Three moles of an ideal gas being initially at a temperature 273 K was isothermally expanded 5.0 times its initial volume and then isochorically heated so that the pressure in the final state became e

Question

Three moles of an ideal gas being initially at a temperature 273 K was isothermally expanded 5.0 times its initial volume and then isochorically heated so that the pressure in the final state became equal to that in the initial state. The total amount of heat transferred to the gas during the process equals 80 kJ. The value of C v ,m for this gas (in cal/K-mol) is

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Isothermal step at 273 K: ΔU = 0, Q₁ = W₁ = nRT ln(V₂/V₁) = 3 × 8.314 × 273 × ln 5 ≈ 10.96 kJ. After 5× expansion P₂ = P₁/5; isochoric heating until P returns to P₁ raises T from 273 K to 5 × 273 = 1365 K (ΔT = 1092 K). Q₂ = nC v ΔT = 3 × C v × 1092 = 80 − 10.96 = 69.04 kJ ⇒ C v ≈ 21.07 J K⁻¹ mol⁻¹ ≈ 5 cal K⁻¹ mol⁻¹ (dividing by 4.184). Matches the expected monatomic value.

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