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# Free energy in free expansion

last edited by 5 years, 1 month ago

7.3.3.P1

In this problem, you will work through making sense of the thermodynamic quantities of entropy (S), internal energy (Uint), 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  pA, VA, TA. When we open the valve the gas spreads out to occupy both bulbs. Once it has settled down we have situation B with parameters pB, VB, TB.   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 = kBlnW, where W is the number of microstates associated with the particular macrostate of the gas we are in.

2.1 Show that WB, the number of microstates after the expansion, is 2N times larger than WA where N is the number of particles. To do this, consider how WB would compare to WA 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 ΔSgas for the free expansion?

2.3 Provide a qualitative argument that explains the sign of ΔSgas by appealing to:

• 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 ΔUgas from before to after the expansion? How do you know?

3.2 How does the change in enthalpy ΔHgas compare to the change in internal energy ΔUgas 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, ΔGgas, 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