1. Kinetic Theory of Gases I

2. Ideal Gas The number of molecules is large
The average separation between
molecules is large
Molecules moves randomly
Molecules obeys Newton’s Law
Molecules collide elastically with each
other and with the wall
Consists of identical molecules

3. in K The Ideal Gas Law n: the number of moles in the ideal gas
total number
of molecules
Avogadro’s number: the number of
atoms, molecules, etc, in a mole of
a substance: NA=6.02 x 1023/mol.
R: the Gas Constant: R = 8.31 J/mol · K

4. Pressure and Temperature
Pressure: Results from collisions of molecules
on the surface
Force
Pressure:
Area
Force:
Rate of momentum
given to the surface
Momentum:
momentum given by each collision
times the number of collisions in time dt

5. Only molecules moving toward the surface hit
the surface. Assuming the surface is normal to
the x axis, half the molecules of speed vx move
toward the surface.
Only those close enough to the surface hit it
in time dt, those within the distance vxdt
The number of collisions hitting an area A in
time dt is
Average density
The momentum given by each collision to the surface

6. Momentum in time dt:
Force:
Pressure:
Not all molecules have the same

7. is the root-mean-square speed
Pressure:
Average Translational Kinetic Energy:

8. Pressure:
From
and
Temperature:
Boltzmann constant:

9. From
and
Avogadro’s number
Molar mass

10. Pressure  Density x Kinetic Energy
Temperature  Kinetic Energy

11. Internal Energy For monatomic gas: the internal energy = sum
of the kinetic energy of all molecules:

12. HRW 16P (5th ed. ). Consider a given mass of an ideal gas
HRW 16P (5th ed.). Consider a given mass of an ideal gas. Compare curves representing constant-pressure, constant volume, and isothermal processes on (a) a p-V diagram, (b) a p-T diagram, and (c) a V-T diagram. (d) How do these curves depend on the mass of gas? p V
constant pressure
isothermal
constant volume T
(d)
Constant temperature
Constant volume
Constant pressure

13. (d) Cyclic process  ∆Eint = 0
HRW 18P (5th ed.). A sample of an ideal gas is taken through the cyclic process abca shown in the figure; at point a, T = 200 K. (a) How many moles of gas are in the sample? What are (b) the temperature of the gas at point b, (c) the temperature of the gas at point c, and (d) the net heat added to the gas during the cycle?
(a) a b c
1.0
3.0
Pressure (kN/m2)
2.5
7.5
(b)
(c)
(d) Cyclic process  ∆Eint = 0
Volume (m3)
Q = W = Enclosed Area= 0.5 x 2m2 x 5x103Pa = 5.0 x 103 J

14. (a) (b) Since for 0.5 vrms for 2 vrms
HRW 30E (5th ed.).(a) Compute the root-mean-square speed of a nitrogen molecule at 20.0 ˚C. At what temperatures will the root-mean-square speed be (b) half that value and (c) twice that value?
(a)
(b) Since
for 0.5 vrms
for 2 vrms

15. HRW 34E (5th ed.). What is the average translational kinetic energy of nitrogen molecules at 1600K, (a) in joules and (b) in electron-volts?
(a)
(b) 1 eV = 1.60 x J

16. Kinetic Theory of Gases II

17. the bigger the molecules
Mean Free Path
Molecules collide elastically with other molecules
Mean Free Path l: average distance between
two consecutive collisions
the bigger the molecules
the more collisions
the more molecules
the more collisions

18. Molar Specific Heat Definition: For constant volume:
For constant pressure:
The 1st Law of Thermodynamics:
(Monatomic)

19. Constant Volume
(Monatomic)

20. Constant Pressure
(Monatomic)

21. 1st Law
Adiabatic Process
Ideal Gas Law
(Q=0)
Divide by pV:

22. Ideal Gas Law

23. Equipartition of Energy
The internal energy of non-monatomic
molecules includes also vibrational and
rotational energies besides the
translational energy.
Each degree of freedom has associated with
it an energy of per molecules.

24. Monatomic Gases
3 translational degrees of freedom:

25. Diatomic Gases 3 translational degrees of freedom
2 rotational degrees of freedom
2 vibrational degrees of freedom
HOWEVER, different DOFs require different
temperatures to excite. At room temperature,
only the first two kinds are excited:

26. HRW 63P (5th ed.). Let 20.9 J of heat be added to a particular ideal gas. As a result, its volume changes from 50.0 cm3 to 100 cm3 while the pressure remains constant at 1.00 atm. (a) By how much did the internal energy of the gas change? If the quantity of gas present is 2.00x10-3 mol, find the molar specific heat at (b) constant pressure and (c) constant volume.
(a) Constant pressure: W = p∆V

27. HRW 63P (5th ed.). Let 20.9 J of heat be added to a particular ideal gas. As a result, its volume changes from 50.0 cm3 to 100 cm3 while the pressure remains constant at 1.00 atm. (a) By how much did the internal energy of the gas change? If the quantity of gas present is 2.00x10-3 mol, find the molar specific heat at (b) constant pressure and (c) constant volume.
(b)
(c)

28. (a) Adiabatic Monatomic
HRW 81P (5th ed.). An ideal gas experiences an adiabatic compression from p =1.0 atm, V =1.0x106 L, T = 0.0 ˚C to p =1.0 x 105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? (e) What is the ratio of the squares of the rms speeds before and after the compression?
(a) Adiabatic
Monatomic

29. HRW 81P (5th ed.). An ideal gas experiences an adiabatic compression from p =1.0 atm, V =1.0x106 L, T = 0.0 ˚C to p =1.0 x 105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? (e) What is the ratio of the squares of the rms speeds before and after the compression?
(b)

30. HRW 81P (5th ed.). An ideal gas experiences an adiabatic compression from p =1.0 atm, V =1.0x106 L, T = 0.0 ˚C to p =1.0 x 105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? (e) What is the ratio of the squares of the rms speeds before and after the compression?
(c)
(Pay attention to the units)
(d) For N/n = 1

31. HRW 81P (5th ed.). An ideal gas experiences an adiabatic compression from p =1.0 atm, V =1.0x106 L, T = 0.0 ˚C to p =1.0 x 105 atm, V =1.0x103 L. (a) Is the gas monatomic, diatomic, or polyatomic? (b) What is its final temperature? (c) How many moles of gas are present? (d) What is the total translational kinetic energy per mole before and after the compression? (e) What is the ratio of the squares of the rms speeds before and after the compression?
(e)