Clausius-Clapeyron Equation
As assigned by Mr. Amendola despite
the fact that he is no longer our
chemistry teacher
The Men Behind the Equation
 Rudolph Clausius
 German physicist and mathematician
 One of the foremost contributors to the science
of thermodynamics
 Introduced the idea of entropy
 Significantly impacted the fields of kinetic theory
of gases and electricity
 Benoit Paul Émile Clapeyron
 French physicist and engineer
 Considered a founder of thermodynamics
 Contributed to the study of perfect gases and
the equilibrium of homogenous solids
The Equation
 In its most useful form for our purposes:
P1 H vap 1 1
ln 
(  )
P2
R T2 T1
In which:
P1 and P2 are the vapor pressures at T1 and T2 respectively
T is given in units Kelvin
ln is the natural log
R is the gas constant (8.314 J/K mol)
∆Hvap is the molar heat of vaporization
Deriving the Equation
We will use this diagram in
deriving the ClausiusClapeyron equation.
This diagram represents a
generalized phase
diagram. The line acts as a
phase line, or a coexistent
curve, separating phases α
and β.
As we know, this indicates
that at all points on the
line, phases α and β are in
equilibrium.
Deriving the Equation
 Since the phases are in equilibrium along the line,
∆G=0
 ∆G=∆H-T∆S
 Since ∆G=0, ∆S=∆H/T
 We want to find the slope of the coexistent curve.
However, since the graph we are examining is a curve
rather than a line, the slope must be found by using
calculus. The slope is represented by dy/dx, or in the
case of the phase diagram, dp/dt. To represent the
derivative along the coexistent curve, we write:
p
( ) G
T
The curved “d” represents the use of a partial derivative.
Deriving the Equation
 We use the cyclic rule, a
rule of calculus, to find:
We previously wrote ∆G=∆H-T∆S. We can also represent this as ∆G=P∆VT∆S (see the thermodynamics chapter of your book). Taking the
derivative of both sides, we find that d∆G=∆VdP-∆SdT
We have two variables in this differential equation: T and P. To solve this,
we treat this in two cases. First, we consider P as a constant. Then, we
consider T a constant.
By manipulation, we find:
Deriving the Equation
 Substituting in the equation
we found through the cyclic
rule, we find:
As ∆S=∆H/T, this can be written as:
We integrate this equation,
assuming ∆H and ∆V to be
constant, to find:
Useful Information
 The Clausius-Clapeyron models the change
in vapor pressure as a function of time
 The equation can be used to model any
phase transition (liquid-gas, gas-solid,
solid-liquid)
 Another useful form of the ClausiusClapeyron equation is:
ln P  
H vap
RT
C
Useful Information
 We can see from this form that the
Clausius-Clapeyron equation depicts a line
ln P  
H vap
RT
C
Can be written as:
H vap  1 
ln P  
 C
R T 
which clearly resembles the model y=mx+b, with ln P representing y,
C representing b, 1/T acting as x, and -∆Hvap/R serving as m.
Therefore, the Clausius-Clapeyron models a linear equation when the
natural log of the vapor pressure is plotted against 1/T, where
-∆Hvap/R is the slope of the line and C is the y-intercept
Useful Information
ln P  
H vap
RT
C
We can easily manipulate this equation to arrive at the more familiar
form of the equation. We write this equation for two different
temperatures:
H vap
H vap
ln
P


C
ln P1  
C
2
RT2
RT1
Subtracting these two equations, we find:
ln P1  ln P2  
H vap
 H vap  H vap  1 1 
 
  
  
RT1 
R2 
R  T2 T1 
Common Applications
 Calculate the vapor pressure of a liquid at
any temperature (with known vapor
pressure at a given temperature and known
heat of vaporization)
 Calculate the heat of a phase change
 Calculate the boiling point of a liquid at a
nonstandard pressure
 Reconstruct a phase diagram
 Determine if a phase change will occur
under certain circumstances
An example of a phase diagram
Real World Applications
 Chemical engineering
 Determining the vapor pressure of a
substance
 Meteorology
 Estimate the effect of temperature on
vapor pressure
 Important because water vapor is a
greenhouse gas
Shortcomings
 The Clausius-Clapeyron can only give
estimations
 We assume changes in the heat of
vaporization due to temperature are
negligible and therefore treat the heat of
vaporization as constant
 In reality, the heat of vaporization does
indeed vary slightly with temperature