Heat capacity

Let’s turn our attention to heat capacity. The heat capacity is the amount of thermal energy required to change the temperature of material. so the heat capacity C is delta Q divide by delta T. The heat capacity at constant volume Cv is dQ over dT at constant volume. And from the first law, dQ at constant volume is the same with dU since dV is zero at constant volume. So Cv is dU over dT at constant volume. Likewise, the heat capacity at constant pressure Cp is dQ over dT at constant pressure and dQp is the same with dH at constant pressure. So Cp is dH over dT at constant pressure. And this heat capacity strongly depends on temperature. Look at this graph.

It is the variation of Cv over temperature for various materials. In general, heat capacity increases with temperature. And also, heat capacity is different for different materials since the heat capacity is related with bonding. In this graph, bond strength decreases from diamond, to silicon and to lead. As you see here, materials with stronger bond have lower heat capacity. To discuss this trend, lets consider the energy of a material. There can be vibrational, translational, rotational and electronic energies in a material. And the Temperature of a material is proportional to the energy of material. Let’s only consider vibrational energy of a material. Then, the temperature of the material represent the total vibrational energy of the material.

The heat capacity is then the energy, the heat, required to induce more vibration. The stronger bond means that it is easier to induce vibration throughout the material since the atoms strongly bind to each other, so the vibration spread easily. Thus it needs smaller energy to induce the same amount of vibration throughout meaning that it needs smaller energy to increase temperature. We have to keep in mind that Cp is always larger than or the same with Cv. Their difference is alpha square VT over k. And this equation will be derived later in this course. Here, alpha is the thermal expansion coefficient defined as 1 over V times dV/dT at constant pressure. K is the isothermal compressibility.

It is minus 1 over V times dV over dP at constant temperature. For condensed phase such as liquids and solids, the expansion coefficient is generally very small, thus Cp-Cv is virtually zero. They have the same value.