Thermodynamics and Statistical
Mechanics
First Law of Thermodynamics
Review of van der Waals Critical
Values
vC  3b
8a
TC 
27 Rb
a
PC 
2
27b
Thermo & Stat Mech - Spring 2006
Class 3
2
van der Waals Results
vc  3b
1 a
Pc 
2
27 b
8 a
Tc 
27 bR
1 a
3b
2
Pc vc 27 b
3

  0.375
RTc R 8 a
8
27 bR
Thermo & Stat Mech - Spring 2006
Class 3
3
van der Waals Results
Substance
Pcvc/RTc
He
0.327
H2
0.306
O2
0.292
CO2
0.277
H2O
0.233
Hg
0.909
Thermo & Stat Mech - Spring 2006
Class 3
4
Configuration Work
đW = PdV
Gas, Liquid, Solid:
Thermo & Stat Mech - Spring 2006
Class 3
5
Kinds of Processes
Often, something is held constant. Examples:
dV = 0
dQ = 0
dP = 0
dT = 0
isochoric or isovolumic process
adiabatic process
isobaric process
isothermal process
Thermo & Stat Mech - Spring 2006
Class 3
6
Work done by a gas
đW = PdV
Vf
W   PdV
Vi
For isochoric process dV = 0,
so W = 0
For isobaric process dP = 0,
so W = PDV
Thermo & Stat Mech - Spring 2006
Class 3
7
Work done by a gas
Thermo & Stat Mech - Spring 2006
Class 3
8
Work done by an ideal gas
Vf
W   PdV
Vi
For isothermal process dT = 0,
so T = constant.
RT
P
V
Thermo & Stat Mech - Spring 2006
Class 3
9
Isothermal Process
Vf
W  RT 
Vi
dV
V
W  RT ln V 
Vf
Vi
V f 
W  RT ln  
 Vi 
Thermo & Stat Mech - Spring 2006
Class 3
10
Heat Capacity
Heat capacity measures the amount of heat that
must be added to a system to increase its
temperature by a given amount. Its definition:
dQ 

Cy   
 dT  y
where y is a property of the system that is kept
constant as heat is added.
Thermo & Stat Mech - Spring 2006
Class 3
11
Heat Capacity
Properties that are usually kept constant for a
hydrostatic system are volume or pressure.
Then,
dQ 
dQ 


CV  
 , or CP  

 dT V
 dT  P
Thermo & Stat Mech - Spring 2006
Class 3
12
Heat Capacity
Clearly, the heat capacity depends on the size
of the system under consideration. To get rid of
that effect, and have a heat capacity that
depends only on the properties of the substance
being studied, two other quantities are defined:
specific heat capacity, and molar heat capacity.
Thermo & Stat Mech - Spring 2006
Class 3
13
Specific Heat Capacity
Specific heat capacity is the heat capacity per
mass of the system. A lower case letter is used
to represent the specific heat capacity. Then, if
m is the mass of the system,
CV 1  dQ 
CP 1  dQ 
cV 
 
 
 , or cP 

m m  dT V
m m  dT  P
Thermo & Stat Mech - Spring 2006
Class 3
14
Molar Heat Capacity
Molar heat capacity is the heat capacity per
mole of the system. A lower case letter is used
to represent the molar heat capacity. Then, if
there are n moles in the system,
CV 1  dQ 
CP 1  dQ 
cV 
 
 
 , or cP 

n n  dT V
n n  dT  P
Thermo & Stat Mech - Spring 2006
Class 3
15
Shorter Version
Use heat per mass.
dq 
dq 


cV    , or cP   
 dT V
 dT  P
Thermo & Stat Mech - Spring 2006
Class 3
16
cP – cV
đq = du + Pdv
where u(T,v)
u 
u 


du    dT    dv
 T v
 v T
u 
 u 


dq    dT     P  dv
 T v
 v T

Thermo & Stat Mech - Spring 2006
Class 3
17
Constant Volume
dq   u 
du

cv       , often
dT
 dT v  T v
Thermo & Stat Mech - Spring 2006
Class 3
18
Constant Pressure
 dq    u    u   P  dv 
     
 

 dT  p  T v  v T
 dT  P
 u 

c p  cv     P  v
 v T

 u 

c p  cv     P  v
 v T

Thermo & Stat Mech - Spring 2006
Class 3
19
Ideal Gas
u is not a function of v.
cP  cv  R
Thermo & Stat Mech - Spring 2006
Class 3
20
Adiabatic Process
For an ideal gas, and most real gasses,
đQ = dU + PdV
đQ = CVdT + PdV,.
Then, when đQ = 0,
PdV
dT  
CV
Thermo & Stat Mech - Spring 2006
Class 3
21
Adiabatic Process
For an ideal gas, PV=nRT, so
PV
PdV  VdP
T
, and dT 
nR
nR
Then,
PdV PdV  VdP


CV
nR
Thermo & Stat Mech - Spring 2006
Class 3
22
Adiabatic Process
pV
PdV  VdP
T
, and dT 
nR
nR
Then,
1
PdV PdV  VdP
1  VdP
0

 PdV    
CV
nR
CV nR  nR
 nR  CV  VdP
0  PdV 


 CV nR  nR
Thermo & Stat Mech - Spring 2006
Class 3
23
Adiabatic Process
 nR  CV 
0  PdV 
 VdP

 CV 
nR  CV  CP
CP 
0  PdV    VdP   PdV  VdP
CV 
Cp
where,  
CV
Thermo & Stat Mech - Spring 2006
Class 3
24
Adiabatic Process
 PdV  VdP  0
dV dP


 0, which can be integrated ,
V
P
 ln V  ln P  constant
ln V  ln P  ln PV


  constant

PV  constant
Thermo & Stat Mech - Spring 2006
Class 3
25
Adiabatic Process

PV  constant
With the help of PV  nRT
this can also be expressed as,
TV
 1
 constant

T
 1  constant
P
Thermo & Stat Mech - Spring 2006
Class 3
26
 for “Ideal Gasses”
  1
2

2
monatomic :   1   1.67
3
2
diatomic :   1   1.40
5
2
polyatomic :   1   1.33
6
Thermo & Stat Mech - Spring 2006
Class 3
27