First Law for isovolumetric processes

When the volume of a gas is constant, no mechanical transfer of energy takes place and therefore no work is done on or by the gas (see Quasistatic expression for work). Hence

𝑑 𝑄 = 𝑑 𝑈 = 𝑛 𝐶 𝑉 𝑑 𝑇

where

$𝐶 𝑉 : J \cdot m o l - 1 \cdot K - 1$ is the molar specific heat for gas of constant volume.

For monatomic ideal gas

Since $𝑈 = 32 𝑛 𝑅 𝑇$

𝑑 𝑈 = 32 𝑛 𝑅 𝑑 𝑇 = 𝑛 𝐶 𝑉 𝑑 𝑇

hence

𝐶 𝑉 = 32 𝑅 \approx 12.5 J \cdot m o l - 1 \cdot K - 1

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