# Heat Capacity and Microscopic Changes

Submitted by ChemPRIME Staff on Thu, 12/16/2010 - 15:05

Let us turn our attention from the macroscopic to the microscopic level. According to the first law of thermodynamicsA formal statement that energy can neither be created nor destroyed; as applied to chemical systems, the change in internal energy is equal to the heat energy transfer into the system plus the work energy transfer into the system., the heatEnergy transferred as a result of a temperature difference; a form of energy stored in the movement of atomic-sized particles. energyA system's capacity to do work. absorbed as we raise the temperatureA physical property that indicates whether one object can transfer thermal energy to another object. of a substanceA material that is either an element or that has a fixed ratio of elements in its chemical formula. cannot be destroyed. But where does it go? In the case of a monatomic gasA state of matter in which a substance occupies the full volume of its container and changes shape to match the shape of the container. In a gas the distance between particles is much greater than the diameters of the particles themselves; hence the distances between particles can change as necessary so that the matter uniformly occupies its container., like neon, this question is easy to answer. All the energy absorbed is converted into the kinetic energy of the neon molecules (atomsThe smallest particle of an element that can be involved in chemical combination with another element; an atom consists of protons and neutrons in a tiny, very dense nucleus, surrounded by electrons, which occupy most of its volume.). In other sections, we found that the kinetic energy of the molecules in a sample of gas is given by the expression

$E_{k}=\tfrac{\text{3}}{\text{2}}nRT$      (1)

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

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

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 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 complexA central metal and the ligands surrounding it; also called coordination complex. the molecule, the higher the molar heat capacityThe quantity of heat energy transfer needed to raise the temperature of a sample of matter by one unit. of the gas. There is a simple reason for this behavior.

TABLE 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 pressureThe pressure (or partial pressure) exerted by the gaseous form of a substance in equilibrium with the liquid form. 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 pointThe temperature at which a solid becomes a liquid. Also called freezing point. are shown in Table 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 2. Molar Heat Capacities at Constant Pressure Cp for Various Solids and Liquids at the MeltingThe process of a liquid forming from a solid. 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 latticeAn orderly, repeating arrangement of points in 3-D space in which each p;oint has surroundings identical to every other point. A crystal's constituent atoms, molecules, and ions are arranged about each lattice point., the individual atoms can also vibrate with respect to each other. Occasionally molecules can rotate in the crystalA solid with a regular polyhedral shape; for example, in sodium chloride (table salt) the crystal faces are all at 90&deg; angles. A solid in which the atoms, molecules, or ions are arranged in a regular, repeating lattice structure., but usually rotation is only possible when the solid melts. As can be seen from the values for molecular liquids in Table 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.