Kinetic Theory of Gases Physics 202 Professor Lee Carkner Lecture 15

PAL #14 Heat Transfer  Heat transfer in a cylinder  No conduction through vacuum    No convection through iron or vacuum   No radiation through iron 

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

Mole  When thinking about molecules it sometimes is helpful to use the mole  6.02 x 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

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 

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 

Work and the Ideal Gas Law   We can use the ideal gas law to solve this equation

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

Isothermal Work

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 

Isotherms

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 

Random Gas Motions

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)

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

Maxwell’s Distribution

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

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 J/K and is called the Boltzmann constant 