Thermodynamics | Lec #5 | Heat Capacity | Specific Heat Capacity | Dependence on Pressure and Volume

Heat capacity and specific heat capacity quantify how much energy a substance needs to change temperature, and the lecture ties those quantities to the thermodynamic constraints that matter most—constant volume versus constant pressure. Heat itself is described as energy in transit from higher-temperature molecules (with higher kinetic energy) to lower-temperature ones, which is why a hot coffee cup warms the hand and an ice cube cools it by absorbing heat.

Heat capacity is defined as the heat required to raise a substance’s temperature by 1°C (or 1 K). It is represented by a capital C and is treated as an amount of heat per unit temperature change. The lecture then distinguishes this from specific heat capacity, which normalizes the heat requirement by mass: it is the heat needed to raise 1 gram of a substance’s temperature by 1°C (or 1 K). The key practical difference is that specific heat capacity makes comparisons across different amounts of material possible, because it explicitly includes the amount of substance (per gram).

The discussion connects these ideas to the three states of matter—solids, liquids, and gases—by focusing on what changes when temperature rises. For solids and liquids, increasing temperature by 1°C does not significantly alter shape or texture; the system primarily changes internal energy. For gases, temperature increase strongly affects molecular motion and intermolecular interactions, leading to expansion. Because gases expand, measuring heat capacity for gases requires specifying the thermodynamic condition under which heating occurs.

For gases, heat capacity depends on whether volume or pressure is held constant. Under constant volume, a piston is effectively prevented from moving (for example, by keeping volume fixed), so the gas cannot do boundary work; the heat supplied goes into changing internal energy. Under constant pressure, heating causes the piston to move as the gas expands, so part of the supplied heat is used to do work against the external pressure; the relevant temperature-dependent quantity becomes enthalpy. The lecture links this to the first law of thermodynamics: heat supplied equals the change in internal energy plus work done, with work expressed as pressure times change in volume (PΔV).

Using these constraints, the lecture states that heat capacity at constant volume corresponds to how internal energy changes with temperature, while heat capacity at constant pressure corresponds to how enthalpy changes with temperature. For ideal gases, the pressure term can be handled using the ideal gas relation, leading to the standard relationship that heat capacity at constant pressure is greater than at constant volume. The reason is physical: at constant pressure, heating includes the extra energy needed for expansion work, whereas at constant volume there is no such work.

Finally, the lecture frames the takeaway as a measurement rule: specify the condition (constant V or constant P) when reporting heat capacity, especially for gases, because the same temperature change can require different amounts of heat depending on whether the system is allowed to expand.