Clausius - Clapeyron Equation Derivation

2. CLAUSIUS–CLAPEYRON EQUATION
relates the latent heat (heat of transformation)
of vaporization or condensation to the rate of
change of vapor pressure with temperature
or, in the case of a solid-liquid transformation, it
relates the latent heat of fusion or solidification
to the rate of change of melting point with
pressure

3. DERIVATION
Imagine a piston containing a vapor in equilibrium
with its liquid. Thus,
A(l) A(g)
If the piston to be pulled out, at constant
temperature; liquid evaporates and the pressure
remains constant. If the piston is pushed in, vapor
condenses, at constant temperature and pressure.

4. DERIVATION
During this process, the Gibbs free energy (G) of
the system remains constant.
In the equilibrium system, ∆G = ∆H - T∆S
where H is the latent heat of vaporization, T the
temperature and S the entropy of the system
Since G of the said system remains constant,
G2 – G1 = ∆G = ∆H - T∆S = 0
Thus,
∆H = T∆S

5. DERIVATION
However, for gases, ∆H = ∆U + P∆V, where U
is the internal energy of the system, making the
previous equation
∆H =T∆S = ∆U + P∆V
Given two states in the compression, the
changes in entropy (∆S), internal energy (∆U)
and volume (∆V) would become constant.

6. DERIVATION
Differentiating both sides with respect to
temperature, we get
dP
∆S = ∆V
dT
dP ∆S S2 – S1
= =
dT ∆V V2 – V1

7. DERIVATION
However, the change in entropy ∆S is given by
∆S = L / T
where L is the latent heat of vaporization
Thus, we arrive at the Clausius – Clapeyron
equation
dP ∆S L
= =
dT ∆V T (V2 – V1)

8. DERIVATION
When the transition is to a gas phase, the final
volume can be many times the size of the initial
volume and thus
ΔV can be approximated
as V2. Furthermore, at low pressures, the gas
phase may be approximated by the ideal gas law
PV = nRT, changing the previous equation to:
dP ∆S LP
= =
dT ∆V nRT2

9. DERIVATION
Since ∆Hvap = L / n,
dP dT ∆Hvap
=
P T2 R
Also since ∆Hvap is independent of pressure and
temperature and R is a constant, integrating both
sides, we get
1 ∆Hvap
ln P = + C
T R

10. DERIVATION
If the vapor pressure was measured at two
separate temperatures, we have two points on the
same line.
1 ∆Hvap
ln P1 = + C
T1 R
1 ∆Hvap
ln P2 = + C
T2 R
Subtracting these two, we get
∆Hvap 1 1
ln P2 - ln P1 =
R
(T T1
)
2

11. DERIVATION
Finally, we get
P2 ∆Hvap 1 1
ln
P1
=
R
(T T1
)
2
the more commonly known form of the Clausius –
Clapeyron equation