E_{1i} + E_{2i}) \(\end{aligned}\)$

Consider a container like that shown in the Figure, with $n_1$ moles of a monatomic gas on one side and $n_2$ moles of a diatomic gas on the other. The monatomic gas has initial temperature $T_{1i}$. The diatomic gas has initial temperature $T_{2i}$. Show that the equilibrium temperature is

$T_f=\frac{3n_1{T}_{1i}+5n_2{T}_{2i}}{3n_1+5n_2}$

A 2.0 mol sample of oxygen gas in a rigid, 15 L container is slowly cooled from 250℃ to 50℃ by being in thermal contact with a large bath of 50℃ water. What is the entropy change of (a) the gas and (b) the universe?

An experiment you're designing needs a gas with γ = 1.50. You recall from your physics class that no individual gas has this value, but it occurs to you that you could produce a gas with γ = 1.50 by mixing together a monatomic gas and a diatomic gas. What fraction of the molecules need to be monatomic?

A thin partition divides a container of volume V into two parts. One side contains n_A moles of gas A in a fraction f_A of the container; that is, V_A = f_AV. The other side contains nB moles of a different gas B at the same temperature in a fraction f_B of the container. The partition is removed, allowing the gases to mix. Find an expression for the change of entropy. This is called the entropy of mixing.

n₁ moles of a monatomic gas and n₂ moles of a diatomic gas are mixed together in a container. Derive an expression for the molar specific heat at constant volume of the mixture.