7.3.3.P1 In this problem, you will work through making sense of the thermodynami

Question

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 (Q) 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 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 8? Be quantitative and explain your reasoning 2. We know the the entropy of the gas is given by S -kglnW, where Wis the number of microstates associated with the particular macrostate of the gas we are in. 2.1 Show that Ws, 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 Wg 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 as for the free expansion? 2.3 Provide a qualitative argument that explains the sign of ASgas by appealing to: .Energy spreading

Explanation / Answer

1. At the beginning we have a given pressure, volume and temperature for A

at state V the volume available is increased since the bulbs have the same shape then we can say that the volume is doubled

so Vb = 2Va

the temperature will remain the same because no energy is entering or leaving the system so

Tb = Ta

2. If there is only 1 particle and you double the volume then the number of microstates will be doubled

wb = 2 wa

for 2 particles this will be

Wb = 2*2Wa = 4 Wa

The number of microstates is found by multiplying the amount of possible configurations of each particle, this number is 2n

The entropy change is The entropy difference of state A and state b

Sb - Sa

Sb = Kb ln Wb

Remember that Wb = 2n Wa so

Sb = Kb ln (2n Wa)

Sb = Kb ln Wa + N Kb ln 2

Sa = Kb Ln Wa

Entropy change

Sb - Sa = Kb ln Wa + N Kb ln 2 - Kb Ln Wa =  N Kb ln 2

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