Question

On a cold, dry morning after a frost, the temperature was -5C and the partial pressure of water in the atmosphere fell to 0.30 kPa. Will the frost sublime? What partial pressure of water would ensure that the frost remained?

Answer 1

The ice sublimates if the vapor pressure exceeds the partial pressure in air.

You can calculate the vapor pressure from the Clausius-Clapeyron relation and triple point data

d(lnp) /dT = ΔH_s / (R·T²)
(ΔH_s molar heat of sublimation, R universal gas constant, temperature in Kelvins)

Integrating with boundary condition p(T_₀) = p_₀ leads to
ln(p_T/p_₀) = (-ΔH_s/R) · (1/T - 1/T_₀)
<=>
p_T = p₀· exp{(-ΔHs/R) · (1/T - 1/T₀)}

Choose triple point as reference point
p_₀ = 611.7Pa
T_₀ = 0.1°C

From the triple point data (see link) you can calculate heat of sublimation:
ΔHs = H_g(p_₀,T_₀) - H_s(p_₀,T_₀)
= 2500kJ/kg - (-333.4kJ/kg)
= 2833.4kJ/kg = 51044J/mol

The vapor pressure at -5°C is
p(268.15K) = 611.7Pa · exp{(51044J/mol / 8.314472J/(molK)) · (1/268.15K - 1/273.14K)}
= 402.6

This exceeds the partial pressure of 300Pa. Hence frost sublimes.

At partial pressure of 402.6 the frost remains. For greater partial pressures the water vapor in gas phase solidifies.