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# 3.5: Heat Capacities of an Ideal Gas

We learned about specific heat and molar heat capacity previously; however, we have not considered a process in which heat is added. We do that in this section. First, we examine a process where the system has a constant volume, then contrast it with a system at constant pressure and show how their specific heats are related.

Let’s start with looking at Figure $$\PageIndex{1}$$, which shows two vessels A and B, each containing 1 mol of the same type of ideal gas at a temperature T and a volume V. The only difference between the two vessels is that the piston at the top of A is fixed, whereas the one at the top of B is free to move against a constant external pressure p. We now consider what happens when the temperature of the gas in each vessel is slowly increased to $$T + dT$$ with the addition of heat.

Figure $$\PageIndex{1}$$: Two vessels are identical except that the piston at the top of A is fixed, whereas that atop B is free to move against a constant external pressure p.

Since the piston of vessel A is fixed, the volume of the enclosed gas does not change. Consequently, the gas does no work, and we have from the first law

$dE_{int} = dQ - dW = dQ.$

We represent the fact that the heat is exchanged at constant volume by writing

$dQ = C_VdT,$ where $$C_V$$ is the molar heat capacity at constant volume of the gas. In addition, since $$dE_{int} = dQ$$ for this particular process,

$dE_{int} = C_VdT.$

We obtained this equation assuming the volume of the gas was fixed. However, internal energy is a state function that depends on only the temperature of an ideal gas. Therefore, $$dE_{int} = C_VdT$$ gives the change in internal energy of an ideal gas for any process involving a temperature change dT.

When the gas in vessel B is heated, it expands against the movable piston and does work $$dW = pdV$$. In this case, the heat is added at constant pressure, and we write $dQ = C_pdT,$ where $$C_p$$ is the molar heat capacity at constant pressure of the gas. Furthermore, since the ideal gas expands against a constant pressure,

$d(pV) = d(RT)$ becomes $pdV = RdT.$

Finally, inserting the expressions for dQ and pdV into the first law, we obtain

$dE_{int} = dQ - pdV = (C_p - R)dT.$

We have found $$dE_{int}$$ for both an isochoric and an isobaric process. Because the internal energy of an ideal gas depends only on the temperature, $$dE_{int}$$ must be the same for both processes. Thus,

$C_VdT = (C_p - R)dT,$

and

$C_p = C_V + R. \label{eq50}$

The derivation of Equation \ref{eq50} was based only on the ideal gas law. Consequently, this relationship is approximately valid for all dilute gases, whether monatomic like He, diatomic like $$O_2$$,  or polyatomic like $$CO_2$$ or $$NH_3$$.

In the preceding chapter, we found the molar heat capacity of an ideal gas under constant volume to be

$C_V = \dfrac{d}{2}R,$

where d is the number of degrees of freedom of a molecule in the system. Table  $$\PageIndex{1}$$ shows the molar heat capacities of some dilute ideal gases at room temperature. The heat capacities of real gases are somewhat higher than those predicted by the expressions of $$C_V$$ and $$C_p$$ given in Equation \ref{eq50}. This indicates that vibrational motion in polyatomic molecules is significant, even at room temperature. Nevertheless, the difference in the molar heat capacities, $$C_p - C_V$$, is very close to R, even for the polyatomic gases.

Table $$\PageIndex{1}$$: Molar Heat Capacities of Dilute Ideal Gases at Room Temperature
$$C_p$$ $$C_V$$ $$C_p - C_V$$
Type of Molecule Gas (J/mol K) (J/mol K) (J/mol K)
Monatomic Ideal $$\frac{5}{2}R = 20.79$$ $$\frac{3}{2}R = 12.47$$ $$R = 8.31$$
Diatomic Ideal $$\frac{7}{2}R = 29.10$$ $$\frac{5}{2}R = 20.79$$ $$R = 8.31$$
Polyatomic Ideal $$4R = 33.26$$ $$3R = 24.04$$ $$R = 8.31$$

molar heat capacity at constant volume
quantifies the ratio of the amount of heat added removed to the temperature while measuring at constant volume

## Contributors

• Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).