Calculation of First-Law Quantities II

The first law of thermodynamics establishes that the change in internal energy of a system is given by ΔU = q + w, where q is the heat exchanged, and w is the work performed. For a perfect gas, both internal energy (U) and enthalpy (H) depend solely on temperature. Consequently, for any change of state, whether reversible or irreversible, the internal energy change is determined by integrating the heat capacity at constant volume, and the enthalpy change by integrating the heat capacity at constant pressure over the relevant temperature limits.

\begin{equation*}\Delta U = \int_{T_i}^{T_f}\!C_v(T)\, dT\end{equation*}

\begin{equation*}\Delta H = \int_{T_i}^{T_f}\!C_p(T)\, dT\end{equation*}

These expressions remain valid regardless of the path taken between the initial and final states.

Although the change in internal energy and the change in enthalpy depend only on temperature, the quantities heat and work depend on the process path. In a reversible process, the work is given by

\begin{equation*}w = -\!\int_{T_i}^{T_f}\!pdV\end{equation*}

Once w is known, the heat transfer follows from the first law.

In a reversible isothermal process, temperature remains constant. Since internal energy and enthalpy depend only on temperature, ΔU = 0 and ΔH = 0, so q = −w.

For a reversible adiabatic process, no heat is exchanged, so q = 0 and ΔU = w. The energy changes are then written as:

\begin{equation*}\Delta U = \int_{T_i}^{T_f}\!C_v(T)\, dT\end{equation*}

\begin{equation*}\Delta H = \int_{T_i}^{T_f}\!C_p(T)\, dT\end{equation*}