1. Thermodynamics Kinetic Theory of Gases (Section 10.6)

2. Physicists try to understand heat and energy in terms of basic physics principles Momentum conservation Energy Conservation Newton’s Laws 10 23 particles moving randomly

3. Molecular Model for Pressure of an Ideal Gas ASSUMPTIONS 1.The number of molecules is large and the particles are small. 2.The particles are all classical, obeying Newton’s laws, but their motion is random (no synchronized swimming) 3.Particles undergo completely elastic collisions with the walls of the container. Thus kinetic energy is constant. 4.The forces between the molecules are negligible except during a collision. 5.The gas is a pure substance. All molecules are identical

4. One at a time… Instead of trying to track all the particles, let’s look at just one particle. –Just the x component –Making an ideal collision with just one wall –Use this average collision and extrapolate to all three directions and all N particles.

5. One collision :  p x = p f – p i = -2(p i ) = -2mv x F 1  t =  p = 2mv x For each collision. The next collision it makes with that same wall, it travels a distance 2d at speed v x so: v x = 2d/  t so  t = 2d/v x is the time interval between collisions. The average force on a wall is therefore: F 1 = 2mv x /  t= 2mv x /(2d/v x ) = mv x 2 /d F 1 = (m/d) v x 2

6. If F 1 = (m/d) v x 2 is the force for 1 object, then N object have a total Force of F = (m/d)  (v x1 2 + v x2 2 + v x3 2 + v x4 2 +… + v xN 2 ) But all the particles are really randomly moving about, so let’s look at the average value of the squares of the velocities. (v x 2 ) av = (v x1 2 +v x2 2 +v x3 2 +v x4 2 +… +v xN 2 )/N F = (m/d) N(v x 2 ) av

7. This was Just the x direction We assume that every direction is the same, and using Pythagorean theorem in 3- dimensions, we get: So all molecules together have an average velocity But, all three directions are equivalent, and so we get: v 2 = v x 2 + v y 2 + v z 2 v 2 av =(v x 2 ) av +(v y 2 ) av +(v z 2 ) av (v x 2 ) av = (v y 2 ) av = (v z 2 ) av v 2 av = 3(v x 2 ) av

8. Putting it together: Total Average Velocity squared of a particle Force on a wall by N particles Total Force on the wall Pressure: P=F/A = F/d 2 Pressure of an Ideal Gas v 2 av = 3(v x 2 ) av F = (m/d) N(v x 2 ) av F =(m/d)N(v 2 av /3) =(N/3)(mv 2 av )/d P=(N/3)(mv 2 av )/d 3 P=2(N/3)(½ mv 2 av )/V P=(2/3)(N/V)(½ mv 2 av )

9. PRESSURE OF AN IDEAL GAS P=(2/3)(N/V)(½ mv 2 av ) Proportional to the number of molecules per unit volume Proportional to the average translational kinetic energy of the molecules We have linked the LARGE WORLD to the SMALL WORLD

10. BUT WAIT!! (there’s more) Now we can understand temperature P=(2/3)(N/V)(½ mv 2 av ) PV=(2/3)(N)(½ mv 2 av ) PV = Nk B T …T = (2/3k B )(½ mv 2 av )  Temperature is a direct measure of average molecular kinetic energy!  The energy of a molecule tells us its temperature! (½ mv 2 av ) = (3/2)k B T

11. Temperature is energy (½ mv 2 av ) = (3/2)k B T Note: 3 stands for 3 dimensions. Each component contributed equally to the total translational kinetic energy v rms =  (v 2 av) =  (3k B T/m ) This is known as the rms speed of a molecule. Notice that as Temp goes up…. v rms goes up. Notice that as mass goes up…. v rms goes down.

12. rms Speeds and escape velocity GasMolar MassV rms at 20 o C H2H2 2.02 x 10 -3 1902 He4.0 x 10 -3 1352 H2OH2O18 x 10 -3 637 N 2 or CO 28 x 10 -3 511 CO 2 44 x 10 -3 408