1. The Ideal Gas Laws
Chapter 14

2. Expectations After this chapter, students will: Know what a “mole” is
Understand and apply atomic mass, the atomic mass unit, and Avogadro’s number
Understand how an ideal gas differs from real ones
Use the ideal gas equation, Boyle’s Law, and Charles’ Law, to solve problems

3. Expectations After this chapter, students will:
understand the connection between the macroscopic properties of gases and the microscopic mechanics of gas molecules

4. Preliminaries: the Mole
A mole is a very large number of discrete objects, such as atoms, molecules, or sand grains.
Specifically, it is Avogadro’s Number (NA) of such things: 6.022×1023 of them.
The mole (“mol”) is not a dimensional unit; it is a label.

5. Amadeo Avogadro 1776 – 1856 Native of Turin, Italy
Hypothesized that equal volumes
of gases at the same temperature
and pressure contained equal
numbers of molecules.
(He was correct, too.)

6. The Mole and Atomic Mass
Mathematical definition: 12 g of C12 contains one mole of carbon-12 atoms.
Mass of one C12 atom:
The mass of one C12 atom is also 12 atomic mass units (amu), so:

7. The Mole and Atomic Mass
Atomic masses for the elements may be found in the periodic table of the elements, located inside the back cover of your textbook.
These are often erroneously called “atomic weights.”
Atomic masses may be added to calculate molecular masses for chemical compounds (or diatomic elements).

8. The Mole: Calculations
If we have N particles, how many moles is that?
If we have a given mass of something, how many moles do we have?
number of moles

9. The Ideal Gas
The notion of an “ideal” gas developed from the efforts of scientists in the 18th and 19th centuries to link the macroscopic behavior of gases (volume, temperature, and pressure) to the Newtonian mechanics of the tiny particles that were increasingly seen as the microscopic constituents of gases.

10. The Ideal Gas
An ideal gas was one whose particles are well-behaved, in terms of the Newtonian theory of collisions: elastic collisions and the impulse-momentum theorem.
An ideal gas is one in which the particles have no interaction, except for perfectly-elastic collisions with each other, and with the walls of their container.

11. The Ideal Gas
An ideal gas has no chemistry. That is, the particles (atoms or molecules) have no tendency to “stick” to other particles through chemical bonds.
Inert gases (He, Ne, Ar, Kr, Xe, Rn) at low densities are very good approximations to the ideal gas.
Our analytic model of the ideal gas gives us insights into the properties of many real gases, inert or not.

12. The Ideal Gas Equation Observations from experience
The pressure of a gas is directly proportional to the number of moles of particles in a given space. Example: blow up a balloon, and you’re adding to n, the number of moles of molecules.
Conclusion:

13. The Ideal Gas Equation Observations from experience
The pressure of a gas is directly proportional to its temperature. Example: toss a spray can into a fire (no, wait, really, don’t do it, just think about it). Increasing pressure will cause the can to fail catastrophically.
Conclusion:

14. The Ideal Gas Equation Observations from experience
The pressure of a gas is inversely proportional to its volume. Example: squeeze the air in a half-filled balloon down to one end and squeeze it tighter. Increased pressure makes the balloon’s skin tight.
Conclusion:

15. The Ideal Gas Equation Combine the observations
A constant of proportionality, R, makes this an equation:

16. The Ideal Gas Equation
The constant of proportionality, R, is called the universal gas constant. Its value and units depend on the units used for P, V, and T.
Value and SI units of R: 8.31 J / (mol K)
pressure
volume
absolute temperature
universal gas constant
number of moles

17. The Ideal Gas Equation
We can also write the ideal gas equation in terms of the number of particles, N, instead of the number of moles, n.
Since N = n·NA, we can both multiply and divide the right-hand side by NA:
Boltzmann’s constant

18. Ludwig Boltzmann
Austrian physicist
1844 – 1906

19. Boyle’s Law
Suppose we hold both n and T constant: how are P and V related?
This is called Boyle’s Law.

20. Robert Boyle
Irish mathematician
1627 – 1691

21. Charles’ Law
Suppose we hold both n and P constant: how are T and V related?
This is called Charles’ Law.

22. Jacques Alexandre Cesar Charles
French scientist
1746 – 1823
Built and flew the first
large hydrogen-filled
balloon.

23. Kinetic Theory of the Ideal Gas
Macroscopic properties of a gas: temperature, pressure, volume, density
Microscopic properties of the particles making up the gas: mass, velocity, momentum, kinetic energy
How are they related?

24. Kinetic Theory of the Ideal Gas
Consider a gas molecule contained in a cube having edge length L.
The molecule’s mass is m, and
its velocity (in the X direction
only) is v.
Time between collisions with the
right-hand wall:

25. Kinetic Theory of the Ideal Gas
The time between collisions with the right-hand wall is just the round-trip time:
From the impulse-momentum
theorem, we can calculate the
average force exerted on the
particle by the wall:

26. Kinetic Theory of the Ideal Gas
Substitute for the time and simplify:
By Newton’s third law, the average
force exerted on the wall is

27. Kinetic Theory of the Ideal Gas
The average force on the wall from one particle is
If there are N particles, and
their directions are random, we
could expect 1/3 of them to be
moving in the X direction.
Total force on the wall:

28. Kinetic Theory of the Ideal Gas
Average pressure on the wall:
But So:

29. Kinetic Theory of the Ideal Gas
Substituting kinetic energy:
So, we see that for an ideal gas,
the average molecular kinetic energy
is directly proportional to the
absolute temperature.

30. Kinetic Theory of the Ideal Gas
This result is true for any ideal gas.
By a similar argument, if an ideal gas is monatomic (the gas particles are single atoms), the internal energy of n moles of the gas at an absolute temperature T is