7.3.3.P1
In this problem, you will work through making sense of the thermodynamic quantities of entropy (S), internal energy (U_{int}), enthalpy (H), and Gibbs free energy (G) in a simple toy model in which you can actually calculate everything explicitly: the free expansion of a thermally isolated ideal gas.
We consider two identical glass bulbs connected by a valve. We start in situation A with 1 mole of an ideal gas in the right bulb with the valve closed. The gas has a pressure, volume, and temperature we symbolize by
p_{A}, V_{A}, T_{A}.
When we open the valve the gas spreads out to occupy both bulbs. Once it has settled down we have situation B with parameters
p_{B}, V_{B}, T_{B}.
Since there is nothing opposing the motion of the gas into the larger volume, this is called a free expansion.


1. How do the volume and temperature compare in systems A and B? Be quantitative and explain your reasoning.
2. We know the the entropy of the gas is given by S = k_{B}lnW, where W is the number of microstates associated with the particular macrostate of the gas we are in.
2.1 Show that W_{B}, the number of microstates after the expansion, is 2^{N} times larger than W_{A} where N is the number of particles. To do this, consider how W_{B }would compare to W_{A} if there was only one particle in the chamber. Then compare for two particles, and three, and so on, until you’ve built up the result for N particles.
2.2 What is the entropy change ΔS_{gas} for the free expansion?
2.3 Provide a qualitative argument that explains the sign of ΔS_{gas} by appealing to:
 Energy spreading
 Microstates
3. Now that we know the amount by which the entropy changes upon expansion, let’s think about how to quantify what happens to the various ways we talk about the system’s energy during the same process:
3.1 What is the change in internal energy ΔU_{gas} from before to after the expansion? How do you know?
3.2 How does the change in enthalpy ΔH_{gas} compare to the change in internal energy ΔU_{gas} during the expansion? Explain what it represents and how it's different from internal energy.
3.3 What's the change in the Gibbs free energy of the gas, ΔG_{gas}, as a result of the free expansion? Explain what it represents and how it's different from internal energy and enthalpy.
Ben Geller, Ben Dreyfus, and Joe Redish 2/14/13
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