ENTROPY AND THE 2nd LAW
1
2
2
1
T bath
dS ≥ 0
dS1 ≥ dQ1 / Tbath
8.044 L10B1
S as a State Function
Note: adiabatic (≡ d
/Q = 0)
⇒ constant S if the
change is quasistatic. This is the origin of the sub­
script S on the adiabatic compressibility.
1 ∂V
κT ≡ −
V ∂P T
1 ∂V
κS ≡ −
V ∂P S
8.044 L10B2
Example A Hydrostatic System
dS =
∂S
∂S
dT +
dV
∂V T
∂T V
1
P
=
dU + dV
T
T
=
by expansion
from dU = T dS − P dV
1 ∂U
1
dT +
T
T ∂T V
∂U
+ P dV
∂V T
by expansion of U
But the cross derivatives of S must be equal.
8.044 L10B3
∂ 1 ∂U
1 ∂ 2U
=
∂V T ∂T V T
T ∂V ∂T
∂ 1
∂T T
∂U
+P
∂V T
1
=− 2
T
V
1 ∂P
∂U
1 ∂ 2U
+P +
+
∂V T
T ∂V ∂T
T ∂T V
Equating these two expressions gives
∂U
∂P
+P =T
∂V T
∂T V
∂U
∂P
=T
−P
∂V T
∂T V
New Information! Does not contain S!
8.044 L10B4
CONSEQUENCES a) γ
�
�
��
�
�
∂U
∂U
d
/Q = dU + P dV =
dT +
+ P dV
T
' ∂T
V. V
' ∂V V.
CV
T ∂P
∂T
�
�
∂P
d
/Q
≡ CP = CV + T
dT P
∂T V
�
V
�
∂V
' ∂T
V. P
αV
�
�
�
�
1 ∂V
1 ∂V
Use α ≡
and κT ≡ −
.
V ∂T P
V ∂P T
8.044 L10B5
∂P
−1
=
∂T
∂V
∂T V
∂V
∂P
P


=
(−1) ∂V
∂T
T
2V
α
T
α
CP − CV = T   αV =
κT
κT
∂V
∂P T
P
α
=
κT
T α2 V
→ γ−1=
κT CV
For an ideal gas P V = N kT ⇒ α = 1/T and κT = 1/P .
Thus
V /T
= Nk
CP − CV =
1/P
This holds for polyatomic as well as monatomic gases.
8.044 L10B6
CONSEQUENCES b) Ideal Gas: CV
N kT
P =
V
∂U
Nk
=T
−P =P −P =0
∂V T
V
∂U
∂U
dU =
dT +
dV = CV dT
∂T
∂Vv- T
v- V
CV
U =
� T
0
0
CV (T ', V ) dT ' + constant
(∂U/∂V )T = 0 for all T ⇒ CV is not f (V ); CV = CV (T ).
8.044 L10B7
CONSEQUENCES c) Ideal Gas: S
�
�
�
�
∂S
∂S
dS =
dT +
dV
∂T V
∂V T
�
�
∂S
d
/Q
≡ CV = T
dT V
∂T V
d
/Q = T dS
dU
P
dU = T dS − P dV ⇒ dS =
+ dV
T
T
�
�
�
�
∂S
1 ∂U
P
=
+
.
∂V T
T - ∂V1L T
T
-1L
0
N k/V
8.044 L10B8
Nk
CV (T )
dS =
dT +
dV
T
V
CV (T ')
V
'
dT + N k ln( ) + S(T0, V0)
S(T, V ) =
'
T0
T
V0
T
For a monatomic gas CV = (3/2)N k.
T
V
S(T, V ) − S(T0, V0) = (3/2)N k ln( ) + N k ln( )
T0
V0

=

V 
N k ln 
T
V0 T0
3/2



8.044 L10B9
V
isentropic (adiabatic) ⇒


T 3/2 








V 2/3T
V 5/3P










are constant
8.044 L10B10
Maxwell Relations
dE(S, V ) =
∂E
∂E
dS +
dV
∂S V
∂V S
= T dS − P dV
expansion
1st and 2nd laws
∂T
∂P
⇒
=−
∂V S
∂S V
8.044 L10B11
dE(S, L) = T dS + FdL
⇒
dE(S, M ) = T dS + HdM ⇒
∂T
∂F
=
∂L S
∂S L
∂T
∂H
=
∂M S
∂S M
Observe:
d(T S) = T dS + SdT
d(P V ) = P dV + V dP
8.044 L10B12
Helmholtz Free Energy
F ≡ E − TS
dF = −SdT − P dV
Enthalpy
⇒
∂S
∂P
=
∂T V
∂V T
H ≡ E + PV
dH = T dS + V dP
⇒
∂T
∂V
=
∂P S
∂S P
8.044 L10B13
Gibbs Free Energy
G ≡ E + PV − TS
dG = −SdT + V dP ⇒
∂S
∂V
−
=
∂T P
∂P T
E, F , H and G are called ”thermodynamic potentials”.
8.044 L10B14
The Magic Square Mnemonic
x
E
-V
F
dE = T dS + Xdx
S
H
-T
P
G
∂S
∂V
(−1)
= (−1)(−1)
��
� ∂T
∂P T
P
(+1)
X
8.044 L10B15
MIT OpenCourseWare
http://ocw.mit.edu
8.044 Statistical Physics I
Spring 2013
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