Question

A well known problem in thermodynamics is the determination of boiling points from the Van’t Hoff equation using the liquid-vapor phase equilibrium. When taking the temperature dependence of the enthalpy of vaporization and entropy of vaporization into account, it is possible to quite accurately predict the boiling temperature as the point at which the liquid’s vapor pressure matches the ambient pressure (assuming the system is open so that the external pressure is fixed, of course).

My question is whether a similar procedure can be done for the solid-liquid phase transition. Unfortunately, the most relevant equilibrium expression would be quite useless since there are just two condensed phases on either side of the reaction. My thought is that the solid and liquid phases each have a characteristic vapor pressure that changes with temperature, and at the point of the phase transition the two vapor pressures must be equal. This is because of Le Chatelier’s principle that if the vapor pressure over the solid exceeded the vapor pressure the liquid phase would have at that temperature, the equilibrium would be shifted such that the liquid would form. I did some quick and dirty calculations using thermochemical data for water, and this method got the melting point wrong by about 7 K without taking the temperature dependence of enthalpy and entropy into account, which seems like a plausible amount of error. Is this line of reasoning correct/is there a simpler line of reasoning that would allow one to calculate the melting transition temperature?

Answer 1

The student is correct in reasoning that vapor pressures of solid and liquid phases can be used to predict the melting point, similar to boiling point predictions. Refinements such as accounting for the temperature dependence of enthalpy and entropy and the use of phase diagrams can enhance the accuracy of the predicted melting point.

Step-by-step explanation:

The student's question revolves around the possibility of predicting the melting point of a substance by equating the vapor pressures of the solid and liquid phases at the transition temperature, akin to the approach used for boiling points. The Clausius-Clapeyron equation effectively describes the relationship between vapor pressure and temperature for phase transitions and can be utilized not only for boiling points but also for other phase changes, such as melting.

The student's approach of calculating the vapor pressures at the melting point is fundamentally correct but may be refined by considering the temperature dependency of the enthalpy and entropy of the phase transition. Since the enthalpy of fusion (∆Hfus) is generally lower than the enthalpy of vaporization (∆Hvap), the temperature change will typically result in a smaller change in vapor pressure for the solid-liquid transition compared to the liquid-gas transition.

Utilizing a phase diagram that delineates the pressure and temperature domains where specific physical states exist can further aid in pinpointing the conditions that define the melting or freezing point. According to Le Chatelier's principle, the solid-liquid equilibrium will adjust so that the higher vapor pressure phase diminishes, maintaining an equilibrium at the melting point.

Therefore, the student's reasoning is well-founded and suggests that by incorporating the temperature dependence of the thermodynamic variables, a more accurate melting point determination using vapor pressure can be achieved.