1. Kinetic Theory of Gases CM2004 States of Matter: Gases

2. A Theory for 10 23 Particles In classical theory a particle’s next move depends upon (equated to) its position, velocity and force acting on it Trying to solve such equations for a mole of gas with 10 23 particles each with x,y,z coordinates and different speeds is almost impossible So we theoretically describe the kinetic system on average in terms of a large set of no-volume “points”, which do not attract or repel each other

3. Pressures on Average On average the speed term is best represented by as given in the Maxwell-Boltzmann distribution. Furthermore a particle is equally likely to hit any one of the 6 available walls of the box. Hence: “Mean- square speed”

4. Microscopic Energies Can be reformulated as: is called the average kinetic energy per particle

5. Macroscopic Energies and Boyle’s Law N 0 is the Total Kinetic Energy of one mole and is called E k, the macroscopic energy: PV=nRT So TEMPERATURE is a direct measure of the INTERNAL ENERGY of moving gas particles

6. Internal Energies T 2 >T 1 COLDHOT Each particle moves with an average kinetic energy of:

7. Root Mean Square Speeds These (v RMS )represent a single chosen speed to associate with every gas particle, as if they were all moving at this rate. START END Molar Mass

8. Thermal Energy: Energy at a Definite Temperature Kinetic Energy of 1 mole is: Define Boltzmann’s constant: Because: Then Kinetic Energy of 1 particle is:

9. Equipartition of Energy The EQUIPARTITION theorem states that a molecule gains ½ k B T of thermal energy for each DEGREE OF FREEDOM (i.e. x,y, z directions). So the total is ³/ 2 k B T

10. Quantifying Collision Rates Collision Rate (Z*) per face of cube p = 2mv x Z/6A Z = 6pvA/ 2mv 2 Z = pvA/(k B T ) A is termed, , the collision cross-section v is termed c rel the relative mean speed NOTE: But, mv 2 = 3k B T TOTAL pressure in the cube volume, where Z=6Z*

11. Relative Mean Speeds, c rel Same Direction Direct Approach Typical “on average” approach

12. Mean Free Path, The average distance between collisions is called the MEAN FREE PATH, Hence if a molecule collides with a frequency, Z, it spends a time, 1/Z in free flight between collisions and therefore travels a distance of [(1/Z) x c] = c/Z Z = p c rel /(k B T) = c k B T/p c rel  c rel = 2 ½ c  Therefore: and = k B T/2 ½ p =d/2) 2 d is the collision diameter

13. Maxwell-Boltzmann and v RMS Probability that particle has specific energy,  INCREASING TEMPERATURE MORE PARTICLES MOVE FASTER

14. Populations We shall return to the importance of Maxwell- Boltzmann Distributions in CM3006 next year Molecules and atoms consist of many “micro” states and the higher the temperature the higher the probability that “excited” states become populated

15. Important Equations (1)

16. Important Equations (2) Z = p c rel /(k B T) = k B T/ 2 ½ p 