Transcript Document

Partial Derivatives and their use in describing equations of state. You need to be able to
obtain the total differential of P:
 P 
 P 
 P 
dP  
dn
 dV    dT   
 V  n,T
 T  n,V
 n V ,T
For an ideal gas PV  nRT
dP  
nRT
V


2
dV 
And for a van der Waals gas, where  P 
The total differential is: dP 
nR
RT
dT 
dn
V
V
a 
V  b   RT
V2

R
RT
2a 
dT  

 dV
2
3
V b
 V  b  V 
2
2


  2U 
1
 y 
     U   , and  z   z
The useful relations:   
,
 VT 
yx xy
 x  a ,b  x 

 n  V  T V ,n  T ,n

y

 a ,b
Know how to prove exact and inexact differentials.
You should all know the basic equation of state dealing with ideal gases. The other
important ones are listed below.
You should be able to derive the cubic form of the van der Waals equation:
RT  2 a
ab

V 3  b 
0
V  V 
P 
P
P

In terms of the compressibility factor:
Z
V
a

V  b RTV
RT
A
 1/ 2
V  B T V V  B 
RT 2  2 BRT
A 
AB
V3
V B 
 1 / 2 V  1 / 2  0
P
P
T P
T P

P
The Redlich-Kwong equation will be given:
The cubic form is:
You should appreciate how cubic equations are solved using the Newton-Raphson
method (you will not be asked to actually solve one in an exam).
xn 1  xn 
f  xn 
f '  xn 
You need to know the most fundamental equation of sate, the virial equation of state. In
particular, we consider the second virial coefficient to be the most important.
Z
and
B T 
PV
 1  2V
 1  B2 P T P
RT
V

B2V T   RTB2 P T   2N A 0 e U r  / k
Note: Although not mentioned,

B
T

 1 r 2 dr  V  VIDEAL
Know the various contributions to the Lennard-Jones potential.
  12    6 
u r   4      
 r  
 r 
Know the graphical form of the above equation and the interpretation of  and ε.
For the long range attractive interactions, these are usually dipole-dipole, dipoleinduced dipole, and London dispersion.
Simple potential models can be used to solve for the second virial coefficient.
For the hard-sphere potential
U r   
U r   0
r 
r 
and B2V ( T ) 
U r    r  
U r      r  
U r   0 r  
For the square-well potential
a B2V ( T ) 
 
2 3 N A
3

2 3 N A
1  3  1 e / k
3
B
T

1
We can write the second virial coefficient in terms of the cubic form of the van der
Waals equation. Once this is done, we can use a hard-sphere/Lennard Jones hybrid
potential to solve the integral for B2V(T). This allows one to determine the molecular
parameters a and b of the van der Waals equations. Please make sure you understand
this process. It’s given in pages 84 and 85 in your textbook.
You should know the molecular interpretation of a and b (including the full derivation
based on a hybrid hard-sphere and Lennard-Jones potential).
a
2N A2 c6
3 3
and b 
You need to learn the complete proof for this.
2 3 N A
3
Ideal Gases and the Compressibility Factor
1 mol, 300K for various gases
A high pressure, molecules are more
influenced by repulsive forces.
Vreal > Videal
Z>1
The effect of molecular attraction causes:
Vreal < Videal
Z<1
Effect of Pressure on Compressibility Factor
CH4, various temperatures
Lower temperatures
Little thermal motion of molecules
Attractive forces dominate:
Vreal < Videal
Z<1
Higher temperatures
More influenced by repulsive forces:
Vreal > Videal
Z>1
Comparing Various Equations of State
Ethane, 400K
van der Waals
Redlich-Kwong
Peng-Robinson
Solid line – experimental
Second Virial Coefficient and Intermolecular Potentials
B2V T 
Z  1
V
The second virial coefficient is the most important.
Can be determined by plotting Z vs. P.
Ammonia B2V = -0.142 dm3 mol-1 at 373 K
Ideal gas
1.000
Z
373 K
slope = B2P(T) = B2V(T)/RT
0.998
0
0.1
P / bar
The effect of temperature
At high T
B2V increases
Boyle temperature: Temperature at which B2V(T) = 0.
Repulsive and attractive interactions cancel and gas behaves ideal


B2V T   2N A  e
0
 u ( r ) / k BT

 1 r dr
2
The Lennard-Jones Potential
  12   6 
u r   4      
 r  
 r 
The short range
1/r12
repulsive term
The long range
1/r6 attractive term
Contributions:
Dipole-dipole (including H-bonding)
Induced dipole
London dispersion attraction
u(r)/ε
0
ε
0
1
2
3
r/σ