1. Lecture 18 Chapter 18 Kinetic Theory of Gases
Now we look at temperature, pressure, and internal energy in terms of the motion of molecules and atoms? Relate to the 1st Law of Thermodynamics
Gas
Solid

2. Todays Material Prelude Comments Ideal Gas Law
First Law of Thermo and the Gas Law
Microscopic kinetic theory of gases
Specific heats for gasses
Equipartition theorem
Entropy – two equivalent definitions

3. Similarity between Hook’s Law and Morse Potential
Solids can be modeled using hooks law

4. Gases (U=0)
Molecules are far apart at reasonable temperatures and do not
interact or stick together when they elastically collide with each other.
They do not interact with walls of the container either. They pretty
much move independently from each other.
Internal energy of a gas is due
to the motion of the molecules.
Namely their kinetic energy.

5. Liquids Are inbetween solids and gases and are not at either extreme.
They are more difficult to analyze from basic principles so we
do not normally study them in detail because their motion is too
complicated for an introductory course.

6. Avogadro’s Number
How many molecules are there in a cubic meter of air at STP?
First, find out how much mass there is.
The mass is the density times the volume.
m = 1.2 kg/m3 x 1 m3 = 1.2 kg. (or 2.5 lbs)
Now find the number of moles in the cubic meter and multiply by Avogadro’s number
1 m

7. Avogadro’s Number
NA=6.02 x atoms or molecules in one mole of any gas.
One mole is the molecular weight in grams. It is also called the molar mass.
1 mole of air contains 29 gms
Then the number of moles in 1.2 kg is
Then the number of molecules or atoms is
Advogado’s number is also related to the Ideal Gas law.

8. Ideal Gases
Experiment shows that 1 mole of any gas, such as helium, air,
hydrogen, etc at the same volume and temperature has almost the
same pressure. At low densities the pressures become even closer
and obey the Ideal Gas Law:
Called the gas constant

9. Ideal Gas Law in terms of Boltzman Constant
At high gas densities you must add the Van der Waal corrections
where a and b are constants. (Nobel prize 1 910)
Lets use apply the 1st law of Thermodynamics for reversible processes
using the an ideal gas

10. What is the work done by an ideal gas
Always keep this picture in
your head. Memorize it!!

11. For an isothermal process or constant temperature.
pV diagram
For an isothermal process or constant temperature.
This is an equation of a hyperbola.

12. What is the work done by an ideal gas when the temperature is constant?
For constant temperature.
constant

13. What is the work done by an ideal gas when the temperature is constant?
Isothermal expansion Vf > Vi W > ln is positive
Isothermal compression Vf < Vi W < ln is negative
Constant volume Vf = Vi W = ln 1=0

14. Four situations where the work done by an ideal gas is very clear.
For a constant volume process:
Important
For a constant pressure process
For a constant temperature process
For an ideal gas undergoing any reversible thermodynamic process.

15. Sample problem 19-2
One mole of oxygen expands from 12 to 19 liters at constant temperature of 310 K. What is the work done?
W = The area under the curve.

16. Sample problem 19-2
One mole of oxygen expands from 12 to 19 liters at constant temperature of 310 K. What is the work done?
(Note volume
units cancel)
W = The area under the curve.

17. This is want we know so far. What’s Next
We now know how to find W. If we can find the change in internal energy, then we know Q. Resort to a microscopic or kinetic theory of a gas.

18. Consider one molecule first
Kinetic theory model of a gas: What is the connection between pressure and speed of molecules? Pressure is force per unit area a force is change in momentum per unit time.
Consider one molecule first
We want to find the x component of force per unit area which is related to pressure.
Assume elastic collisions between walls and molecules and neglect collisions between molecules.
Only want the x component of the momentum since the change in y is parallel to surface and does not contribute to the pressure.

19. Find p due to one molecule first and then sum them up
Assume elastic collisions between walls and molecules and neglect collisions between molecules.
Only want the x component of the momentum since the change in y is parallel to surface and does not contribute to the pressure.
due to one
molecule

20. Find pressure due to all molecules Due to one molecule
Assume elastic collisions between walls and molecules and neglect collisions between molecules.
Only want the x component of the momentum since the change in y is parallel to surface and does not contribute to the pressure.
Due to all molecules
where M is the molar mass and V is L3
Now find the average molecular velocity.

21. What is the root mean square of the velocity of the Molecules?
and
then
Define the root mean square of v
Since we have trillions of molecules moving randomly in all directions the square of the speed in any direction is equal. So we can equate the squares of the components and multiply by 3 to get the magnitude. Actually we use the root-mean-square of the speed.

22. Average Molecular speeds at 300 K
Gas m/s
Hydrogen
Helium
Water vapor 645
Nitrogen
Oxygen
This is the average speed. Some molecules are going much faster and some slower. On the sun T= 2*106 K and hydrogen is
82 times faster than at vroom temperature. Molecules would break up at such collisions.
Notice that the speed decreases with mass
What is the kinetic energy of the molecules?

23. Average Kinetic Energy of the Molecule
When you measure the temperature of the gas, you are measuring the translational energy.
All ideal gas molecules have the same translational energy at a given T
independent of their mass. Remarkable result.

24. Mean Free Path mean free path = average distance
molecules travel in between collision
d is the diameter of the molecule
N/V is the density of molecules

25. Problem. Suppose we have a oxygen molecule at 300 K at p =1 atm
with a molecular diameter of d= 290 pm. What is λ, v, and f where
f is defined as the frequency of collisions in an ideal gas?
Find λ

26. Problem. Suppose we have a oxygen molecule at 300 K at p =1 atm
with a molecular diameter of d= 290 pm. What is λ, v, and f where
f is defined as the frequency of collisions in an ideal gas?
Find λ

27. What is v, the speed of a molecule in an ideal gas?
Use the rms speed.
What is f, the frequency of collision?
f = vrms /λ
What is the time between collisions?

28. Demo comments Jug O' Air 2. Boiling by Cooling (Ice on beaker)
3. Boiling by Reducing Pressure(Vacuum in Bell jar)
4. Dipping Duck Toy
5. Leslie cube and the laser themometer.
Demo comments

29. 1. Jug of Air (Inflate with bike pump and watch temp
1. Jug of Air (Inflate with bike pump and watch temp. rise) p/T = nR/V = constant
Increase pressure using bicycle pump. To keep P/T constant
temperature rises. Volume in jug remains constant.

30. 2. Boiling by Reducing Pressure (Vacuum in Bell jar)
See velocity distribution of molecules next subject

31. Heat up until water boils
3. Boiling by Cooling
Heat up until water boils
Remove flame then cork flask quickly
Put cubes of ice on flask
Wait until temperature and pressure is lowered
at constant volume until boiling starts again.
P/T = constant

32. 4. Dipping Duck Toy Pressure drops inside head
Wet head cools as water evaporates off of it
Pressure drops inside head
Contained pressure pushes water up tube
When center of gravity is exceeded head tips
Exposes bottom of tube then pressure equalized
Starts all over again

33. 5. Leslie cube and the laser thermometer.
Mostly infrared - colors are misleading in optical regime

34. Maxwell’s speed distribution law: Explains boiling.vp
Area under red or green curve = 1
Note there are three velocities vp
It is the faster moving molecules in the tail of the distribution that escape from the surface of a water.
It is also the protons in the tail in the sun that have high enough energy to overcome the Coulomb barrier to cause nuclear fusion. vp
is the most probable speed
It is the faster moving molecules in the tail of the distribution that escape from the surface of a water.

35. Vp is the most probable speed
It is the faster moving molecules in the tail of the distribution that escape from the surface of a water.
It is also the protons in the tail in the sun that have high enough energy to overcome the Coulomb barrier to cause nuclear fusion.
Vp is the most probable speed