Transcript ________ Gases • have non-zero volume at low T and high P • have repulsive and attractive forces between molecules short range, important at.

________ Gases
• have non-zero volume at low T and high P
• have repulsive and attractive forces between molecules
short range,
important at ________ P
longer range,
important at ________ P
At low pressure, molecular volume and intermolecular
forces can often be neglected, i.e. properties  ideal.
________ Equations
 B C
PV  RT 1   2 
 V V


PV  RT 1  BP  C P 2 
V  Vm 
V
n

B is the second ________ ________.
C is the third ________ ________.
They are temperature dependent.
_____ ___ ______ Equation
a 

P

V  b   RT

2 
V 

© Paul Percival
Modified by Jed Macosko
11/6/2015
________ Factor
also known as ________ factor
Z
PV
V

RT Videal
C2H4
CH4
2.0
H2
NH3
Z
1.0
0
P
The curve for each
gas becomes more
________ as T  
T1
T2
Z
T3
1.0
P
© Paul Percival
Modified by Jed Macosko
11/6/2015
The van der Waals Equation 1
a 

P

V  b   RT

2 
V 

Intermolecular attraction
= “________ pressure”
“molecular volume”
 ________ volume
4
3
P
RT
a
 2
V b V
Z
PV
V
a


RT V  b RTV
 1
  2r  / 2  23 3
3
1 
a 
a 
a  2
b

b 
P 

P 
3
RT 
RT 
RT

 RT  
(do the algebra)
1 
a 
 Z 
   
b


  ...
RT 
 P T RT 
The initial slope depends on a, b and T:
• ______ for
b  a / RT
________ size dominant
• ______ for
b  a / RT
T  a / Rb
________ dominant
• ______ at
© Paul Percival
________ Temperature
~ ideal behaviour over wide range of P
Modified by Jed Macosko
11/6/2015
________ of Gases
Real gases ________ … don’t they?
P
supercritical
fluid
Pc
P2
Tc
T2
T1
P1
liquid
Vc
V
gas
Tc, Pc and Vc are the ________ constants of the gas.
Above the ________ temperature the gas and liquid phases
are continuous, i.e. there is no interface.
© Paul Percival
Modified by Jed Macosko
11/6/2015
The van der Waals Equation 2
The van der Waals Equation is not exact, only a model.
a and b are ________ constant.
RT  2 a
ab

V 3  b 
V

V

0

P 
P
P

P
The ________ form
of the equation
predicts
3 solutions
P
RT
a
 2
V b V
0
b
V
There is a point of ________ at the critical point, so…
RT
__ a
 P 



0


__
__
 V T
V  b  V
slope:
curvature:

 2 P 
__ RT
__ a


0
 V 2 
__
__
V

T V  b 
Pc 
a
__ b 2
Zc 
PV
c c
 __
RTc
© Paul Percival
Vc  __ b Tc 
__ a
__ Rb
TB 
Modified by Jed Macosko
a __
 Tc
Rb __
11/6/2015
The Principle of Corresponding States
__________ variables are dimensionless variables
expressed as fractions of the critical constants:
Pr 
P
Pc
Vr 
V
Vc
Tr 
T
Tc
Real gases in the same state of _______ volume and _________
temperature exert approximately the same _________ pressure.
They are in corresponding states.
If the van der Waals Equation is written in reduced variables,

3 
 Pr  V 2   3Vr  1  8Tr

r 
Since this is __________ of a and b, all gases follow the
same curve (approximately).
Tr = 1.5
1.0
Tr = 1.2
Z
Tr = 1.0
Pr
© Paul Percival
Modified by Jed Macosko
11/6/2015
Partial Differentiation
for functions of more than one variable: f=f(x, y, …)
A  xy
Take _______as an example
For an increase x in x,
A1  yx
y constant
For an increase y in y,
A2  xy
x constant
For a simultaneous increase
____
x
A   x  x  y  y   xy
 ____  ____  ____
A
A
 1 ____  2 ____  ____
x
y
____
A2
A  xy
A1 y
In the limits x  0, y  0
 ____ 
 ____ 
A  dA  
dx


 y  dy

x

y

x
total differential
__________ differential
for a real single-value function f of two independent variables,
 f  x  x, y   ________ 
 f 

lim

  x0 

x

x
 y


© Paul Percival
Modified by Jed Macosko
11/6/2015
Partial Derivative Relations
Consider f ( x, y, z )  0, so z  z ( x, y )
 z 
 z 
dz    dx    dy
 x  y
 y  x
• Partial derivatives can be taken in __________ .
   z  
   z  
       
 x  y  x  y  y  x  y  x
2 z
2 z

xy yx
1
• Taking the inverse:
 z  
 x 

 
  

x


 z  y
y

• To find the __________ partial derivative:
 ___ 
 ___ 
dz  ___  
dy



 dx

 x  y
 y  x
 z / y  x  z   ___ 
 ___ 
 y     z / x     ___   z 
y

z

x 
y
 x   ___   ___ 
• __________ Rule: 

  1



 ___  z  ___  x  x  y
and
© Paul Percival
 y   ___   ___ 
 ___   ___   y   1

z 
y 
x
Modified by Jed Macosko
11/6/2015
Partial Derivatives in Thermodynamics
From the __________ equation of state for a __________ system,
f  P,V ,T   0
__________ partial derivatives can be written:
1
but given the ______ inverses, e.g
and the __________ rule
 V  
 T 



 T  
 V  P

P
 V   T   P 

   
  1
 T  P  P V  V T
there are only two __________ “basic properties of
matter”. By convention these are chosen to be:
the coefficient of __________
expansion (isobaric), and

the coefficient of __________ __________ .
1  V 


V  T  P
1  V 
 

V  P T
The third derivative is simply
 V / T  P 
 P 

  

T

V
/

P

T 
 V
© Paul Percival
Modified by Jed Macosko
11/6/2015
The __________ Relation
Suppose
z  A x, y  dx  B  x, y  dy
Is z an exact differential, i.e. dz?
dz is exact provided
because then
 A   B 
 y    x 
 x   y
crossdifferentiation
 ___ 
A

 ___  y
___
 A 

 y  yx
 x
 ___ 
B

 ___  x
___
 B 

 
 x  y xy
The corollary also holds (if exact, the above relations hold).
__________ functions have exact differentials.
__________ functions do not.
New thermodynamic relations may be derived from the
__________ relation.
e.g. given that
dU  TdS  PdV
it follows that
 ___ 
 ___ 






 V  ___
 S  ___
© Paul Percival
Modified by Jed Macosko
11/6/2015