Specific heat of an ideal gas

The molar specific heat of an ideal gas at constant volume and constant pressure respectively are given by

𝑐 𝑉 = 𝛼 𝑅 𝑐 𝑝 = 𝑐 𝑉 + 𝑅

where $𝑅 = 𝑁 𝐴 𝑘 𝐵$ is the molar gas constant.

Thermodynamic derivation

By ^Quasistatic we have $đ 𝑄 = 𝑑 𝐸 + 𝑝 𝑑 𝑉$ . At constant volume $𝑑 𝑉 = 0$ and thus $đ 𝑄 = 𝑑 𝐸$ , so the Energy of an ideal gas gives $đ 𝑄 = 𝛼 𝜈 𝑅 𝑑 𝑇$ and thus

𝑐 𝑉 = đ 𝑄 𝜈 𝑑 𝑇 = 𝛼 𝑅

whence $𝐸 = 𝜈 𝑐 𝑉 𝑇$ Similarly at constant pressure $𝑑 𝑝 = 0$ so applying the Ideal gas law and noting $𝑑 𝜈 = 0$

đ 𝑄 = 𝑑 𝐸 + 𝑝 𝑑 𝑉 = 𝜈 𝑐 𝑉 𝑑 𝑇 + 𝑑 (𝑝 𝑉) = 𝜈 𝑐 𝑉 𝑑 𝑇 + 𝑑 (𝜈 𝑅 𝑇) = 𝜈 (𝑐 𝑉 + 𝑅) 𝑑 𝑇

wherefore

𝑐 𝑝 = đ 𝑄 𝜈 𝑑 𝑇 = 𝑐 𝑉 + 𝑅

as claimed.

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