Heat Capacities of an Ideal Gas II

At constant volume, all absorbed heat increases internal energy since no work occurs. At constant pressure, absorbed heat increases both internal energy and performs work on surroundings. Since more heat must be added to achieve the same temperature increase at constant pressure, the molar heat capacity Cp is always greater than Cv for an ideal gas.

The difference between molar heat capacities at constant pressure and constant volume equals the universal gas constant R. Mathematically, Cp - Cv = R, where R ≈ 8.314 J/mol·K. This relationship derives directly from the first law of thermodynamics and applies to all ideal gases regardless of their molecular composition.

Internal energy is a state function, so its change depends only on temperature change, not the process path. In both constant-volume and constant-pressure processes, if temperature increases by the same amount, the internal energy change is identical. However, the heat absorbed differs because work is performed only in the constant-pressure process.

In a constant-volume process, all absorbed heat is used exclusively to increase the system's internal energy. No work is done because volume remains fixed, so the heat absorbed equals the change in internal energy. This relationship simplifies calculations for isochoric processes involving ideal gases.

In a constant-pressure process, work done by the gas can be expressed using the ideal gas equation in differential form. The work is proportional to the temperature change and the amount of gas. By combining this with the first law of thermodynamics, the relationship between Cp and Cv emerges through algebraic cancellation of common factors.

Molar heat capacity is the amount of heat required to increase the temperature of one mole of gas by one unit (1 K or 1°C). It quantifies how much thermal energy a substance absorbs per mole per degree of temperature change. Different processes—constant volume or constant pressure—require different amounts of heat for identical temperature increases.

Yes. For helium, measured values are Cp = 20.78 J/mol·K and Cv = 12.47 J/mol·K. Their difference is 8.31 J/mol·K, which closely matches R ≈ 8.314 J/mol·K. This experimental agreement confirms the theoretical prediction derived from the first law of thermodynamics for ideal gases.