Chapter 19

The Kinetic Theory of Gases

From the macro-world to the micro-world

Key contents:

Ideal gasesPressure, temperature, and the RMS speedMolar specific heatsAdiabatic expansion of ideal gases

19.2 Avogadro’s Number

One mole is the number of atoms in a 12 g sample of carbon-12.The number of atoms or molecules in a mole is called Avogadro’s Number, NA.

If n is the number of moles contained in a sample of any substance, N is the number of molecules, Msam is the mass of the sample, m is the molecular mass, and M is the molar mass, then

Italian scientist Amedeo Avogadro (1776-1856) suggested that all gases occupy the same volume under the condition of the same temperature, the same pressure, and the same number of atoms or molecules. => So, what matters is the ‘number’ .

19.3: Ideal Gases

The equation of state of a dilute gas is found to be

Here p is the pressure, n is the number of moles of gas present, and T is its temperature in kelvins. R is the gas constant that has the same value for all gases.

Or equivalently,

Here, k is the Boltzmann constant, and N the number of molecules.

(# The ideal gas law can be derived from the Maxwell distribution; see slides below.)

19.3: Ideal Gases; Work Done by an Ideal Gas

Example, Ideal Gas Processes

Example, Work done by an Ideal Gas

Considering , we have

Defining , we have

Comparing that with , we have

The temperature has a direct connection to the RMS speed squared.

19.4: Pressure, Temperature, and RMS Speed

The momentum delivered to the wall is +2mvx

Translational Kinetic Energy

19.4: RMS Speed

Example:

19.7: The Distribution of Molecular Speeds

Maxwell’s law of speed distribution is:

The quantity P(v) is a probability distribution function: For any speed v, the product P(v) dv is the fraction of molecules with speeds in the interval dv centered on speed v.

Fig. 19-8 (a) The Maxwell speed distribution for oxygen molecules at T =300 K. The three characteristic speeds are marked.

Example, Speed Distribution in a Gas:

Example, Different Speeds

19.8: Molar Specific Heat of Ideal Gases: Internal Energy

The internal energy Eint of an ideal gas is a function of the gas temperature only; it does not depend on any other variable.

For a monatomic ideal gas, only translational kinetic energy is involved.

19.8: Molar Specific Heat at Constant Volume

where CV is a constant called the molar specific heat at constant volume.

But,

Therefore,

With the volume held constant, the gas cannot expand and thus cannot do any work. Therefore,

# When a confined ideal gas undergoes temperature change T, the resulting change in its internal energy is

A change in the internal energy Eint of a confined ideal gas depends on only thechange in the temperature, not on what type of process produces the change.

19.8: Molar Specific Heat at Constant Pressure

Example, Monatomic Gas:

CV (J/mol/K)

CP-CV (J/mol/K)

=CP/CV

monatomic 1.5R=12.5 R=8.3

He 12.5 8.3 1.67

Ar 12.5 8.3 1.67

diatomic 2.5R=20.8

H2 20.4 8.4 1.41

N2 20.8 8.3 1.40

O2 21.0 8.4 1.40

Cl2 25.2 8.8 1.35

polyatomic 3.0R=24.9

CO2 28.5 8.5 1.30

H2O(100°C) 27.0 8.4 1.31

Molar specific heats at 1 atm, 300K

19.9: Degrees of Freedom and Molar Specific Heats

Every kind of molecule has a certain number f of degrees of freedom, which are independent ways in which the molecule can store energy. Each such degree of freedom has associated with it —on average — an energy of ½ kT per molecule (or ½ RT per mole). This is equipartition of energy.

Recall that

CV (J/mol/K) CP-CV (J/mol/K) =CP/CV

monatomic 1.5R=12.5 R=8.3

He 12.5 8.3 1.67

Ar 12.5 8.3 1.67

diatomic 2.5R=20.8

H2 20.4 8.4 1.41

N2 20.8 8.3 1.40

O2 21.0 8.4 1.40

Cl2 25.2 8.8 1.35

polyatomic 3.0R=24.9

CO2 28.5 8.5 1.30

H2O(100°C) 27.0 8.4 1.31

Example, Diatomic Gas:

19.10: A Hint of Quantum Theory

# Oscillations are excited with 2 degrees of freedom (kinetic and potential energy) for each dimension.# Hidden degrees of freedom; minimum amount of energy# Quantum Mechanics is needed.

A crystalline solid has 6 degrees of freedom for oscillations in the lattice. These degrees of freedom are frozen (hidden) at low temperatures.

19.11: The Adiabatic Expansion of an Ideal Gas

with Q=0 and dEint=nCVdT , we get:

From the ideal gas law,

and since CP-CV = R,

we get:

With = CP/CV, and integrating, we get:

Finally we obtain:

19.11: The Adiabatic Expansion of an Ideal Gas

19.11: The Adiabatic Expansion of an Ideal Gas, Free Expansion

A free expansion of a gas is an adiabatic process with no work or change in internal energy. Thus, a free expansion differs from the adiabatic process described earlier, in which work is done and the internal energy changes.

In a free expansion, a gas is in equilibrium only at its initial and final points; thus, we can plot only those points, but not the expansion itself, on a p-V diagram.

Since ΔEint =0, the temperature of the final state must be that of the initial state. Thus, the initial and final points on a p-V diagram must be on the same isotherm, and we have

Also, if the gas is ideal,

Example, Adiabatic Expansion:

Four Gas Processes for an Ideal Gas

Homework:

Problems 13, 24, 38, 52, 59