Calculation of First Law Quantities I

Thermodynamic systems undergoing phase transitions or temperature changes experience energy transfer in the form of heat (q) and work (w). For a reversible phase change at constant temperature (T) and pressure (p), the process involves no chemical reaction but results in energy exchange between distinct phases.

The heat transferred during this process corresponds to the latent heat of transition, which is the amount of heat energy absorbed or released by a substance when it changes from one physical state to another at constant temperature and pressure. It is a measurable quantity characterizing the enthalpy change (ΔH). This enthalpy change is equal to the heat exchanged at constant pressure (q_p)​, and the work done is given by w = −pΔV, where ΔV is the volume change between phases. The internal energy change (ΔU) is derived from the first law of thermodynamics as ΔU = q + w, simplifying to ΔU = ΔH − pΔV.

For systems undergoing constant-pressure heating without a phase transition, energy input alters both the internal energy and the volume of the system, resulting in pressure–volume work. In such cases, the heat added is determined by the integral of the temperature-dependent heat capacity at constant pressure (C_P​(T)) over the temperature range (dT). Even if the process is irreversible, this relation still holds as long as the initial and final pressures are the same, because enthalpy is a state function.

Conversely, during constant-volume heating, the volume remains fixed, eliminating pressure–volume work (w = 0). All heat added increases the internal energy, which is obtained by integrating the heat capacity at constant temperature (ΔU = ∫C_v dT). This result applies to both reversible and irreversible processes. The enthalpy change can then be obtained from ΔH = ΔU + Δ(pV), accounting for changes in the pressure–volume term.

These principles provide a foundational understanding of energy changes in closed systems under various thermal conditions.