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

This problem contains multiple parts, and I will write my initial attempts at solving the problem under it's corresponding section

If we're given that the vapor pressures (described in Atm) of both the liquid and solid phases are as follows:

ln(p_sat) = A - B/T + Dln(T)

We are also given a table of coefficients for the two cases of the Solid-Gas and the Liquid-Gas transition:

[Situation, A, B(K), D;

Solid-Gas, 19.25, 15773, -0.755;

Liquid-Gas, 21.79, 15246, -1.255]

(There are no units given, just those numbers)

And we are expected to determine:

a.) The normal boiling point

• My initial thought was to work with the given equation and solve it for T, assuming p_sat to be 1atm (the normal boiling point) but ran into obvious algebraic difficulty in isolating T, and without introducing the Lambert W function (which I feel is beyond the scope of an introductory thermodynamics course) I'm not entirely sure what the best method to proceed is.

b.) The triple point temperature

• In this case, my initial thought is to find the temperature at which the equation as written for both of the situations. That is, if there is no particular affinity for either the Solid-Gas or the Liquid-Gas transition it makes sense to me that the corresponding temperature would be the triple point.

c.) The difference between the heat capacities of the liquid and the solid

d.) The heat of fusion at the triple point

f.) The slope of the liquid-solid boundary line at the triple point given densities of liquid and solid phases to be 5.66 and 7.14g/cm^3 respectively and if the atomic weight of the metal is 65.4.

Any assistance would be very much appreciated. Even just pointers towards other equations I might be able to bring in.

Best, -Richard

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.
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).