Chapter 20 The kinetic Theory of Gases

20-2 Avogadro’s Number The mole is one of the seven SI base units and is defined as follows: One mole is the number of atoms in a 12 g sample of carbon – 12. The number of moles n is

20-3 Ideal Gases At low enough densities,all real gases tend to obey the relation The gas constant R The Boltzmann constant k

Work Done by an Ideal Gas at Constant Temperature On a p-v diagram,an isotherm is a curve that connects point that have the same temperature.

Work Done at Constant Volume and at Constant Pressure If the volume of the gas is constant If the pressure of the gas is constant Sample Problem 20-1

Sample Problem 20-2

20-4 Pressure , Temperature , and RMS Speed The only change in the particle’s momentum is along the x axis: The average rate at which momentum is delivered to the shaded wall by this single molecule is The pressure is

With Combining Eq.20-21 with the ideal gas law leads to

20-5 Translational Kinetic Energy Sample Problem 20-3 (a) (b) 20-5 Translational Kinetic Energy Its average translational kinetic energy over the time that we watch it is

At a given temperature T, all ideal gas molecules – no matter what their mass – have the same average translational kinetic energy ,namely , kT .When we measure the temperature of a gas ,we are also measuring the average translational kinetic energy of its molecules. Something unexpected:

20-6 Mean Free Path The expression for the mean free path

Sample Problem 20-4 (a) (b)

20-7 The Distribution of Molecular Speeds Maxwell’s speed distribution law is The value of this total area is unity The fraction (frac) of molecules with speed in an interval of,say, v1 to v2 is:

Average,RMS,and Most Probable Speeds The average speed is: The average of the square of the speed is The root – mean – square speed is :

The most probable speed is Sample Problem 20-5

Sample Problem 20-6 (a) (b) (c)

20-8 The Molar Specific Heats of an Ideal Gas Internal Energy Eint The internal energy Eint of the sample is The internal energy Eint of an ideal gas is a function of the gas temperature only;it does not depend on any other variable.

Molar Specific Heat at Constant Volume The heat Q is related to the temperature change by is a constant called the molar specific heat at constant volume. W=0

The internal energy of any ideal gas by substituting Cv for A change in the internal energy Eint of a confined ideal gas depends on the change in the gas temperature only;it does not depend on what type of process process the change in the temperature.

Molar Specific Heat at Constant Pressure is a constant called the molar specific heat at constant pressure.

Sample Problem 20-7 (a) (b) (c) or

20-9 Degrees of Freedom and Molar Specific Heats The equipartition of energy 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 per molecule (or per mole) .

Sample Problem 20-8 20-10 A Hint of Quantum Theory

20-11 The Adiabatic Expansion of an Ideal Gas The relation between the pressure and the volume during such an adiabatic process is the ratio of the molar specific heats for

Proof of Eq. 20-53 The first law of thermodynamics can then be written as

Free Expansions From the ideal gas law,we have The initial and final points on a p-v diagram must be on the same isotherm,and instead of Eq.20-56

Sample Problem 20-9 (a) (b)

REVIEW & SUMMARY Avogadro’s Number The number of moles n is Ideal Gas

Work in an Isothermal Volume Change The Boltzmann constant k Work in an Isothermal Volume Change Pressure,Temperature,and Molecular Speed

Temperature and Kinetic Energy Mean Free Path Maxwell Speed Distribution

Molar Specific Heats

Degrees of Freedom and Cv Adiabatic Process