kinetic theory of gases physics 202 professor lee carkner lecture 13
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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
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
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
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