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.

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}) \(\end{aligned}\)$

n moles of a diatomic gas with C_v = 5/2 R has initial pressure p_i and volume V_i. The gas undergoes a process in which the pressure is directly proportional to the volume until the rms speed of the molecules has doubled. How much heat does this process require? Give your answer in terms of n, p_i and V_i.

The rms speed of the molecules in 1.0 g of hydrogen gas is 1800 m/s. 500 J of work are done to compress the gas while, in the same process, 1200 J of heat energy are transferred from the gas to the environment. Afterward, what is the rms speed of the molecules?

The 2010 Nobel Prize in Physics was awarded for the discovery of graphene, a two-dimensional form of carbon in which the atoms form a two-dimensional crystal-lattice sheet only one atom thick. Predict the molar specific heat of graphene. Give your answer as a multiple of R.

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.