Kinetic Model

1. THE KINETIC MODEL
PHYSICAL CHEMISTRY FOR ENGINEERS 1

2. Assumptions
• Gases are composed of rigid particles.
• Particles are in constant motion and may collide with another or
walls of container.
• Collisions are perfectly elastic.
• The absolute temperature is proportional to their average kinetic
energy.
𝑣𝑟𝑚𝑠 =
3𝑅𝑇
𝑀
root mean squared speed

3. Assumptions
• 𝑣𝑝 =
2𝑅𝑇
𝑀
most probable speed
• ⊽=
8𝑅𝑇
𝜋𝑀
mean speed

4. Assumptions
• At relatively low P, IMFA is negligible.
• The volume of particles is negligible compared to the volume of
the entire gas.

5. IDEAL GAS
• Achieved at high T and low P in order to neglect the attractive and
repulsive forces between particles, as well as the volume of the
particle themselves.
𝑃 =
𝑛𝑅𝑇
𝑉

6. DALTON’S LAW OF PARTIAL PRESSURES
• The pressure exerted by an ideal gas is the sum of the partial
pressures of the individual gases.
𝑃𝑇 = 𝑃𝑖 = 𝑃𝐴 + 𝑃𝐵
𝑦𝑖 =
𝑛𝑖
𝑛𝑡
=
𝑃𝑖
𝑃𝑇

7. Example
• Sulfur, in gaseous form at 500ºC and 699 torr has a density of 37.1
g/L. What is the form of sulfur?
Ans. 𝑆8
• What are the partial pressures of 1g of oxygen and 1g hydrogen at
27ºC and occupying a 20 d𝑚3 container?

8. GRAHAM’S LAW OF EFFUSION
• Movement of gas particles through a small hole.
𝑣𝐴
𝑣𝐵
=
𝑡𝐵
𝑡𝐴
=
𝑀𝐵
𝑀𝐴
=
𝜌𝐵
𝜌𝐴

9. APPARENT MOLAR MASS
• Average molar mass of a mixture of gases
𝑀 = 𝑦𝑖𝑀𝑖

10. HUMIDITY
• Indicates the fraction of water vapor present in moist air.
• Relative Humidity
fraction of water vapor over the amount of saturation
𝑅𝐻 =
𝑛𝑎𝑐𝑡𝑢𝑎𝑙
𝑛𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛
=
𝑃𝑎𝑐𝑡𝑢𝑎𝑙
𝑃𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛

11. Example
• What is the apparent molar mass of air if its molar mass
components are nitrogen, oxygen and argon (mole percentage are
78.10, 20.94 and 0.96)?

12. Barometric Distribution Law
• For a column of fluid
𝑑𝑃 = −𝜌𝑔𝑑𝑧
𝑃 = 𝑃𝑂𝑒
−𝑀𝑔𝑧
𝑅𝑇
Where:
Z= column height
𝑥𝑜= parameters at sea level (z=0)
M = molar mass

13. Real Gas
• The IMFA and the volume of particles are significant. Conditions:
high P and low T.
• Deviations from Ideality can be accounted using:
1. Modified Equation of State
2. Compression Factor (Z)
3. Fugacity coefficient (∅)

14. MODIFIED EQUATION OF STATE
• Account for the pressure and volume changes
1. Van der Waals
2. Berthelot
3. Redlich-Kwong
4. Dieterici
5. Virial

15. Empirical Constants
• For Van der Waals
𝑎 =
27𝑅2𝑇𝑐2
64𝑃𝑐
𝑏 =
𝑅𝑇𝑐
8𝑃𝑐

16. Critical Parameters
• Conditions wherein the difference between gas and liquid starts to
disappear.
(
𝜕𝑃
𝜕𝑉
)𝑇 = 0
Table 2-141 (8th edition)

17. Principle of Corresponding States
• Substances behave alike at the same reduced states.
• Pr =
𝑃
𝑃𝑐
• Tr =
𝑇
𝑇𝑐
• 𝑉r =
𝑉
𝑉c

18. Compressibility Factor (Z)
• Indicates the ability of a gas to be compressed at a certain
condition
• 𝑍 =
𝑉act𝑢𝑎𝑙
𝑉ideal
=
𝑃𝑉actual
𝑅𝑇
• Truncated Z expression (for VdW-EOS)
𝑍 = 1 +
𝑏𝑃
𝑅𝑇
−
𝑎𝑃
(𝑅𝑇)2

19. Example
• Calculate the pressure exerted by a 25g argon in a 1.5L vessel at
30ºC if it behaved as ideal gas and a Van der Waals gas.

20. BOYLE TEMPERATURE
• Temperature at which property of real gas become ideal
• lim
𝑃→0
(
𝜕𝑍
𝜕𝑃
)𝑇 = 0

21. FUGACITY COEFFICIENT
• Tendency to flee or escape
• Fugacity- effective pressure
• ∅ =
𝑓
𝑃
• f< P
• f >P

22. Joule-Thomson Coefficient(𝜇𝐽𝑇)
• Indicates a temperature change upon isenthalpic expansion of gas.
• 𝜇𝐽𝑇 = (
𝜕𝑇
𝜕𝑃
)𝐻= −(
𝜕𝐻
𝜕𝑃
)𝑇(
𝜕𝑇
𝜕𝐻
)𝑃

23. SUMMARY OF BEHAVIOR
Dominating
Forces
Favors Range Z ∅ 𝝁𝑱𝑻
Attractive Compression Relatively long <1 <1 >0
Repulsive Expansion Short >1 >1 <0
None( ideal) None - 1 1 0