Chemistry 232 Kinetic Theory of Gases

Kinetic Molecular Theory of Gases Macroscopic (i.e., large quantity) behaviour of gases – pressure, volume, and temperature. The kinetic molecular theory of gases attempts to explain the behaviour of gases on a molecular level.

Assumptions of Kinetic Theory Total energy of the system Intermolecular attractive interactions are negligible.

Postulates of Kinetic Theory of Gases Gases consist of molecules of mass m and diameter d. Gas molecules are in constant, rapid, straight-line motion. Collisions are elastic. The gas molecules interact only when they collide.

Kinetic Theory Postulates (Cont’d) Average kinetic energy (K.E.) of molecules depends on absolute temperature (T) only. All collisions are elastic.

Kinetic Theory of Gases

Explanation of Pressure Gas pressure - collisions of gas molecules with the container walls. The force of a collision depends on the number of collisions per unit time how hard gas molecules strike the container wall!

The greater the momentum of gas molecules, the greater the effect of the impact on the walls. Force/A = P

The Momentum Change During a Collision Particle of mass m i collides with the wall with only the x component of the momentum changing. + m v ix - m v ix

Not All Particles Reach the Wall! How many particles actually reach the wall during a specified time interval  t? +v J,x  t These molecules don’t reach the wall! These molecules come into contact with the wall!

The Total Momentum Change The total momentum change is calculated form the sum of the momentum changes for the individual particles.

The Definition of Pressure The pressure exerted by the gas is calculated as follows

Distribution of Molecular Speeds This speed in the above equation should be an average speed (some will always be fast, some slow). Replace with the ensemble average

The Mean Square Speed Kinetic Molecular Theory of Gases allows us to relate macroscopic measurements to molecular quantities P, V are related to the molar mass and mean square seed

The Root Mean Square Speed 1/3 M J 2 = RT 2 = 3RT / M J ( 2 ) 1/2 = v rms = (3RT/M J ) 1/2 v rms = the root mean square speed

The Maxwell Probability Distribution In kinetic theory, we are interested in the fraction of molecules having a particular range of speeds. The probability distribution of speeds

The Maxwell Distribution for Typical Gases

Other Speed Equations In addition to the root-mean-square speed, we have the Most probable speed The mean speed

The Root Mean Square Speed

Collisions With Walls and Surfaces Rate at which molecules collide with a wall of area A

Effusion Rate at which molecules pass through a small hole of area A o, r

Effusion (Cont’d) Effusion. A gas under pressure goes (escapes) from one compartment of a container to another by passing through a small opening.

Effusion

The Effusion Equation Graham’s Law - estimate the ratio of the effusion rates for two different gases. Effusion rate of gas 1  r 1.

Effusion Equation (Cont’d) Effusion rate of gas 2  r 2.

Effusion Ratio Ratio of effusion rates.

Intermolecular Collisions in Hard- sphere Gases Quantitative picture of the events that take place in a collection of gaseous molecules. Frequency of collisions? Distance between successive collisions? Rate of collisions per unit volume?

The Definition of a Collision A pair of molecules will collide whenever the centres of the two molecules come within a distance d (the collision diameter) of one another. No collision.Collision occurs. d

The Collision Cylinder 2d d Stationary particles inside the collision tube.

The Number Density For N-1 stationary particles, the number of molecules per unit volume

Collision Frequency We count the total number of molecules with centres inside the collision tube. # Inside tube = N d   t

Collision Frequency (cont’d) For N-1 stationary particles The collision frequency - z 1 Examine the case where all the molecules inside the collision tube are moving.

Collision Frequency (Cont’d) Relative speed of the colliding particles.

The Mean Collision Time The mean collision time is average time elapsed between successive collisions. Define  coll = 1/z 1

The Mean Free Path Gas molecules encounter collisions with other gas molecules and with the walls of the container Define the mean free path as the average distance between successive molecular collisions Note -  - the collision cross section  =  d 2

The Mean Free Path The mean free path - the average distance traveled between successive collisions.

The Mean Free Path

The Collision Density We define the collision density as the total rate of collisions per unit volume.

Collisions in Heteronuclear Systems Modify the above discussion to include collisions between unlike molecules. The mean collision diameter. The reduced mass of the colliding molecules. The collision zone.

The Mean Collision Diameter Define in terms of the collision diameters of the colliding species. d1d1 d2d2 Mean collision diameter d 12 = ½ (d 1 +d 2 )

The Collision Zone For a collision occurring along the x and y axis. Impact Zone X 1 =t c x y y 2 =t c t c = time yet to elapse before the collision occurs

Mean Relative Speed The mean relative speed.

The Reduced Mass The reduced mass of two particles 1 and 2 is defined as follows

Mean Free Paths in Heteronuclear Collisions For substance 1 colliding with substance 2

Mean Free Paths (Cont’d) For substance 2 colliding with substance 1

Heteronuclear Collision Frequencies The collision frequency of molecule 1 with molecule 2 is given by

Heteronuclear Collision Frequencies (cont’d) The collision frequency of molecule 2 with molecule 1 is given by

Heteronuclear Collision Density The total rate of heteronuclear collisions per unit volume