Contents

**Introduction**

The enthalpy of vaporization and the enthalpy of fusion are two important thermodynamic quantities that describe the energy changes involved in phase transitions. A phase transition is a process in which a substance changes its physical state, such as from solid to liquid or from liquid to gas. The enthalpy of vaporization is the amount of energy required to convert one mole of a liquid into a gas at constant pressure and temperature. The enthalpy of fusion is the amount of energy required to convert one mole of a solid into a liquid at constant pressure and temperature. Both these quantities are positive, meaning that they are endothermic processes, or processes that absorb heat from the surroundings.

**Factors Affecting the Enthalpy of Vaporization and Fusion**

The enthalpy of vaporization and fusion depend on several factors, such as the nature of the substance, the intermolecular forces, the temperature, and the pressure. Generally, substances with stronger intermolecular forces have higher enthalpies of vaporization and fusion, because more energy is needed to overcome these attractions and separate the molecules. For example, water has a high enthalpy of vaporization (40.67 kJ/mol) and a high enthalpy of fusion (6.01 kJ/mol) because it has strong hydrogen bonds between its molecules. On the other hand, methane has a low enthalpy of vaporization (8.18 kJ/mol) and a low enthalpy of fusion (0.94 kJ/mol) because it has weak dispersion forces between its molecules.

The temperature also affects the enthalpy of vaporization and fusion, because it determines the kinetic energy and the degree of disorder of the molecules. As the temperature increases, the kinetic energy and the disorder increase, making it easier for the molecules to escape from the liquid or solid phase. Therefore, the enthalpy of vaporization and fusion decrease with increasing temperature. For example, according to Chemistry LibreTexts, the enthalpy of vaporization of water decreases from 45.07 kJ/mol at 25 °C to 40.65 kJ/mol at 100 °C.

The pressure also affects the enthalpy of vaporization and fusion, because it determines the volume change and the work done by or on the system during the phase transition. As the pressure increases, the volume change decreases, making it harder for the molecules to expand from the liquid or solid phase. Therefore, the enthalpy of vaporization and fusion increase with increasing pressure. For example, according to Wikipedia, the enthalpy of vaporization of water increases from 40.65 kJ/mol at 1 atm to 41.51 kJ/mol at 2 atm.

**Relationship Between the Enthalpy of Vaporization and Fusion**

The enthalpy of vaporization and fusion are related by a simple mathematical equation known as Trouton’s rule. This rule states that for most liquids, the ratio of their enthalpy of vaporization to their normal boiling point (in kelvins) is approximately constant and equal to 85 J/mol·K. This means that liquids with higher boiling points have higher enthalpies of vaporization, and vice versa.

Using Trouton’s rule, we can estimate the enthalpy of vaporization of a substance if we know its normal boiling point, or vice versa. For example, if we know that ethanol has a normal boiling point of 78.4 °C (351.6 K), we can estimate its enthalpy of vaporization as:

Enthalpy_of_vaporization = 85 J/mol·K × 351.6 K = 29.9 kJ/mol

Similarly, if we know that benzene has an enthalpy of vaporization of 30.8 kJ/mol, we can estimate its normal boiling point as:

Normal_boiling_point = Enthalpy_of_vaporization / 85 J/mol·K = 30.8 kJ/mol / 85 J/mol·K = 362 K (88.9 °C)

However, Trouton’s rule is not valid for all liquids, especially those with strong intermolecular forces or large molecular sizes. For example, water has an unusually high enthalpy of vaporization (40.67 kJ/mol) and a low normal boiling point (373 K), giving it a ratio of 109 J/mol·K, which is much higher than 85 J/mol·K.

The relationship between the enthalpy of vaporization and fusion can also be derived from another thermodynamic equation known as Clausius-Clapeyron equation. This equation relates the vapor pressure of a substance to its temperature and enthalpy of vaporization. It can be written as:

ln(P) = -Enthalpy_of_vaporization / R × 1 / T + C

where P is the vapor pressure, R is the gas constant, T is the temperature, and C is a constant.

By integrating this equation between two temperatures, we can obtain the following expression:

ln(P2 / P1) = -Enthalpy_of_vaporization / R × (1 / T2 – 1 / T1)

where P1 and P2 are the vapor pressures at temperatures T1 and T2, respectively.

If we apply this equation to the triple point and the normal boiling point of a substance, we can obtain the following relationship:

ln(1 atm / P_triple) = -Enthalpy_of_vaporization / R × (1 / Normal_boiling_point – 1 / Triple_point)

where P_triple is the vapor pressure at the triple point, and Triple_point is the temperature at the triple point.

Similarly, if we apply this equation to the triple point and the normal melting point of a substance, we can obtain the following relationship:

ln(P_triple / P_solid) = -Enthalpy_of_fusion / R × (1 / Normal_melting_point – 1 / Triple_point)

where P_solid is the vapor pressure of the solid phase at the triple point, and Normal_melting_point is the temperature at the normal melting point.

By combining these two equations, we can eliminate P_triple and obtain the following relationship between the enthalpy of vaporization and fusion:

ln(1 atm / P_solid) = (Enthalpy_of_vaporization – Enthalpy_of_fusion) / R × (1 / Normal_boiling_point – 1 / Normal_melting_point)

This equation shows that the difference between the enthalpy of vaporization and fusion is proportional to the difference between the inverse of the normal boiling point and the inverse of the normal melting point. This means that substances with higher normal boiling points and lower normal melting points have larger differences between their enthalpies of vaporization and fusion, and vice versa.

**Conclusion**

The enthalpy of vaporization and fusion are two important thermodynamic quantities that describe the energy changes involved in phase transitions. They depend on several factors, such as the nature of the substance, the intermolecular forces, the temperature, and the pressure. They are related by two simple mathematical equations: Trouton’s rule and Clausius-Clapeyron equation. These equations can be used to estimate one quantity from another, or to compare different substances based on their phase transition properties.