________ Gases • have non-zero volume at low T and high P • have repulsive and attractive forces between molecules short range, important at.
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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 BP 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 yx
y constant
For an increase y in y,
A2 xy
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
x0
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
xy yx
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 yx
x
___
B
___ x
___
B
x y xy
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