1. Properties of Gases Kinetic Molecular Model Speed of gas
Combining Kinetic Molecular Model and Ideal gas Law
Real gas
Virial equation
PV-plot for real gas
Van der Waals equation
Comparing ideal and real gas

2. Kinetic molecular model of Gases
Assumption
1) gas molecules in ceaseless random motion ( no phase change)
2) molecular size is negligible (no repulsion)
3) molecules do not interact (no attraction)
All gases at same T and P have the same properties

3. Definition of pressure
“ gas pressure is caused by collective collision of gas molecules with container’s wall”
Container’s wall

4. gas molecule has mass m and velocity v
t = 0 s
v meter
t = 1 s
gas molecule has mass m and velocity v

5. Middle of container
Let V = Volume of container and N = Number of molecules
At equilibrium N/2 travel left with +IvI speed
N/2 travel right with -IvI speed
Number of molecules that hit wall per second
= number of molecules at distance IvxI from walls
In volume V there are N molecules
“ IvxIA there are NIvxIA “ V
A  surface area of a wall

6. 2 V V P = F = Nm|vx|2 A V force exerted on right wall per second
F = 1 N|vx|A (2 m|vx|)
2 V
only 1 N|vx|A hit right wall (N/2 out of N travel right)
2 V
F = mN|vx|2A V
P = F = Nm|vx|2
A V

7. c2 = |v|2 = vx 2 + vy 2 + vz 2 c = gas speed vx 2 = 1/3 c2 P = 1 Nmc2
Vetcor v can be written as
v = vx i + vy j + vz k
c2 = |v|2 = vx 2 + vy 2 + vz 2
c = gas speed
assume velocity in all 3 axes are equal
vx = vy = vz
or c2 = 3 vx 2
vx 2 = 1/3 c2
P = 1 Nmc2
3 V
or PV = 1/3Nmc n = number of moles
= 1/3nMc M = molar mass

8. Speed of gas every gas does not travel at the same speed
The Maxwell distribution of speeds and its variation with the temperature.

9. Average velocity
wil  frequency (weight)
root mean square (rms) velocity
More accurate than average if molecules do not travel in same direction
most probable velocity (highest population) vmp

10. Maxwell distribution of gas velocity
distribution of gas speed can be presented by a mathematical function call “Maxwell distribution”

13. Combining Kinetic Molecular Model and Ideal Gas Law
average kinetic energy
from equipartition theory
thus

14. from
Ideal gas Law

15. Real gas molecules not always in motion (condense phase can be formed)
molecular size is non-negligible (there is molecular repulsion)
Molecules do interact (there is molecular attraction)

16. - Compression factor (Z)
Vm = molar volume or volume per mole
If Z < 1 molar volume of real gas is less than ideal gas molecular attraction is dominant
Z > 1 molar volume of real gas is larger than ideal gas molecular repulsion is dominant
Substitute

17. The variation of the compression factor, Z, with pressure for several gases at 0 oC

18. Equation of state is equation that represents the state of system
Ideal gas equation is the equation of state for gas

19. Virial equation
in search for equation of state for real gas, one looks at P dependent of Z or
Since
and are 2nd, 3rd, …, virial coefficient respectively

20. Replacing Z by
virial equation of state
virial coefficients vary with gas type
So does the virial equation of state

21. PV plot for real gas
The experimental isotherms of CO2 at several temperatures.

22. Critical constants

23. Reduced properties
The compression factors of four gas.

24. Van der waals equation
another attempt to find equation of state for real gas
To correct what’s wrong in kinetic molecular model, Van der waals proposed
molecular attraction reduces impact force to wall
volume of gas is the space where gas can travel in a container
Ideal gas has no molecular volume, volume of gas is volume of container
real gas has molecular volume (repulsion), space that gas can travel is
less than volume of container

25. (rate of collision) x (impact force
Attraction reduces rate of collision and impact force
Reduction in pressure (due to attraction)
From ideal gas equation

26. real gas volume is less than ideal gas
Given b as molecular volume
Substitute V in (1) with V nb
then
“Van der waals ‘s equation”
a, b = van der waals parameter (obtained from experiment)

27. Since Van der Waals equation comes from modification of kinetic molecular model (theory) while a, b comes from experiment
Van der waals equation is “semi-empirical”
virial equation is “empirical”

28. Van der waals parameters of gas.

29. Van der waals loop
Isotherms calculated by using the van der waals equation of state.

30. Comparing ideal and real gas
V = 10 L , n = 1 mol
P (atm)
Ideal gas
V d w
Virial*
273 K N2
2.239
2.234
2.236
CO2
2.212
2.205 O2
2.232
Air**
2.233
600 K
4.920
4.926
4.931
4.905
4.914
4.922
Air
4.925 *
** AIR = 80%N %O2

31. Example 2.1 Estimate the molar volume of CO2 at 500 K and 100 atm by treating it as a van der waals gas.
From

32. must solve for roots of cubic equation
a = L2 atm mol-2 ; b = 4.29 x 10-2 L mol-1

34. assume
Vm = L mol-1