Entropy of an ideal gas

Consider $𝜈$ moles of an Ideal gas with temperature $𝑇$ and molar volume $𝑣 = 𝑉 / 𝜈$ . Its entropy is then $𝑆 = 𝜈 𝑠$ where

𝑠 = (𝑐 𝑉 + l n 𝑇 + 𝑅 l n 𝑣 + ˜ 𝐶)

is its molar entropy.

Thermodynamic derivation

đ 𝑄 = 𝜈 𝑐 𝑉 𝑑 𝑇 + 𝑝 𝑑 𝑉

for a quasistatic process with a fixed number of moles. The heat added per mole is then

đ 𝑞 = 𝑐 𝑉 𝑑 𝑇 + 𝑝 𝑑 𝑣

Applying the Ideal gas law $𝑝 𝑣 = 𝑅 𝑇$ and dividing by temperature gives

𝑑 𝑠 = đ 𝑞 𝑇 = 𝑐 𝑉 𝑑 𝑇 𝑇 + 𝑅 𝑑 𝑣 𝑇 = 𝑑 (𝑐 𝑉 l n 𝑇 + 𝑅 l n 𝑣 + ˜ 𝐶)

where $˜ 𝐶$ is an undetermined constant. Note the unitful logarithms are dealt with by the constant.

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