Chapter 2The Kinetic Theory of Gases

DefinitionThe Kinetic theory of Gases (KTG) is model used to explain the relation between the macroscopic properties (pressure, temperature, internal energy) and microscopic properties (mass of particles, momentum, force, kinetic energy).

Some Assumption in TKGIn TKG, gases is assume as a ideal gasThe characteristics of ideal gas: The number molecules is large and the gas is very dilute.The gases move randomly. The distance of each particle>> size of particle so is assume as a pointThe molecules do not interact except when they undergo collisionsThe collisions are elastic.

The Ideal Gas LawExperimentally, it is found that pressure, volume and absolute temperature of a gas obey the following equation of state, called the ideal gas law; P= pressureV=Volumen=number of mole gasR=The Universal Gas Constant R=8.31 J/KT=TemperatureN=number of particleK= boltzamann constant 1.38 x 10-23 J/K

Example A motorist starts trip on cold morning when the temperature is 4oC, and checks her tire pressure at a gas station and finds the pressure gauge reads 32 psi. After driving all day, her tires heat up, and by afternoon the tire temperature has risen to 50oC. Assuming the volume of tire is constant, to what pressure will the air in the tires have rise?Standard temperature and pressure (stp) for a gas is defined as 0oC (273 K) and 1 atm (1.013 x 105 Pa). What volume does 1 mole of ideal gas occupy?

Molecular Basic of Pressure and Temperature Assume a particle in a cubical container of side d

A particle moving along the positive x axis with velocity vx will collide elastically with a wall and bounce back with velocity vx. Its momentum in the x direction will change from +mvx to mvx. After striking the wall, the particle will bounce back and travel in the negative x direction until it strikes the opposite wall and rebounds. Again it moves in the positive x direction until it strikes the first wall a second time.

The change of particle momentum can be found as;

The time between collisions with the first wall is:

The force exerted on the particle by the wall in order to change its momentum is, from newtons second law,

By the newtons third law, the force exterted on the wall by the particle is The total force for exerted on the wall is the sum of the forces exerted by each particleBut the average value of vx2 for N particles is

Thus: If one of particle has velocity component vx, vy,vzSince the motion is random v2x= v2y= v2z=1/3v2So,The total force on the wall is thus

The pressure on the wall isIf We can find the temperatureThe absolute temperature of gas is proportional to the average molecular kinetic energy

SinceThusThe last equation is called the theorem of equipartion of energy that says that eachDegree of freedom of gas contributes an amount of energy KBT to the totalenergy

Example of degree of freedom 3 degrre of freedom7 degrre of freedom

For particle the total kinetic energy is

This equation is represented the total internal energy of n particleFrom equation the total energy we can solve the root mean square (rms) molecular speed R=NAKBM=NAmM= mass of molecules

ExampleWhat is the rms speed of a nitrogen molecule (N2) in air at 300 K.

The Maxwell-Boltzmann DistributionThe molecules in a gas travel at wide range of speeds.Distribution of speed of particles is stated by maxwell

where m is the mass of a gas molecule, kB is Boltzmanns constant, and T is the absolute temperature. Observe the appearance of the Boltzmann factor with

From the figure we can findRms speedAverage speedMost propable speed

Molar specific Heat of GasMolar specific heat of gas is the amount of energy that must be added to 1 mole of gas to increase its temperature by 1 degree.In constant volume the specific heat is defined as cv , where cvdT=dE, so:For monoatomic gas Cv=3/2R

At constant pressure, for monatomic gas specific heat (Cp)Cp=3R/2 +RCp=5/2 RIn general, for any gas ideal,Cp=Cv+R

Example4.0 moles of argon gas is contained in a cylinder at 300 K. How much heat must be added to the gas to raise its temperature to 600 K atConstant volumeConstant pressure

**********************