Kinetic Theory of Gases

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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

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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 Avogadros number (NA)
• M
• M mNA
• A mole of any gas occupies about the same volume

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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

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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

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Work and the Ideal Gas Law

• pnRT (1/V)

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Isothermal Process

• If we hold the temperature constant in the work
  equation
• W nRT ln(Vf/Vi)
• Work for ideal gas in isothermal process

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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

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Constant Volume or Pressure

• W0
• 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

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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 Newtons second law Fd(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)

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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

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MaxwellsDistribution
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Translational Kinetic Energy

• 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

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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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Next Time

• Read 19.8-19.11