Thermodynamics and Statistical
Mechanics
Equations of State
Thermodynamic quantities
Internal energy (U): the energy of atoms or
molecules that does not give macroscopic
motion.
Temperature (T): a measure of the internal
energy of a system.
Heat (Q): a way to change internal energy,
besides work. (Energy in transit.)
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Laws of Thermodynamics
First law:
đQ – đW = dU
Q – W = DU
Energy is conserved
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Work done by a gas
dW  Fds
F
dW  Ads
A
dW  PdV
Vf
W   PdV
Vi
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Work done by a gas
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Work done by a gas
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Configuration Work
Product of intensive variable times
corresponding extensive variable:
đW = xdY
Gas, Liquid, Solid: PdV
Magnetic Material: BdM
Dielectric Material: EdP
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Equation of State
For an ideal gas: PV = nRT
P = pressure (N/m²)(or Pa)
V = volume (m³)
n = number of moles
T = temperature (K)
R = gas constant (8.31 J/(K·mole))
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Ideal gas law
Ideal gas law:
PV = nRT
In terms of molar volume, v = V/n,
this becomes:
Pv = RT, or P = RT/v
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Real Substance
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Real Substance
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van der Waals equation of state
v cannot be decreased indefinitely, so replace
v by v – b. Then,
RT
P
vb
Next account for intermolecular attraction
which will reduce pressure as molecules are
forced closer together. This term is
proportional to v-2
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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 .
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van der Waals equation of state
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Critical Values
vC  3b
8a
TC 
27 Rb
a
PC 
2
27b
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van der Waals equation of state
This can be expressed in term of dimensionless
coordinates, P', v', and T ' with the following
Substitutions:
v  vC v' ,
P  PC P' ,
T  TCT '
Then,
 P' 3  v' 1   8 T '


2 
v'  3  3

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van der Waals equation of state
This can also be written,
8T '
3
P' 
 2
3v'1 v'
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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  VbDT
 T  P
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Thermal Expansion
Usually, b is positive.
An exception is water in the temperature range
between 0° C and 4° C.
Range of b is about:
b  10-3 for gasses.
b  10-5 for solids.
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Linear Expansion
Coefficient of Linear Expansion, a.
1  X 
a  
X  T  P
a (T , p )
X 

DX  
 DT  XaDT
 T  P
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Relationship Between a and b
V '  V  DV  V  VbDT  V (1  bDT )
V  XYZ
V '  X (1  aDT )Y (1  aDT ) Z (1  aDT )
 XYZ (1  aDT )  V (1  3aDT )
3
b  3a
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Compressibility
Volume also depends on pressure.
Isothermal Compressibility:
1  V 
   
 (T , P)
V  P T
V 

DV    DP  VDP
 P T
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Bulk Modulus
P 

Bulk Modulus   V  

 V T
1
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A Little Calculus
Consider, V (T , P)
V 
V 


dV    dT    dP
 T  P
 P T
If, V  const, dV  0
 V  dT   V  dP  0
  V   V
 T  P
 P T
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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
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Application
Suppose you need:
 P 
 
 T V
 V   P   T   1
     
 P T  T V  V  P
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Application
V 
1  V 

 
 
1
T  P V  T  P b
 P  




 
1  V  
 V 
 T V  V   T 
  
   
 
V  P T
 P T  V  P  P T
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