PHYSICS 7B QUIZ 1 SOLUTIONS SECTION 104 CORY SCHILLACI 1a) Because of the speciﬁc form for the ﬁrst law, the heat capacity is simply given by (1) C = ∂E ∂T . Since we have T as a function of M , it’s easiest to compute (2) 1 C = ∂T ∂E = 1 c 2 ∂T ∂M =-~ c 8 πGk 1 M 2 1b) Substituting back in using the temperature formula (1) and doing a small bit of algebra leads immediately to (3) C =-~ c 5 8 πGk 1 T 2 . As many of you noticed, this heat capacity is negative! 2a) The key fact here is that dE = dq = TdS . Armed with this we can just integrate
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