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omg4 - Statistical Physics Example Sheet 4 David Tong March...

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Statistical Physics: Example Sheet 4 David Tong, March 2012 1i. By examining variations in E , F , H and G , derive the four diFerent Maxwell relations for the partial derivatives of S, p, T and V . ii. Obtain the partial derivative identity ∂S ∂T v v v v p = v v v v V + ∂V v v v v T v v v v p iii. Obtain the partial derivative identity ∂p v v v v V v v v v p v v v v T = 1 2. Consider a gas with a ±xed number of molecules. Two experimentally accessible quantities are C V , the heat capacity at ±xed volume and C p , the heat capacity at ±xed pressure, de±ned as C V = T v v v v V , C p = T v v v v p Using the results of the previous question, show that: i . C p C V = T v v v v p v v v v V = T v v v v 2 p v v v v T ii . ∂E v v v v T = T v v v v V p iii . v v v v T = T v v v v p p v v v v T iv . ∂C V v v v v T = T 2 p 2 v v v v V v . p v v v v T = T 2 V 2 v v v v p 1
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3. Consider a classical ideal gas with equation of state pV = Nk B T and constant heat capacity C V = B α for some α . Use the results above to show that C p = B ( α +1), and that the entropy is S = B log p V N P + B
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