Chapter 10 - Gases

Chemistry 232Kinetic Theory of GasesKinetic Molecular Theory of GasesMacroscopic (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 TheoryTotal energy of the system

Intermolecular attractive interactions are negligible. Postulates of Kinetic Theory of GasesGases 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 (Contd)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 CollisionParticle of mass mi collides with the wall with only the x component of the momentum changing. + m vix - m vixNot All Particles Reach the Wall!How many particles actually reach the wall during a specified time interval t? +vJ,xtThese molecules dont reach the wall!These molecules come into contact with the wall!The Total Momentum ChangeThe total momentum change is calculated form the sum of the momentum changes for the individual particles.

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

Distribution of Molecular SpeedsThis 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 SpeedKinetic 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 Speed1/3 MJ2 = RT2 = 3RT / MJ(2 )1/2 = vrms = (3RT/MJ)1/2vrms = the root mean square speedThe Maxwell Probability DistributionIn 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 EquationsIn 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 SurfacesRate at which molecules collide with a wall of area A

EffusionRate at which molecules pass through a small hole of area Ao, r

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

The Effusion EquationGrahams Law - estimate the ratio of the effusion rates for two different gases.Effusion rate of gas 1 r1.

Effusion Equation (Contd)Effusion rate of gas 2 r2.

Effusion RatioRatio of effusion rates.

Intermolecular Collisions in Hard-sphere GasesQuantitative 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 CollisionA 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.dThe Collision Cylinder2ddStationary particles inside the collision tube. The Number DensityFor N-1 stationary particles, the number of molecules per unit volume

Collision FrequencyWe count the total number of molecules with centres inside the collision tube. # Inside tube = NdtCollision Frequency (contd)For N-1 stationary particlesThe collision frequency - z1

Examine the case where all the molecules inside the collision tube are moving. Collision Frequency (Contd)Relative speed of the colliding particles.

The Mean Collision TimeThe mean collision time is average time elapsed between successive collisions.Define coll = 1/z1

The Mean Free PathGas molecules encounter collisions with other gas molecules and with the walls of the containerDefine the mean free path as the average distance between successive molecular collisionsNote - - the collision cross section = d2The Mean Free PathThe 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 SystemsModify 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 DiameterDefine in terms of the collision diameters of the colliding species.d1d2Mean collision diameterd12 = (d1+d2) The Collision ZoneFor a collision occurring along the x and y axis.Impact ZoneX1=tcxyy2=tc tc = time yet to elapse before the collision occurs

Mean Relative SpeedThe mean relative speed.

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

Mean Free Paths in Heteronuclear CollisionsFor substance 1 colliding with substance 2

Mean Free Paths (Contd)For substance 2 colliding with substance 1

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

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

Heteronuclear Collision DensityThe total rate of heteronuclear collisions per unit volume