Chemistry
Lecture 10
Maxwell Relations
NC State University
Thermodynamic state functions
expressed in differential form
We have seen that the internal energy is
conserved and depends on mechanical (dw) and
thermal (dq) energy. Given that pressure-volume
work done by the system is dw = -PdV and that the
entropy change can be expressed as dS = dq/T (i.e. dq
= TdS) we can express the internal energy as
dU = TdS – pdV
By a transformation of variables we define H = U + PV
or dH = d(U+PV) = dU + PdV + VdP. Therefore,
dH = TdS + VdP
These two expressions are known as total derivatives.
The total derivative
The total derivative permits more than one
variable to change at one time. We can write
the state functions U and H in the following
form to make it clearer what the variable
dependencies are:
U
U
which implies that
dU =
dS +
dV
T = U
S
T = H
S
, P = – U
V S
, V = H
P S
V S
S V
dH = H dS + H dP
P S
S P
V
P
Definition of an exact differential
In the general case we can write a total derivative as:
dF = Fxdx + Fydy
or
dF = F dx + F dy
x y
y x
If the total derivative is an exact differential then the
second derivatives of both terms in the differential are
equal.
Fy
Fx
F
F
=
=
Fyx =
=
= Fxy
y
yx xy x
Second derivatives of the
internal energy function
The differential form of the internal energy is:
dU = TdS -PdV
Using the definitions we determined above:
T = U
S
V
, P = – U
V
S
The second derivatives can be written as:
T = U = U = – P
V VS SV
S
Second derivatives of the
enthalpy function
The differential form of the enthalpy is:
dH = TdS + VdP
Using the definitions we determined above:
T = H
S
,V =
P
H
P
S
The second derivatives can be written as:
T = H = H = V
P PS SP S
Second derivatives of the Gibbs
free energy function
The differential form of the Gibbs energy is:
dG = VdP - SdT
Using the definitions we determined above:
V = G
P
, –S =
T
G
T
P
The second derivatives can be written as:
– S = G = G = V
P
PT
TP
T
Second derivatives of the
Helmholtz free energy function
The differential form of the Gibbs energy is:
dA = - PdV - SdT
Using the definitions we determined above:
– P = A
V
T
, –S = A
T
V
The second derivatives can be written as:
– P = A = A = – S
T
TV
VT
V
Thermodynamics
Applications of Exact Differentials
Heat Capacity of Liquids and Solids
NC State University
Thermodynamic equation of state
From the master equation for internal energy:
We can take the derivative with respect to volume at constant
temperature to obtain:
We can use one of the Maxwell relations to obtain:
The equation is known as a thermodynamic equation of state.
It relates the internal energy dependence on volume only
to temperature and pressure.
The significance of the internal
pressure
The calculation of the internal pressure is:
We can see that this quantity is a correction to the ideal
gas law that depends on intermolecular interactions
described by the a coefficient in the van der Waal’s equation
of state.
Dependence of internal energy
on changes in volume and temperature
We have seen from the first law that the internal energy
change in the system is equal to the work done and the
heat transferred. Work depends on volume changes and
heat transfer leads to changes in temperature. Thus, it is
logical that the initial description of internal energy depends
on volume and temperature. Thus, for example the dependence of U on volume at constant temperature is:
U = U + U dV
V T
And similarly the dependence of U on temperature at
constant volume is:
U = U + U dT
T V
Dependence of internal energy
on changes in volume and temperature
Putting these together we have:
U = U + U dV + U dT
V T
T V
which can be written as the total derivative:
dU = U dV + U dT
V T
T V
This called the total derivative. The idea is that it includes the
variables in a space. Here the space is V,T space. We shall
see that these are not the only possible variables that can
describe the dependence of U.
Dependence of internal energy
on changes in volume and temperature
The partial derivatives are slopes in the space.
U
V
= pV ,
T
U = C
V
T V
We have already seen CV and we know it as the heat capacity
At constant volume. However, pV is new. pV is the change in
the internal energy when the volume of a substance is
changed at constant temperature.
If the intermolecular interactions are zero (ideal gas), then pV
will be zero. However, real interactions between molecules
can give rise to non-zero pV. Joules set out to measure pV
in an expansion experiment, but was not successful.
Expansion coefficient and
isothermal compressibility
In order to proceed with the next level of analysis we introduce the expansion coefficient:
 = 1 V
V T
P
The expansion coefficient is a measure of the change in
molar volume (i.e. also inverse density) of a material with
temperature. Many substances expand as the temperature
is increased, hence the name, expansion coefficient.
The isothermal compressibility is
 = – 1 V
V P T
a measure of the degree to which a substance will have a
higher density (smaller molar volume) at high pressure.
Heat capacity relationships
The molar heat capacity at constant pressure and constant
Volume are related by:
The above expression applies only to an ideal gas. We have
seen the utility of that expression in that it applies to
monatomic, diatomic and polyatomic gasses.
For liquids and solids we have the formula:
Heat capacity relationships
To derive the expression for the heat capacity difference
We return to the definition. Note that we will assume that
All extensive quantities are molar quantiites in this derivation.
Therefore
Heat capacity relationships
We can also write this as
We use Euler’s chain relation,
We find that
which gives us the result
The Joules-Thompson coefficient
The differential of the enthalpy can be expressed as:
dH = – mCPdP + CPdT
Where m is the Joules-Thomson coefficient. The enthalpy
change be written:
Therefore:
dH = H dP + H
P T
T
H
P
= – T
P
T
H
H
T
dT
P
= – mCP
P
The derivation of the above expression relies on the
permutation relation:
T
P
H
P
H
T
H
T
=–1
P
Modern measurement of the
Joules-Thompson coefficient
It is relatively difficult to perform measurements under
constant H (isenthalpic or adiabatic) conditions. Therefore,
the modern measurements are at constant temperature.
In other words,
H
P
= mT
T
is measured. Based on the expression on the previous
slide:
H = – mC
m T = – T
P
P H T P