1. Kinetic Theory of Gases Physics 202 Professor Lee Carkner Lecture 13

2. 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

3. 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 10 23 molecules  6.02 x 10 23 is called Avogadro’s number (N A )  M = M = mN A    A mole of any gas occupies about the same volume

4. 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

5. 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

6. Work and the Ideal Gas Law   p=nRT (1/V)

7. 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

8. 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

9. Constant Volume or Pressure  W=0  W =  pdV = p(V f -V i ) W = p  V  For situations where T, V or P are not constant, we must solve the integral  The above equations are not universal

10. 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 = (nMv 2 rms )/(3V)  Where M is the molar mass (mass of 1 mole)

11. 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 (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

12. Maxwell’s Distribution

13. Translational Kinetic Energy   Using the rms speed yields: 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  Temperature is a measure of the average kinetic energy of a gas

14. 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 

15. Next Time  Read: 19.8-19.11