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Kinetic Theory of Gases

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Physics 202

Professor Lee Carkner

Lecture 13

- 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

- A gas is composed of molecules
- m =

- N =

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

- 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

- 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

p=nRT (1/V)

- If we hold the temperature constant in the work equation:
W = nRT ln(Vf/Vi)

- Work for ideal gas in isothermal process

- 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

W=0

W = pdV = p(Vf-Vi)

W = pDV

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

- 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 = (nMv2rms)/(3V)

- Where M is the molar mass (mass of 1 mole)

- 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

- Using the rms speed yields:
Kave = ½mvrms2

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

- 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

- Read: 19.8-19.11