Thermodynamics and Statistical
Mechanics
Review for Quiz 1
Laws of Thermodynamics
First law:
đQ – đW = dU
Energy is conserved
Thermo & Stat Mech - Spring 2006
Class 11
2
Laws of Thermodynamics
Second Law: The entropy of an isolated
system increases in any irreversible process
and is unaltered in any reversible process.
This is the principle of increasing entropy.
DS  0
Thermo & Stat Mech - Spring 2006
Class 11
3
Laws of Thermodynamics
Third Law: The entropy of a true equilibrium
state of a system at a temperature of absolute
zero is zero.
Equivalent to: It is impossible to reduce the
temperature of a system to absolute zero
using a finite number of processes.
Thermo & Stat Mech - Spring 2006
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4
Second Law Variations
No series of processes is possible whose sole
result is the absorption of heat from a thermal
reservoir and the complete conversion of this
energy to work.
There are no perfect engines!
Thermo & Stat Mech - Spring 2006
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5
Second Law Variations
No series of processes is possible whose sole
result is the transfer of heat from a reservoir at
a given temperature to a reservoir at a higher
temperature.
There are no perfect refrigerators!
Thermo & Stat Mech - Spring 2006
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Zeroth Law
If two systems are separately in thermal
equilibrium with a third system, they are
in thermal equilibrium with each other.
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Work done by a gas
dW  Fds
F
dW  Ads
A
dW  PdV
Vf
W   PdV
Vi
Thermo & Stat Mech - Spring 2006
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8
Ideal gas law
Ideal gas law:
PV = nRT
In terms of molar volume, v = V/n,
this becomes:
Pv = RT, or P = RT/v
Thermo & Stat Mech - Spring 2006
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van der Waals equation of state
RT
a
Then, P 
 2 , or
vb v
 P  a v  b   RT

2
v 

This equation has a critical value of T which
suggests a phase change. The next slide shows
graphs for several values of T .
Thermo & Stat Mech - Spring 2006
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10
Thermal Expansion
Expansivity or Coefficient of Volume
Expansion, b.
1  V 
1  v 
b     
V  T  P v  T  P
b (T , P )
V 

DV    DT  Vb DT
 T  P
Thermo & Stat Mech - Spring 2006
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11
Compressibility
Volume also depends on pressure.
Isothermal Compressibility:
1  V 
   
 (T , P)
V  P T
V 

DV    DP  VDP
 P T
Thermo & Stat Mech - Spring 2006
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Cyclical Relation
 V    V   P   0
     
 T  P  P T  T V
 V    V   P 
 
   
 T  P
 P T  T V
 V   P   T   1
     
 P T  T V  V  P
Thermo & Stat Mech - Spring 2006
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13
Carnot Cycle
A Carnot cycle is an idealized reversible cycle
that operates between two heat reservoirs at
temperatures T1 and T2, where T2 > T1. It can
operate as a heat engine, or a refrigerator.
Thermo & Stat Mech - Spring 2006
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Thermal Efficiency (h)
W Q2  Q1
Q1
h

 1
Q2
Q2
Q2
T1 T2  T1
h  1 
T2
T2
If T1 = 0, h = 1 (100%)
Thermo & Stat Mech - Spring 2006
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For a Carnot Engine
Q2 T2

Q1 T1
Q2
T2

Q1
T1
Q T
or Q1  Q2  0
T1
Thermo & Stat Mech - Spring 2006
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T2
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Entropy
dQi
dQ
i T   T  0
i
dQ
 dS
T
For reversible processes.
Entropy is a state variable.
Thermo & Stat Mech - Spring 2006
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First and Second Laws
First Law: dU = đQ – đW
First law, combined with the second law:
dU = TdS – PdV
Thermo & Stat Mech - Spring 2006
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Tds Equations
Tb
 P 
Tds  cv dT  T 
dv
 dv  cv dT 

 T  v
 v 
Tds  cP dT  T 
 dP  cP dT  TvbdP
 T  P
cv
cP
 T 
 T 
Tds  cP 
dv 
dP
 dv  cv 
 dP 
bv
b
 v  P
 P v
Thermo & Stat Mech - Spring 2006
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Ideal Gas
c p  cv 
Tvb
2

Pv
1
c p  cv  Tv 2 P 
T
T
c p  cv  R
Thermo & Stat Mech - Spring 2006
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Properties
From first law: TdS = dU + PdV, or
Internal Energy dU = TdS – PdV
U(S, V)
Enthalpy:
H = U + PV
dH = TdS + VdP
Thermo & Stat Mech - Spring 2006
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H(S, P)
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New Potentials
Helmholtz Function:
F = U – TS
Gibbs Function:
G = U – TS + PV
G = H – TS
G = F + PV
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All Four
dU = TdS – PdV
dH = TdS + VdP
dF = – PdV – SdT
dG = – SdT + VdP
Thermo & Stat Mech - Spring 2006
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U(S, V)
H(S, P)
F(V, T)
G(T, P)
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Maxwell Relations
 T 
 P 

   
 V  S
 S V
 S   P 

 

 V T  T V
 T   V 

 

 P  S  S  P
 S   V 
   

 P T  T  P
Thermo & Stat Mech - Spring 2006
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Clausius-Clapeyron Equation
 23
 dP 

 
 dT  23 T (v  v)
 13
 dP 

 
 dT 13 T (v  v)
 12
 dP 

 
 dT 12 T (v  v)
Liquid - vapor
Solid - vapor
Solid - liquid
Thermo & Stat Mech - Spring 2006
Class 11
25