# Heat Capacities of an Ideal Gas

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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.

\(C_p\) | \(C_V\) | \(C_p - C_V\) | ||
---|---|---|---|---|

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).