Kinetic Theory of Gases

Physics 202Professor Lee

CarknerLecture 15

PAL #14 Heat Transfer Heat transfer in a cylinder No conduction through vacuum

No convection through iron or vacuum

No radiation through iron

What is a Gas?

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 When thinking about molecules it sometimes is helpful

to use the mole

6.02 x 1023 is called Avogadro’s number (NA)

M = mNA Where m is the mass per molecule or atom

Gasses with heavier atoms have larger molar masses

Ideal Gas

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

Gases that obey this relation are called ideal gases

Ideal Gas Law

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

Where:

R is the gas constant 8.31 J/mol K

Work and the Ideal Gas Law

We can use the ideal gas law to solve this equation

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

Isothermal Work

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

Isotherms

Constant Volume or Pressure

In a constant volume process no work is done so:

In a constant pressure process the work

equation becomes

W = pV For situations where T, V or P are not

constant, we must solve the integral

Random Gas Motions

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

A bigger box means fewer collisions

The final result is:

Where M is the molar mass (mass contained in 1 mole)

RMS Speed Not all the molecules have the same speed even if the

temperature is constant

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

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

Translational Kinetic Energy

If the molecules have a velocity then they also have kinetic energy (K=½mv2)

Kave = ½mvrms

2

Kave = (3/2)kT

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