Entropy Change for Ideal Gas

\[\Delta S=nC_v\ln\left(\frac{T_2}{T_1}\right)+nR\ln\left(\frac{V_2}{V_1}\right)\]

Variables

dS[ΔS]entropy change (J/K)

namount of substance (mol)

Cv[C_v]molar heat capacity at constant volume (J/mol·K)

Rideal gas constant (J/mol·K)

T2final absolute temperature (K)

T1initial absolute temperature (K)

V2final volume (m³)

V1initial volume (m³)

Description

What is this formula?

This formula calculates the entropy change of an ideal gas when both temperature and volume may change. It combines the thermal contribution from temperature variation with the expansion or compression contribution from volume variation.

When to use it

Use this formula for ideal gas processes where the initial and final temperatures and volumes are known. It is useful for thermodynamic state changes, cycle analysis, and reversible reference calculations.

Example

Given:

n = 1 mol

Cv = 20.8 J/mol·K

R = 8.314 J/mol·K

T1 = 300 K

T2 = 450 K

V1 = 0.020 m³

V2 = 0.040 m³

Formula:

ΔS = nCv ln(T2/T1) + nR ln(V2/V1)

Substitution:

ΔS = 1 × 20.8 × ln(450/300) + 1 × 8.314 × ln(0.040/0.020)

Result:

ΔS = 14.20 J/K

Applications

Ideal gas thermodynamics, entropy balance, heat engine analysis, compressor and turbine calculations, thermodynamic cycles, and engineering energy systems.