# 19.1: Heat Capacities

- Page ID
- 595

The * constant volume* heat capacity is defined by:

\[C_{v}\left(\frac{\partial U}{\partial T}\right)_{V} \label{19.1}\]

To see the physical significance of the constant volume heat capacity, let us consider a 1 lbmol of gas within a rigid-wall (constant volume) container. Heat is added to the system through the walls of the container and the gas temperature rises. It is evident that the temperature rise (\(ΔT\)) is proportional to the amount of heat added,

\[Q∝ΔT \label{19.2}\]

Introducing a constant of proportionality “cv”,

\[Q=c_{v} \Delta T \label{19.3}\]

In our experiment, no work was done because the boundaries (walls) of the system remained unchanged. Applying the first law of thermodynamics to this closed system, we have:

\[Q \propto \Delta T \label{19.4}\]

Therefore, for infinitesimal changes,

\[C_{v}=\left(\frac{\partial U}{\partial T}\right)_{V} \label{19.5}\]

As we have seen, constant volume heat capacity is the amount of heat required to raise the temperature of a gas by one degree *while retaining its volume*.

Let us now consider the same 1 lbmol of gas confined in a piston-cylinder equipment (i.e., a system with non-rigid walls or boundaries). When heat is added to the system, the gas temperature rises and the gas expands so that *the pressure in the system remains the same at any time*. The piston displaces a volume \(V\) and the gas increases its temperature in \(\Delta T\) degrees. Again, the temperature rise (\(\Delta T\)) is proportional to the amount of heat added, and the new constant of proportionality we use here is “c_{p}”,

\[Q=c_{P} \Delta T \label{19.6}\]

This time, some work was done because the boundaries (walls) of the system changed from their original position. Applying the first law of thermodynamics to this closed system, we have that:

\[\Delta U=Q-W \label{19.7}\]

If the pressure remained the same both inside and outside the container, the system made some work against the surroundings in the amount of W=P\(W=P_{\Delta} V\). Introducing Equation \ref{19.7} into Equation \ref{19.6},

\[\Delta U+P \Delta V=c_{P} \Delta T \label{19.8}\]

The left hand side of this equation represents the definition of *enthalpy change *(\(\Delta H\)) for a constant-pressure process. Therefore:

\[\Delta H=c_{P} \Delta T \label{19.9}\]

Finally, for infinitesimal changes,

\[c_{P}=\left(\frac{\partial H}{\partial T}\right)_{V} \label{19.10}\]

The function “\(c_p\)” is called the *constant pressure heat capacity*. The constant pressure heat capacity is the amount of heat required to raise the temperature of a gas by one degree *while retaining its pressure*.

The units of both heat capacities are (Btu/lbmol-°F) and (cal/gr-°C). Their values are *never equal to each other*, not even for ideal gases. In fact, the ratio “c_{p}/c_{v}” of a gas is known as “k” — the heat capacity ratio — and it is never equal to unity. This ratio is frequently used in gas-dynamics studies.

\[k=\frac{c_{p}}{c_{v}} \label{19.11}\]

Heat capacities can be calculated using equations of state. For instance, Peng and Robinson (1976) presented an expression for the departure enthalpy of a fluid mixture, shown below:

\[\ddot{H}=H-H^{*}=R T(Z-1)+\frac{T \frac{d(a \alpha)_{m}}{d T}-(a \alpha)_{m}}{2 \sqrt{2 b_{m}}} \ln \left(\frac{Z+(\sqrt{2}+1) B}{Z-(\sqrt{2}-1) B}\right) \label{19.12}\]

The value of the enthalpy of the fluid (H) is obtained by adding up this enthalpy of departure (shown above) to the ideal gas enthalpy (H*). Ideal enthalpies are sole functions of temperature. For hydrocarbons, Passut and Danner (1972) developed correlations for ideal gas properties such as enthalpy, heat capacity and entropy as a function of temperature. Therefore, an analytical relationship for “c_{p}” can be derived taking the derivative of Equation \ref{19.12}, as shown below:

\[C_{P}=C_{\dot{P}}+R\left(T\left(\frac{\partial Z}{\partial T}\right)_{P}+Z-1\right)+\frac{T \frac{\text { dacom }}{a x}-(a \alpha)_{m}}{2 \sqrt{2 b_{m}}} \left[\frac{\left(\frac{\partial z}{\partial T}\right)_{P}+2.414\left(\frac{\partial B}{\partial T}\right)_{P}}{Z+2.414 B}-\frac{\left(\frac{\partial Z}{\partial T}\right)_{P}-0.414\left(\frac{\partial B}{\partial T}\right)_{P}}{Z-0.414 B}\right]+\frac{T \frac{d^{2}(a \alpha)_{m}}{d T^{2}}}{2 \sqrt{2 b_{m}}} \ln \left(\frac{Z+2.414 B}{Z-414 B}\right) \label{19.13} \]

where:

\[C_{P}=\left(\frac{\partial H}{\partial T}\right)_{P}\]

and is the ideal gas \(C_P\), also found in the work of Passut and Danner (1972).

The second derivative of \((a \alpha)_{m}\) with respect to temperature can be calculated through the expression:

\[\frac{d^{2}}{d T^{2}}(a \alpha)_{m}=-\frac{0.45244 R^{\circ}}{2 \sqrt{T}} \sum_{i} \sum_{j} c_{i} c_{j}\left(1-k_{i j}\right) \left[f\left(w_{j}\right)\left(\frac{\alpha_{i}^{0.5} T_{ci}}{P_{ci}^{0.5}}\right)\left(\frac{T_{cj}}{P_{cj}}\right)^{0.5} \psi_{i}+f\left(w_{i}\right)\left(\frac{\alpha_{j}^{0.5} T_{cj}}{P_{cj}^{0.5}}\right)\left(\frac{T_{ci}}{P_{ci}}\right)^{0.5} \psi_{j}\right] \label{19.14a}\]

where,

\[\psi_{i}=-\frac{f\left(w_{i}\right)}{2 \sqrt{T_{c i} T \alpha_{i}}}-\frac{1}{2 T} \label{19.14b}\]

For the evaluation of expression \ref{19.13}, the derivative of the compressibility factor with respect to temperature is also required. Using the cubic version of Peng-Robinson EOS, this derivative can be written as:

\[\left(\frac{\partial Z}{\partial T}\right)_{P}=-\left(\frac{\left(\frac{\partial \Omega_{2}}{\partial T}\right)_{P} Z^{2}+\left(\frac{\partial \Omega_{3}}{\partial T}\right)_{P} Z+\left(\frac{\partial \Omega_{4}}{\partial T}\right)_{P}}{3 Z^{2}+2 \Omega_{2} Z+\Omega_{3}}\right) \label{19.15}\]

where

\[\begin{aligned}

&\left(\frac{\partial \Omega_{2}}{\partial T}\right)_{P}=\left(\frac{\partial B}{\partial T}\right)_{P}\\

&\left(\frac{\partial \Omega_{3}}{\partial T}\right)_{P}=\left(\frac{\partial A}{\partial T}\right)_{P}-6 B\left(\frac{\partial B}{\partial T}\right)_{P}-2\left(\frac{\partial B}{\partial T}\right)_{P}\\

&\left(\frac{\partial \Omega_{4}}{\partial T}\right)_{P}=-\left[A\left(\frac{\partial B}{\partial T}\right)_{P}+B\left(\frac{\partial A}{\partial T}\right)_{P}-2 B\left(\frac{\partial B}{\partial T}\right)_{P}-3 B^{2}\left(\frac{\partial B}{\partial T}\right)_{P}\right]\\

&\left(\frac{\partial A}{\partial T}\right)_{P}=\frac{A}{(a \alpha)_{m}} \frac{d(a \alpha)_{m}}{d T}-2 \frac{A}{T}\\

&\left(\frac{\partial B}{\partial T}\right)_{P}=-\frac{B}{T}

\end{aligned}\]

“c_{p}” and “c_{v}” values are thermodynamically related. It can be proven that this relationship is controlled by the P-V-T behavior of the substances through the relationship:

\[c_{p}-c_{v}=T\left(\frac{\partial V}{\partial T}\right)_{P}\left(\frac{\partial P}{\partial T}\right)_{V} \label{19.16}\]

For ideal gases, \(P V=n R T\) and Equation 18.28 collapses to:

\[c_{p}-c_{v}=R \label{19.17}\]

## Contributors and Attributions

Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.