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Chemistry Question on Thermodynamics

Consider the following liquid-vapour equilibrium Liquid<=>Vapour{Liquid <=> Vapour } Which of the following relations is correct ?

A

d In PdT=−ΔHvRT{ \frac{d \, In \, P}{dT} = \frac{ - \Delta H_v}{RT}}

B

d In PdT2=−ΔHvT2{ \frac{d \, In \, P}{dT^2} = \frac{ - \Delta H_v}{T^2}}

C

d In PdT=ΔHvRT2{ \frac{d \, In \, P}{dT} = \frac{ \Delta H_v}{RT^2}}

D

d In GdT2=ΔHvRT2{ \frac{d \, In \, G}{dT^2} = \frac{ \Delta H_v}{RT^2}}

Answer

d In PdT=ΔHvRT2{ \frac{d \, In \, P}{dT} = \frac{ \Delta H_v}{RT^2}}

Explanation

Solution

Acc. To Clausius Claperon equation
P=Ae−ΔHRTP = Ae^{\frac{-\Delta H}{RT}}
lnP=lnA+lne−ΔHRT⇒lnp=lnA−ΔHRTlnel n P = l n A + ln e^{\frac{-\Delta H}{RT}} \Rightarrow lnp = lnA - \frac{\Delta H}{RT} lne
or lnp=lnA−ΔHRTlnp = lnA - \frac{\Delta H}{RT}
ddT(lnP)=0+−ΔHRddT(T−1)\frac{d}{dT} \left(ln P\right) = 0 + \frac{-\Delta H}{R} \frac{d}{dT} \left(T^{-1}\right)
ddTlnP=ΔHVRT2\frac{d}{dT} lnP = \frac{\Delta H_{V}}{RT^{2}}