# 15.3: Heat Capacity and Microscopic Changes

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Let us turn our attention from the macroscopic to the microscopic level. According to the first law of thermodynamics, the heat energy absorbed as we raise the temperature of a substance cannot be destroyed. But where does it go? In the case of a monatomic gas, like neon, this question is easy to answer. All the energy absorbed is converted into the kinetic energy of the neon molecules (atoms). In other sections, we found that the kinetic energy of the molecules in a sample of gas is given by the expression

$E_{k}=\tfrac{\text{3}}{\text{2}}nRT \nonumber$

Thus if the temperature of a sample of neon gas is raised from T1 to T2, the kinetic energy of the molecules increases from 3/2 nRT1 to 3/2 nRT2, a total change of

$\tfrac{3}{2}nR\left( T_{2}-T_{1}\right) =\left( \tfrac{3}{2}R\right) n\left( T_{2}-T_{1}\right) \nonumber$

Inserting the value of R in appropriate units, we obtain

\begin{matrix}\frac{3}{2}\left(8.314\frac{\text{J}}{\text{K mol}}\right)n\left(T_{2}-T_{1}\right)=\left(12.47\frac{\text{J}}{\text{K mol}}\right)n\left(T_{2}-T_{1}\right)\end{matrix}

This is the same quantity that is obtained by substituting the experimental value of CV for neon (calculated in Example 2 from Heat Capacities) into Eq. (4) from Heat Capacities. In other words the quantity of heat found experimentally exactly matches the increase in kinetic energy of the molecules required by the kinetic theory of gases.

Table $$\PageIndex{1}$$ lists the CV values not only for neon but for some other gases as well. We immediately notice that only the noble gases and other mon-atomic gases such as Hg and Na have molar heat capacities equal to 3/2R, or 12.47 J K–1 mol–1. All other gases have higher molar heat capacities than this. Moreover, as the table shows, the more complex the molecule, the higher the molar heat capacity of the gas. There is a simple reason for this behavior.

Table $$\PageIndex{1}$$: Molar Heat Capacities at Constant Volume (CV) for Various Gases (Values at 298 K Unless Otherwise Stated).
Gas Cv /J K-1 mol-1 Gas Cv /J K-1 mol-1
Monoatomic Gases Triatomic Gases
Ne 12.47 CO2 28.81
Ar 12.47 N2O 30.50
Hg 12.47 (700K) SO2 31.56
Na 12.47 (1200K)
Diatomic Gases Alkanes
N2 20.81 CH4 27.42
O2 21.06 C2H6 44.32
Cl2 25.62 C3H8 65.20
C4H10 89.94

A molecule which has two or more atoms is not only capable of moving from one place to another (translational motion), it can also rotate about itself, and it can change its shape by vibrating. When we heat a mole of Cl2 molecules, for example, we not only need to supply them with enough energy to make them move around faster (increase their translational kinetic energy), we must also supply an additional quantity of energy to make them rotate and vibrate more strongly than before. For a mole of more complex molecules like n-butane even more energy is required since the molecule is capable of changing its shape in all sorts of ways. In the butane molecule there are three C—C bonds around which segments of the molecule can rotate freely. All the bonds can bend or stretch, and the whole molecule can rotate as well. Such a molecule is constantly flexing and writhing at room temperature. As we raise the temperature, this kind of movement occurs more rapidly and extra energy must be absorbed in order to make this possible.

When we heat solids and liquids, the situation is somewhat different than for gases. The rapid increase of vapor pressure with temperature makes it virtually impossible to heat a solid or liquid in a closed container, and so heat capacities are always measured at constant pressure rather than at constant volume. Some Cp values for selected simple liquids and solids at the melting point are shown in Table $$\PageIndex{2}$$. In general the heat capacities of solids and liquids are higher than those of gases. This is because of the intermolecular forces operating in solids and liquids. When we heat solids and liquids, we need to supply them with potential energy as well as kinetic energy. Among the solids, the heat capacities of the metals are easiest to explain since the solid consists of individual atoms. Each atom can only vibrate in three dimensions. According to a theory first suggested by Einstein, this vibrational energy has the value 3RT, while the heat capacity is given by 3R = 24.9 J K–1 mol–1.

Table $$\PageIndex{2}$$: Molar Heat Capacities at Constant Pressure Cp for Various Solids and Liquids at the Melting Point.
Substance Cp (solid)/J K-1 mol-1 Cp (liquid)/J K-1 mol-1
Monoatomic Substances
Hg 27.28 27.98
Pb 29.40 30.33
Na 28.20 31.51
Diatomic Substances
Br2 53.8 75.7
I2 54.5 80.7
HCl 50.5 62.2
HI 47.5 68.6
Polyatomic Substances
H2O 37.9 76.0
NH3 49.0 77.0
Benzene 129.0 131.0
n-Heptane 146.0 203.1

As can be seen from the table, most monatomic solids have Cp values slightly larger than this. This is because solids expand slightly on heating. The atoms get farther apart and thus increase in potential as well as vibrational energy.

Solids which contain molecules rather than atoms have much higher heat capacities than 3R. In addition to the vibration of the whole molecule about its site in the crystal lattice, the individual atoms can also vibrate with respect to each other. Occasionally molecules can rotate in the crystal, but usually rotation is only possible when the solid melts. As can be seen from the values for molecular liquids in Table $$\PageIndex{2}$$, this sudden ability to rotate causes a sharp increase in the heat capacity. For monatomic substances, where there is no motion corresponding to the rotation of atoms around each other, the heat capacity of the liquid is only very slightly higher than that of the solid.