Derive Clausius-Clapeyron Equation

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation was proposed by a German physics Rudolf Clausius in 1834 and later on developed by French physicist Benoît Clapeyron in 1850. The Clausius-Clapeyron Equation describes the relationship between the vapor pressure and temperature of a pure substance. This equation is extremely useful in characterizing a discontinuous phase transition between two phases of a single constituent.

We know that-
dG = VdP − SdT -----(equation-1)
Let us consider a single-constituent equilibria-
𝑃ℎ𝑎𝑠𝑒-1 ⇌ 𝑃ℎ𝑎𝑠𝑒-2
Where phase-1 may be solid, liquid, or gas; whereas phase-2 may be liquid or vapor depending upon the nature of the transition whether it is melting, vaporization or sublimation, respectively.

For phase-1, change in free energy is-
dG₁ = V₁dP − S₁dT -----(equation-2)
and for Phase-2, change in free energy is-
dG₂ = V₂dP − S₂dT -----(equation-3)

At equilibrium- dG₁ = dG₂ (i.e. ΔG = 0
so, V₂dP − S₂dT = 𝑉₁dP − S₁dT
V₂dP − V₁dP = S₂dT − S₁dT
(V₂ − V₁)dP = (S₂ − S₁)dT
ΔV. ΔP = ΔS. dT
dP/dT = ΔS/ΔV -----(equation-4)

Now, if the ΔH is the latent heat of phase transformation takes place at temperature (𝑇), then the entropy change is-
ΔS = ΔH/T -----(equation-5)

Now, putting the value of ΔS from (equation-5) into (equation-4), we get-
The (equation-6) is known as Calpeyron equation.

Now if phase-1 is solid while phase-2 is vapor (i.e. solid ⇌ melt), then the equation-6 becomes-
dP/dT = Δ_fusH/T_f.ΔV -----(equation-7)
where Δ_fus is latent heat of fusion and T_f is melting point.
For vaporisation, equilibrium (i.e. liquid ⇌ vapour),
dP/dT = Δ_vapH/T.V_v -----(equation-8)
If the vapor act as an ideal gas-
then,V = RT/P

so, the above equation becomes-
dP/dT = Δ_vapH.P/RT² -----(equation-9)
or, 1/P(dP/dT) = Δ_vapH/RT² -----(equation-10)
The equation-11 is known as the Clausius-Clapeyron equation.

Another form of Clausius-Clapeyron equation-
from equation-11
dlnP = (Δ_vapH/RT²).dT
If the temperature changes from T₁ to T₂ and pressure is varied from P₁ to P₂, then -
∫dlnP = ∫(Δ_vapH/RT²).dT
lnP₂/P₁ = Δ_vapH/R ∫dT/T²
The equation-12 is another form of Clausius-Clapeyron equation.

Converting ln into log-
2.303 logP₂/P₁ = (Δ_vapH/R) [1/T₁ − 1/T₂] -----(equation-13)
or, logP₂/P₁ = (Δ_vapH/ 2.303 R) [1/T₁ − 1/T₂] -----(equation-14)

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