Clausius-Clapeyron Equation

Clausius-Clapeyron Equation

Clausius-Clapeyron Equation

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation was initially proposed by a German physics Rudolf Clausius in 1834 and further developed by French physicist Benoît Clapeyron in 1850. 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-

dP/dT = ΔH/T.ΔV -----(equation-6)

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 = ΔfusH/Tf.ΔV -----(equation-7)
where Δfus is latent heat of fusion and Tf is melting point.
For vaporisation, equilibrium (i.e. liquid ⇌ vapour),
dP/dT = ΔvapH/T.Vv -----(equation-8)
If the vapor act as an ideal gas-
then,V = RT/P
so, the above equation becomes-
dP/dT = ΔvapH.P/RT2 -----(equation-9)
or, 1/P(dP/dT) = ΔvapH/RT2 -----(equation-10)

or, dlnP/dT = ΔvapH/RT2 -----(equation-11)

The equation-11 is known as the Clausius-Clapeyron equation.

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

lnP2/P1 = (ΔvapH/R) [1/T1 − 1/T2] -----(equation-12)

The equation-12 is another form of Clausius-Clapeyron equation.
converting ln into log-
2.303 logP2/P1 = (ΔvapH/R) [1/T1 − 1/T2] -----(equation-13)

or, logP2/P1 = (ΔvapH/ 2.303 R) [1/T1 − 1/T2] -----(equation-14)