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Kinetic Theory of Gases Physics 202 Professor Lee Carkner Lecture 15
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PAL #14 Heat Transfer Heat transfer in a cylinder No conduction through vacuum No convection through iron or vacuum No radiation through iron
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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
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Mole When thinking about molecules it sometimes is helpful to use the mole 6.02 x 10 23 is called Avogadro’s number (N A ) M = mN A Where m is the mass per molecule or atom Gasses with heavier atoms have larger molar masses
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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
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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
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Work and the Ideal Gas Law We can use the ideal gas law to solve this equation
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Isothermal Process If we hold the temperature constant in the work equation: W = nRT ln(V f /V i ) Work for ideal gas in isothermal process
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Isothermal Work
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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
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Isotherms
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Constant Volume or Pressure In a constant volume process no work is done so: In a constant pressure process the work equation becomes W = p V For situations where T, V or P are not constant, we must solve the integral
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Random Gas Motions
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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 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)
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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 (v rms ) We can find an expression for v rms from the pressure and ideal gas equations v rms = (3RT/M) ½ For a given type of gas, velocity depends only on temperature
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Maxwell’s Distribution
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Maxwellian Distribution and the Sun The v rms 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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Translational Kinetic Energy If the molecules have a velocity then they also have kinetic energy (K=½mv 2 ) K ave = ½mv rms 2 K ave = (3/2)kT Where k = (R/N A ) = 1.38 X 10 -23 J/K and is called the Boltzmann constant
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