Kinetic Theory of GasesWhat is a Gas?MoleIdeal GasIdeal Gas LawWork and the Ideal Gas LawIsothermal ProcessIsothermsConstant Volume or PressureGas SpeedRMS SpeedMaxwell’s DistributionTranslational Kinetic EnergyMaxwellian Distribution and the SunNext TimeKinetic Theory of GasesPhysics 202Professor Lee CarknerLecture 13What 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 behaviorMoleA gas is composed of moleculesm = N = When thinking about molecules it sometimes is helpful to use the mole1 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 volumeIdeal 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 gasesIdeal 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 metersWork and the Ideal Gas Law  p=nRT (1/V)VfVipdVWVfVidVV1nRTWIsothermal Process If we hold the temperature constant in the work equation: W = nRT ln(Vf/Vi)Work for ideal gas in isothermal processIsotherms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 temperatureConstant Volume or Pressure W=0 W = pdV = p(Vf-Vi)W = pVFor situations where T, V or P are not constant, we must solve the integralThe above equations are not universalGas 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 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 equationsvrms = (3RT/M)½ For a given type of gas, velocity depends only on temperatureMaxwell’sDistributionTranslational 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 gasMaxwellian 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Read: