Kinetic Theory of Gases

Physics 202Professor Lee

CarknerLecture 13

What is a Gas?

But where do pressure and temperature come from?

A gas is made up of molecules (or atoms)

The pressure is a measure of the force the molecules exert when bouncing off a surface

We need to know something about the microscopic properties of a gas to understand its behavior

Mole A gas is composed of molecules

m = N =

When thinking about molecules it sometimes is helpful to use the mole

1 mol = 6.02 X 1023 molecules 6.02 x 1023 is called Avogadro’s number (NA) M =

M = mNA

A mole of any gas occupies about the same volume

Ideal Gas

Specifically, 1 mole of any gas held at constant temperature and constant volume will have almost the same pressure

Gases that obey this relation are called ideal gases A fairly good approximation to real gases

Ideal Gas Law

The temperature, pressure and volume of an ideal gas is given by:

pV = nRT Where:

R is the gas constant 8.31 J/mol K V in cubic meters

Work and the Ideal Gas Law

p=nRT (1/V)

Vf

VipdVW

Vf

VidVV1

nRTW

Isothermal Process

If we hold the temperature constant in the work equation:

W = nRT ln(Vf/Vi) Work for ideal gas in

isothermal process

Isotherms From the ideal gas law we

can get an expression for the temperature

For an isothermal process temperature is constant so:

If P goes up, V must go down

Lines of constant temperature One distinct line for each

temperature

Constant Volume or Pressure

W=0

W = pdV = p(Vf-Vi)W = pV

For situations where T, V or P are not constant, we must solve the integral The above equations are not universal

Gas Speed

The molecules bounce around inside a box and exert a pressure on the walls via collisions

The pressure is a force and so is related to velocity

by Newton’s second law F=d(mv)/dt The rate of momentum transfer depends on

volume

The final result is:p = (nMv2

rms)/(3V) Where M is the molar mass (mass of 1 mole)

RMS Speed

There is a range of velocities given by the Maxwellian velocity distribution

We take as a typical value the root-mean-squared velocity (vrms)

We can find an expression for vrms from the pressure and ideal gas equations

vrms = (3RT/M)½

For a given type of gas, velocity depends only on temperature

Maxwell’sDistribution

Translational Kinetic Energy

Using the rms speed yields:Kave = ½mvrms

2

Kave = (3/2)kT

Where k = (R/NA) = 1.38 X 10-23 J/K and is called the Boltzmann constant

Temperature is a measure of the average kinetic energy of a gas

Maxwellian Distribution and the Sun

The vrms of protons is not large enough for

them to combine in hydrogen fusion

There are enough protons in the high-speed tail of the distribution for fusion to occur

Next Time

Read: 19.8-19.11