CHAPTER 1. THERMODYNAMICS
1.4
1.4.1
19
Lecture 4: Thermodynamic potentials, Third
law
Thermodynamic potentials
As we have seen the extrema (maximum) of entropy corresponds to an equilibrium state of an isolated system (i.e. dQ = dW = 0)
dS
= 0.
(1.43)
dt
Can this concept be generalized to open systems? For this purpose in
addition to U we define three new thermodynamic potentials: Enthalpy,
Helmholtz free energy, Gibbs free energy and Landau free energy.
• Enthalpy
H = U + PV
(1.44)
When there is no heat exchange (i.e. dQ = 0) and the external force is
constant (i.e. P = const),
dH = dU + P dV + V dP Æ dU ≠ dW = 0.
(1.45)
where the equality corresponds to quasi-static processes (i.e. dW = ≠P dV ).
Then, it is convenient to express H as a function of S and P since
• Helmholtz free energy
dH = T dS + V dP.
(1.46)
A = U ≠ TS
(1.47)
When there is no external work (i.e. dW = 0) and the temperature remains
constant (i.e. T = const),
dA = dU ≠ SdT ≠ T dS Æ dU ≠ dQ = 0.
(1.48)
dA = ≠P dV ≠ SdT.
(1.49)
where the equality corresponds to reversible processes (i.e. dQ = T dS).
Then, it is convenient to express A as a function of V and T since
As an example of the variational principle consider gas at a constant
temperature T in a box of volume V divided by a sliding piston into V1 and
CHAPTER 1. THERMODYNAMICS
20
V2 . In the equilibrium state
0 = dA =
=
=
or
A
A
ˆA
ˆV1
ˆA
ˆV1
AA
A
B
T
ˆA
dV1 +
ˆV2
T
ˆA
dV1 +
ˆV2
B
B
ˆA
ˆV1
T
B
ˆA
ˆV1
A
A
A
=
T
ˆA
ˆV2
dV2
(1.50)
T
B
T
B B
ˆA
≠
ˆV2
A
B
d(V ≠ V1 )
dV1
T
B
T
and using (1.49) we can conclude that in the equilibrium state the pressures
on both sides of the piston must be equal.
• Gibbs free energy
G = U ≠ TS + PV
(1.51)
When the external force (i.e. P = const) and temperature (i.e. T = const)
remains constant,
dG = dU ≠SdT ≠T dS+P dV +V dP = dU ≠T dS+P dV Æ dU ≠dQ≠dW = 0.
(1.52)
where the equality corresponds to reversible (i.e. dQ = T dS) and quasistatic (i.e. dW = ≠P dV ) processes. Then, it is convenient to express G as
a function of P and T since
dG = V dP ≠ SdT.
(1.53)
Note that the exactness of differential dH, dA, and dG, implies
A
ˆT
ˆP
B
S
A
ˆV
ˆS
B
A
ˆP
ˆT
B
V
A
ˆS
ˆV
B
=
=
.
P
.
T
and
A
ˆV
ˆT
B
P
A
ˆS
=≠
ˆP
B
T
.
CHAPTER 1. THERMODYNAMICS
21
Other thermodynamic relations are conveniently summarized by the socalled Maxwell or thermodynamic square:
For a thermodynamic potential of interest (in bold) whose arguments are
placed in the neighboring corners (in italic) the derivative of the potential
with respect to one of its argument with other argument held fixed is determined by going along a diagonal line either with or against the direction
1 2of
ˆU
the arrow. Going against the arrow yields a minus sign (e.g. P = ≠ ˆV
,
S=≠
1.4.2
1
2
S
ˆG
).
ˆT P
Third Law
Entropy of a system is defined for all states using a reference state by connecting it with a reversible transformation. However, if the surface defined
by the equation of state is disconnected then the reversible transformations
might not exist. Moreover it is important to have definition of entropy for
distinct systems which may later come into contact with each other. The
role of the Third Law is to defines an absolute scale of entropy which defines
uniquely the entropy of an arbitrary equilibrium state of any system.
Third Law in words: The entropy of any system at
absolute zero is zero.
Third Law in symbols:
lim S(T ) = 0
(1.54)
T æ0
Consequences of the Third Law:
1. Partial derivative of S at T = 0 with respect to every thermodynamic
parameter X vanishes:
lim
T æ0
A
ˆS
ˆX
B
T
= 0.
(1.55)
CHAPTER 1. THERMODYNAMICS
22
2. Specific heat capacity CX at T = 0 with fixed thermodynamic parameter X vanishes:
lim CX = lim
T æ0
T æ0
A
ˆQ
ˆT
B
X
= lim T
T æ0
A
ˆS
ˆT
B
= 0.
(1.56)
X
3. Absolute zero cannot be reached in a finite number of steps. This
statement is often used as an alternative definition of the third law.
We will come back to it at the end of the course when the quantum
statistical mechanics is introduced.
Physical theories are always based on assumptions which very often turn out
to be false (sooner or later). Now, that you have seen all of the theoretical assumption (laws of thermodynamics) which go into a phenomenological
theory of thermodynamics it is a good time to ask which of the assumption
is likely to be wrong?
Zeroth Law (Universality of Temperature):
A ≥ B and C ≥ B ∆ A ≥ C
(1.57)
First Law (Conservation of Energy):
dU = dQ ≠ dW.
(1.58)
Second Law (Arrow of Time):
dS
Ø0
dt
(1.59)
Third Law (Quantum Mechanics):
lim S(T ) = 0
T æ0
(1.60)
Question To Go: Which law of thermodynamics is most likely to
fail?