Given a known relation between the saturation pressure and temperature of an unspecified metal, how can we determine the following properties?

In order:

a) The normal boiling point is when the pressure of the liquid is 1 atm. Graph the equation $Y = ln(p) - (RHS)$ (right hand side) to find the temperature where $Y = 0$. Alternatively, graph $p_{lv}$ versus $T$ and find when $p_{lv} = 1$.

b) The triple point occurs when the temperature and pressure of the solid, liquid, and vapor are all equal. Create a graph with $p_{sv}$ and $p_{lv}$ as a function of $T$. They will cross at the triple point.

c) I would first challenge whether the question is asking for the difference in specific enthalpies not specific heat capacities. Consider X as either solid or liquid. Vapor pressures of X-gas transitions are defined by the Clausius-Clapeyron equation. An empirical form is also found using the Antoine equation. The transition enthalpy can be determined from a Clausius-Clapeyron form of $ln(p)$ as shown here. Alternatively, heat capacity differences appear in the first reference in this paragraph near the end for a discussion on the second derivative of the phase transition lines.

d) The transition enthalpies from above will depend on temperature. Find the value at the triple point temperature.

e) The Clapeyron equation applies for a solid-liquid transition. This question asks for the value for $dp/dT = Delta_{fus}bar{H}/(T_{fus}Delta_{fus}bar{V})$, where molar values are used for the fusion enthalpy and volume changes. Molar volume is the inverse of moles per volume, which itself is the (mass) density divided by molar mass. The important hint here is to always write a $Delta$ as FINAL - INITIAL. Density decreases going from liquid -> solid (fusion), so molar volume increases. Fusion is endothermic, so the enthalpy is positive. Review the differences between the slopes of the $dp/dT$ lines for fusion of water (where solid ice floats) and almost any other liquid (where its solid sinks).