Week 2 Wednesday January 16, 2013 page 1
Chapter 8
z = compressibility factor
Consider a graph with z in the vertical axis and P in atm in the horizontal axis. The horizontal line at z=1
is where an ideal gas would be. For CH4 at 200K, the curve starts at (0,1) and curves down below 1 then
back up above 1. For N2 at 200K, the curve starts at (0,1) and slopes upward. For H2 at 200K, the curve
starts out at (0,1) and slopes upward above the N2 curve. Below the horizontal line z=1 is where z<1.
Above the horizontal line z=1 is where z>1.
For ideal gas: PVm = RT where Vm = V/n
That’s valid at high T and low P.
z=
PVm
RT
High T or low P means molecules are far apart so we can neglect intermolecular forces and the volume
of the molecules.
κ – kappa – isothermal compressibility, different from z
For ideal gas, z = 1.
Where z<1, intermolecular attractions are dominant.
Where z>1, repulsions are dominating.
CH4 deviates a lot because of the size of the molecule (strength of intermolecular attractions).
With nonpolar molecules, intermolecular attractions increase with molecule size.
Consider a graph with z in the vertical axis and P in atm in the horizontal axis. The horizontal line at z=1
is where an ideal gas would be. For CH4 at 200K, the curve starts at (0,1) and curves way down below 1,
then steeply back up above 1. For CH4 at 500K, the curve starts at (0,1) and curves below 1, but not as
far as it did at 200K, then curves back up above 1, but not as steeply as it did at 200K. For CH4 at 2000K,
the curve starts at (0,1) and doesn’t go below z=1. Instead, it gradually curves upward. This shows that
as T increases, it behaves more like an idea gas.
H2 is tiny so it can be compressed to really high P (intermolecular forces are weak so repulsions
dominate)
Pideal = Preal + correction terms
Preal = Pideal – correction terms
Preal =
nRT n2 a
V-nb V 2
volume of real gas is less than volume of ideal gas
nb is the space occupied by molecules
V-nb is available volume
attractions reduce pressure, hence the – sign
a,b are constants, characteristic of the specific gas
P = Preal
divide 1st term by n on top and bottom
divide 2nd term by n2 on top and bottom
P=
(P+
RT
a
- 2
Vm -b Vm
a
) (Vm -b)=RT
Vm2
lim Vm
P→∞
lim P
Vm →∞
van der Waals equation
Vm =
RT
+ b=b
a
(P+ 2 )
Vm
P=
RT 1
RT
- 2=
Vm -b Vm Vm
Increasing Vm means decreasing P
Assume we have spherical molecules, such as Ne. r is the radius of one molecule. σ is sigma, the
diameter, which is 2 times r. The excluded volume around one molecule is the volume where another
molecule can’t be.
4
1
4
1
4
excluded volume for one molecule= ( πσ3 ) = ( π(2r)3 ) =4 ( πr 3 )
3
2
3
2
3
So the excluded volume is 4 times the volume of each molecule
4
b=4 ( πr 3 ) NA
3
NA is Avagadro' s number
b = 4 * atomic volume/mol or 4 * molar volume/mol
b is normally found experimentally (curve fitting from data)
van der Waals is semi-empirical equation (partly theoretical, partly experimental)
In addition to the van der Waals equation:
P=
nRT
V-nb
n2 a
1
T 2 (V+nb)V
Redlich-Kwong equation
That has different a and b from the van der Waals equation for any gas.
P=
nRT n2 a
Berthelot Equation
V-nb TV 2
Focus on van der Waals equation:
van der Waals constants:
Gas
a (
L
b(
)
mol
L2 atm
)
mol2
He
0.034
Ne
0.244
CO2
3.59
bigger molecule means larger a, due to bigger intermolecular forces
0.0237
0.0266
0.0427
bigger molecule means larger b, since the molecule is bigger
The virial series (equation)
virial means force (from Latin)
equation based on forces
z=
PVm
=1 ideal gas
RT
expand z as a power series
n/V is the reciprocal of molar volume
n
n 2
n 3
z=1+B(T) ( ) +C(T) ( ) +D(T) ( ) …
V
V
V
z=z+
1 is 1st virial coefficient
B is 2nd virial coefficient
C is 3rd virial coefficient
B(T) C(T) D(T)
+ 2 + 3 …
Vm
Vm
Vm
D is 4th virial coefficient
† is cross, means not the same
also, z = 1 + B†(T)P + C†(T)P2 + D†(T)P3…
same as other equation, but expanded based on P instead of Vm
test
B† =
B
RT
C†=
(C-B)2
R2 T 2
For P ≈ 2atm or less and moderate T, then we just have to consider the first few terms:
z = 1 + B†(T)P + C†(T)P2 ignore the rest