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Kinetic Theory of Gases


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 surfaceWe need to know something about the microscopic properties of a gas to understand its behavior. Mole.

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Kinetic Theory of Gases

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Kinetic theory of gases

Kinetic Theory of Gases

Physics 202

Professor Lee Carkner

Lecture 13


What is a gas

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

    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

    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

    Work and the Ideal Gas Law

    p=nRT (1/V)


    Isothermal process

    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

    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

    Constant Volume or Pressure

    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


    Gas speed

    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 = (nMv2rms)/(3V)

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


    Rms speed

    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 s distribution

    Maxwell’sDistribution


    Translational kinetic energy

    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


    Maxwellian distribution and the sun

    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

    Next Time

    • Read: 19.8-19.11