# Enthalpy and energy

Phase change processes are always accompanied by a change of enthalpy, which we will now consider at the molecular level. Phase change phenomena can be viewed as the destruction or formation of intermolecular bonds as the result of changes in intermolecular forces. The intermolecular forces between the molecules in a solid are greater than those between molecules in a liquid, which are in turn greater than those between molecules in a gas. This reflects the greater distance between molecules in a gas than in a liquid, and the greater distance between molecules in a liquid than in a solid. As a result, the intermolecular bonds in a solid are stronger than those in a liquid. In a gas, which has the weakest intermolecular forces of all three phases, intermolecular bonds do not exist between the widely separated molecules.

Energies of the H2O molecule in the vicinity of 273 K
 Types of energy Approximate magnitude per molecule(eV) Lattice vibration 0.0054 Intermolecular hydrogen bond breaking 0.58 Enthalpy of melting 0.06 Enthalpy of vaporization 0.39 Enthalpy of sublimation 0.49

When the intermolecular bonds between the molecules in a solid are completely broken, sublimation occurs. The approximate energy levels identified for the H2O molecule near 273 K are summarized in the table on the right. Since it takes 0.29 eV (electron-volts) to break a hydrogen bond in the ice lattice, and there are two bonds per H2O molecule, the energy required to completely free a H2O molecule from its neighbor should be 0.58 eV. The energy required to break two hydrogen bonds, 0.58 eV, should be of the same order of magnitude as the enthalpy of sublimation, which is 0.49 eV, as shown in table on the right. The energy required to melt ice is 0.06 eV per molecule, while it takes 0.39 eV per molecule to vaporize liquid water. The difference between the enthalpy of melting and vaporization at the molecular level explains the difference in the latent heats of fusion and of vaporization. The internal energy of a substance with molecules containing more than one atom (such as H2O) is the sum of the kinetic, rotational, and vibrational energies. Since the molecules in a solid are held in a fixed pattern and are not free to move or rotate, the lattice vibrational energy is the primary contributor to the internal energy of the ice. As can be seen from on the right, the enthalpy of melting and vaporization are much larger than the lattice vibrational energy, which explains why the latent heat is usually much greater than the sensible heat.

Different phases are characterized by their bond energy and their molecular configurations. For example, the intermolecular bonds in a solid are very strong, and thus able to hold the molecules in a fixed pattern. The intermolecular bonds in a liquid are strong enough to hold the molecules together but not strong enough to hold them in a fixed pattern. The intermolecular bonds in a gas are completely broken and the molecules can move freely. Therefore, phase change can be viewed as conversion from one type of intermolecular ordering to another, i.e., a reordering process. From a microscopic point of view, the entropy of a system, S, is related to the total number of possible microscopic states of that system, known as the thermodynamic probability, P, by the Boltzmann relation:

$S = - {k_b}\ln \left( P \right)\qquad \qquad(1)$

where kb is the Boltzmann constant, $1.3806 \times {10^{ - 23}}{\rm{J/K}}$. Therefore, the entropy of a system increases when the randomness or thermodynamic probability of a system increases. Sublimation, melting, and vaporization are all processes that increase the randomness of the system and therefore produce increases of entropy. Since phase changes occur at constant temperature, one can express the increases of entropy in these processes as:

$\Delta s = \frac{{\Delta h}}{T} = \operatorname{constant} \qquad \qquad(2)$

where Δh is the change of enthalpy during phase change, i.e., the latent heat, and T is the phase change temperature. The constant in eq. (2) depends on the particular phase change process but is independent of the substance. For vaporization and condensation, Trouton’s rule is applicable

${s_v} - {s_\ell } = \frac{{{h_{\ell v}}}}{{{T_{sat}}}} \simeq 83.7{\rm{J/(mol - K)}}\qquad \qquad(3)$

while Richards’ rule is valid for melting and solidification

${s_\ell } - {s_s} = \frac{{{h_{s\ell }}}}{{{T_m}}} \simeq 8.37{\rm{J/(mol - K)}}\qquad \qquad(4)$

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.