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First law of thermodynamics, heat capacity

Problem 2.50

One mole of an ideal gas whose adiabatic exponent equals \(\gamma\) undergoes a process in which the gas pressure relates to the tem perature as \(p=a T^{\alpha},\) where \(a\) and \(\alpha\) are constants. Find: (a) the work performed by the gas if its temperature gets an increment \(\Delta T\) (b) the molar heat capacity of the gas in this process; at what value of \(\alpha\) will the heat capacity be negative?

Reveal Answer
2.50. (a) \(A=(1-\alpha) R \Delta T\) (b) \(C=R /(\gamma-1)+R(1-\alpha)\) \(C<0\) for \(\alpha>\gamma /(\gamma-1)\)
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