Question

Consider the following liquid – vapour equilibrium \[Liquid \leftrightharpoons
Vapour\]. Which of the following relations is correct?
A. \[\dfrac{{d\ln G}}{{d{T^2}}} \Rightarrow \dfrac{{\Delta {H_V}}}{{R{T^2}}}\]
B. \[\dfrac{{d\ln P}}{{dT}} \Rightarrow - \dfrac{{\Delta {H_V}}}{{RT}}\]
C. \[\dfrac{{d\ln P}}{{d{T^2}}} \Rightarrow - \dfrac{{\Delta {H_V}}}{{{T^2}}}\]
D. \[\dfrac{{d\ln P}}{{dT}} \Rightarrow \dfrac{{\Delta {H_V}}}{{R{T^2}}}\]

Answer 1

Hint: This given condition is for Clausius - Clapeyron's equation. The equilibrium between water and water vapor depends upon the temperature of the system.

Step by step answer: According to our question, correct answer is \[\dfrac{{d\ln P}}{{dT}} \Rightarrow \dfrac{{\Delta {H_V}}}{{R{T^2}}}\] as this equation is Clausius - Clapeyron's. The rate at which the regular logarithm of the fume weight of a fluid changes with temperature is dictated by the molar enthalpy of vaporization of the fluid, the ideal gas steady, and the temperature of the system.

Clausius - Clapeyron's equation:
If the temperature expands the immersion weight of the water fume increases. The pace of increment in fume pressure per unit increment in temperature is given by Clausius-Clapeyron's condition. Let \[P\] be the saturation vapor pressure and \[T\] the temperature. The Clausius-Clapeyron’s condition for the balance among fluid and fume is at that point
\[\dfrac{{dP}}{{dT}} \Rightarrow \dfrac{L}{{T({V_V} - {V_1})}}\]
Where\[L\] is the latent heat of evaporation, and \[{V_V}\]and\[{V_1}\] are the specific volumes at temperature \[T\] of the vapor and liquid phases, respectively.
The ideal gas condition applies to the fume; i.e.
\[p{V_V} \Rightarrow RT\]
And hence
\[{V_V} \Rightarrow \dfrac{{RT}}{p}\]
Where \[R\]is the universal gas constant.
It is critical to not utilize the Clausius-Clapeyron’s condition for the strong to fluid change. That requires the use of the more general Clapeyron’s rquation
\[\dfrac{{dP}}{{dT}} \Rightarrow \dfrac{{\Delta H}}{{T\Delta V}}\]
Where \[\Delta H\] and \[\Delta V\] is the molar change in enthalpy (the enthalpy of fusion in this case) and volume respectively between the two phases in the transition.

Hence the correct option is (D).

Note: \[Liquid \leftrightharpoons Vapour\] equilibrium states that when liquid is heated, it converts into vapour but on cooling, it further converts into liquid, which is derived by Clausius - Clapeyron's equation.