1. The Kinetic Theory of Gases
Chapter 21
The Kinetic Theory of Gases

2. Ideal gases The gas under consideration is a pure substance
All molecules are identical
Macroscopic properties of a gas: P, V, T
The number of molecules in the gas is large, and the average separation between the molecules is large compared with their dimensions – the molecules occupy a negligible volume within the container
The molecules obey Newton’s laws of motion, but as a whole they move randomly (any molecule can move in any direction with any speed)

3. Ideal gases
The molecules interact only by short-range forces during elastic collisions
The molecules make elastic collisions with the walls and these collisions lead to the macroscopic pressure on the walls of the container
At low pressures the behavior of molecular gases approximate that of ideal gases quite well

4. Ideal gases

5. Ideal gases
Root-mean-square (RMS) speed:

6. Translational kinetic energy
Average translational kinetic energy:
At a given temperature, ideal gas molecules have the same average translational kinetic energy
Temperature is proportional to the average translational kinetic energy of a gas

7. Internal energy For the sample of n moles, the internal energy:
Internal energy of an ideal gas is a function of gas temperature only

8. Work done by an ideal gas at constant temperature
Isothermal process – a process at a constant temperature
Work (isothermal expansion)

9. Work done by an ideal gas at constant volume and constant pressure
Isovolumetric process – a process at a constant volume
Isobaric process – a process at a constant pressure

10. Molar specific heat at constant volume
Heat related to temperature change:
Internal energy change:

11. Molar specific heat at constant pressure
Heat related to temperature change:
Internal energy change:

12. Adiabatic expansion of an ideal gas

13. Adiabatic expansion of an ideal gas

14. Free expansion of an ideal gas

15. Degrees of freedom and molar specific heat
3 translations, 3 rotations, + oscillations

16. Degrees of freedom and molar specific heat
3 translations, 3 rotations, + oscillations
In polyatomic molecules different
degrees of freedom contribute at
different temperatures

17. Chapter 21
Problem 13
A 1.00-mol sample of hydrogen gas is heated at constant pressure from 300 K to 420 K. Calculate (a) the energy transferred to the gas by heat, (b) the increase in its internal energy, and (c) the work done on the gas.

18. Chapter 21
Problem 27
Consider 2.00 mol of an ideal diatomic gas. (a) Find the total heat capacity at constant volume and the total heat capacity at constant pressure, assuming the molecules rotate but do not vibrate (b) Repeat part (a), assuming the molecules both rotate and vibrate.

19. Distribution of molecular speeds
Not all the molecules have the same speed
Maxwell’s speed distribution law:
Nvdv – fraction of molecules with speeds in the range from v to v + dv
James Clerk Maxwell
( )

20. Distribution of molecular speeds
Distribution function is normalized to 1:
Average speed:
RMS speed:
Most probable
speed:

21. Chapter 21
Problem 44
As a 1.00-mol sample of a monatomic ideal gas expands adiabatically, the work done on it is – 2500 J. The initial temperature and pressure of the gas are 500 K and 3.60 atm. Calculate (a) the final temperature and (b) the final pressure.

22. Questions?

23. Answers to the even-numbered problems
Chapter 21
Problem 8
2.28 kJ
(b) 6.21 × 10−21 J

24. Answers to the even-numbered problems
Chapter 21
Problem 12
209 J
(b) 0
(c) 317 K

25. Answers to the even-numbered problems
Chapter 21
Problem 14
(a) 118 kJ
(b) 6.03 × 103 kg

26. Answers to the even-numbered problems
Chapter 21
Problem 42
(a) 3.65v
(b) 3.99v
(c) 3.00v
(d) 106 m0 v2/V
(e) 7.98 m0 v2