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Question: The Clausius-Clapeyron equation may be given as: \({\log _{10}}\dfrac{{{P_2}}}{{{P_1}}} = \dfrac{...

The Clausius-Clapeyron equation may be given as:
log⁡10P2P1=ΔHvap2.303R[1T1A.1T2]{\log _{10}}\dfrac{{{P_2}}}{{{P_1}}} = \dfrac{{\Delta {H_{vap}}}}{{2.303R}}\left[ {\dfrac{1}{{{T_1}}} A.\dfrac{1}{{{T_2}}}} \right]
B.dPdT=ΔSΔV\dfrac{{dP}}{{dT}} = \dfrac{{\Delta S}}{{\Delta V}}
C.dPdT=qTΔV\dfrac{{dP}}{{dT}} = \dfrac{q}{{T\Delta V}}
D.dPdT=ΔVΔS\dfrac{{dP}}{{dT}} = \dfrac{{\Delta V}}{{\Delta S}}

Explanation

Solution

We have studied that Claus-Clapeyron relation is a distinct way of characterizing a discontinuous phase transition between two phases of matter of a single constituent. We must have to remember that the Claus-Clapeyron equation basically gives the relation between the pressure and the latent heat of the constituent.

Complete step by step answer:
The equation has been derived and mathematically represented using a graph of pressure-temperature. On the graph there is a line that separates the two phases of the constituent which is known as the coexistence curve.
From the graph, the equation can be deduced as
dPdT=LTΔV=qTΔV=ΔSΔV\dfrac{{dP}}{{dT}} = \dfrac{L}{{T\Delta V}} = \dfrac{q}{{T\Delta V}} = \dfrac{{\Delta S}}{{\Delta V}}
From ideal gas approximation it can be written as,
log⁡10P2P1=ΔHvap2.303R[1T1−1T2]{\log _{10}}\dfrac{{{P_2}}}{{{P_1}}} = \dfrac{{\Delta {H_{vap}}}}{{2.303R}}\left[ {\dfrac{1}{{{T_1}}} - \dfrac{1}{{{T_2}}}} \right]
Where, dPdT\dfrac{{dP}}{{dT}} is the slope of the tangent, L & q is the specific latent heat, delta V is the volume change and delta S is the entropy change of phase transition, T is the temperature of the system.
We must remember that the ideal gas equation of this form is very useful as it gives the direct relation of equilibrium or saturation vapour pressure and temperature to the latent heat of the phase change.
Therefore, the correct answer to the question option A, B and C.

Note:
This equation has many applications especially in chemistry. In chemistry and chemical engineering, it is used for transition between a gas and a condensed phase with specific approximation. It is used in meteorology and climatology as well. The atmospheric water vapour drives many important meteorological phenomena, raising the interest of dynamics. It is also useful for evaluation of heat sublimation of ice.