1
5. Statistical Thermodynamics
The objective of this section (and Chapter 15 of the text) is to show how the macro
scopic properties of molecular systems which you learned about in your study of
classical thermodynamic may be described in terms of the quantities and variables of
statistical mechanics. In doing this, we will need to draw on the definitions of and
relationships among the properties and variables of classical thermodynamics.
Recall that
•
U
is the total internal energy of the system
•
The
First Law of Thermodynamics
states that changes in
U
are defined as
Δ
U
=
q
heat
+
w
where
q
heat
is the amount of heat energy which passively flows into (positive
q
heat
) or out of (negative
q
heat
) the system, and
w
is the amount of energy which
flows into (positive
w
) or out of (negative
w
) the system due to some type of work
(mechanical,
PV
expansion/compression, electrical, . . . etc.) being done by/on
the system.
•
While
U
is a
function of state
,
q
heat
and
w
are not.
The value of
w
associated with an infinitesimal expansion or compression of the
system is
dw
=

P
opp
dV
where
P
opp
is the pressure which is
opposing
the volume change
dV
.
•
Enthalpy
is defined as
H
=
U
+
PV
is also a function of state.
•
Entropy
is a function of state defined by the fact that
dS
=
q
rev
T
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•
The Second Law of Thermodynamics
states that in any “spontaneous”
(or “natural” or “irreversible”) process
dS
>
dq
T
•
The
Helmholtz free energy
is defined as
A
=
U

TS
•
The
Gibbs free energy
is defined as
G
=
H

TS
•
For infinitesimal reversible changes in a closed system with only
PV
work:
dU
=
T dS

P dV
=
∂U
∂S
V,N
dS
+
∂U
∂V
S,N
dV
(1)
dH
=
T dS
+
V dP
=
∂H
∂S
P,N
dS
+
∂H
∂P
S,N
dP
(2)
dA
=

S dT

P dV
=
∂A
∂T
V,N
dT
+
∂A
∂V
T,N
dV
(3)
dG
=

S dT
+
V dP
=
∂G
∂T
P,N
dT
+
∂G
∂P
T,N
dP
(4)
5.1
Internal Energy
U
and Heat Capacity
C
V
We recall from Unit 3 that over an ensemble of
N
equivalent systems, the population
in state
i
is defined by the canonical distribution function
a
i
=
N
q
e

β ε
i
the fraction of systems in state
i
is
p
i
=
a
i
N
=
1
q
e

β ε
i
and that the ensemble average of any property
i
if state
i
is
h
f
i
=
f
=
X
states
i
p
i
f
i
=
1
q
(
N, V, β
)
X
states
i
(
f
i
e

β ε
i
)
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 Winter '10
 Prof.Djasd
 Thermodynamics, Mole, Statistical Mechanics, Entropy, WMP

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