Heat Capacities of an Ideal Gas I

A sealed cylinder's gas cannot expand, so all applied heat increases internal energy. A cylinder with a movable piston allows gas expansion, meaning some heat performs work against external pressure. Since internal energy change depends only on temperature, more total heat must be added to the expandable cylinder to achieve the same temperature rise as the sealed one.

CV is the molar heat capacity at constant volume, measuring heat needed to raise one mole's temperature by 1 K without volume change. Cp is the molar heat capacity at constant pressure, measuring heat needed under constant external pressure. Cp exceeds CV because additional heat performs expansion work. For example, air's Cp is approximately 40% greater than its CV.

The first law states that heat added equals internal energy change plus work done. At constant volume, no work occurs, so all heat increases internal energy. At constant pressure, heat is divided between increasing internal energy and performing expansion work. Therefore, reaching the same temperature requires more heat at constant pressure, defining two distinct molar heat capacities.

Molar heat capacity quantifies the heat required to change one mole of a substance's temperature by 1 K, expressed in J/mol·K. It can be measured at constant volume or constant pressure. Measuring at constant volume is easiest because no work is performed. If neither pressure nor volume remains constant, the system acquires infinite possible heat capacities.

When gas expands against external pressure, it performs work, converting some applied heat into mechanical energy. According to the first law, heat input is partitioned between increasing internal energy and doing work. In a sealed container, all heat increases internal energy; in an expandable system, only the portion not used for work contributes to internal energy change.

At constant pressure, applied heat must both increase internal energy and perform expansion work against external pressure. At constant volume, all heat increases internal energy without work. Since the same temperature increase requires identical internal energy change in both cases, constant pressure conditions demand additional heat to account for work performed during expansion.

Internal energy depends only on temperature, so both cylinders experience identical internal energy changes when heated to the same final temperature. However, the sealed cylinder converts all applied heat into internal energy change, while the expandable cylinder diverts some heat to perform work. This difference in heat distribution, not internal energy change, defines the two distinct molar heat capacities.