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THERMAL PHYSICS

 

The Kinetic Theory of Gases

 

assumptions

derivation

KE & temperature

 

 

 

Main Assumptions of the Kinetic Theory

 

1. all the molecules of a particular gas are equal

 

2. collisions between molecules and their container are completely elastic

 

3. collisions between molecules themselves are completely elastic

 

4. the size of actual molecules is negligible compared to molecular separation

 

5. the laws of Newtonian mechanics apply

 

6. extremely large numbers of molecules mean that statistical methods can be applied

 

7. between collisions molecules move in straight lines at constant speed

 

8. the motion of molecules is random

 

9. gravitational effects are negligible

 

10. the time for each collision is negligible

 

 

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Derivation of Equations

 

The first step in understanding this theory is to appreciate the momentum change when a particle rebounds from a collision with a wall.

 

 

particle colliding with a wall

 

 

initial mtm. in the x-direction       = mu1

 

final mtm. in the x-dir.                = m( - u1) = - mu1

 

momentum change in the x-dir.   = mu1 - ( - mu1) = 2mu1

 

 

 

Consider a volume of gas in a cuboidal shape of side L.

 

 

particle in a box

 

 

We have seen how the change in momentum of a molecule of gas when it rebounds from one face , is 2mu1 .

 

The distance the molecules travels between collisions is 2L.

 

It collides with face A. Moves a distance L to collide with opposite face B, before returning to face A .

 

Since,

speed = distance/time,           time = distance /speed

 

Therefore the time (t) it takes for the molecule to traverse this distance 2L is given by:

 

 

The rate of momentum change (dp/dt) in this time interval is given by:

 

rate of momentum change - word equation

 

rate of change of momentum equation

 

From Newton's 2nd Law, applied force is equal to the rate of change of momentum.

 

The molecule therefore exerts a force F on the wall, given by:

 

force in terms of rate of change of momentum

 

Since pressure (p) = force/area , the pressure on wall A produced by a single molecule is,

 

kinetic theory derivation of pressure on one face from a single molecule

 

Now, consider all the molecules (N) in the cube and each of their x-component velocities (u1 u2 u3 . . . uN).

 

The pressure on wall A becomes:

 

kinetic pressure due to all molecules

 

where,

 

u squared bar is the mean square velocity of molecules in the x-direction

 

 

The density ( ρ rho ) of the gas is given by:

 

kinetic theory - gas density

Substituting for density of gas into the equation for pressure p

kinetic pressure due to all molecules

 

pressure in terms of density    

 

 

Up to now we have considered the velocity of a molecule in one direction.

 

Now, let us consider the resultant velocity of a molecule in three dimensions.

 

 

molecule component velocities

 

 

In the diagram, molecule velocity (c1) is resolved into x, y and z directions. The value of each component is respectively, u1, v1 and w1.

 

Side d is the projection of c1 on the x-y plane.

 

 

molecular velocity components

 

 

Using Pythagoras' Theorem,

 

components equation #1

velocity compenents equation #2

 

substituting for d2 in the second equation,

 

component velocities for a single molecule

 

The equation represents component velocities for one molecule.

 

Replacing these velocities with mean square velocities, the equation now applies for all molecules.

 

jinetic theory - mean square velocities

 

We make the assumptions that there are very large numbers of molecules and their motion is random. So we can say that mean square velocity components are equal to one another.

 

equivalence of mean square velocities

 

So molecular velocity c in terms of components becomes,

 

component velocity equation

 

Substituting for u squared bar into the equation for pressure previously obtained,

 

pressure in terms of density

pressure in terms of density and mean square velocity

 

 

 

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Molecular KE and Temperature

 

kinetic theory pressure equation

 

Multiplying the Kinetic Theory equation for pressure by V, (the volume of the gas) we obtain:

 

molecular KE & temperature equation #1                  (i            

 

but density ρ (rho) is given by:

 

density in terms of mass and volume

 

making the mass M the subject, (where M is the mass of gas)

 

mass in terms of density and volume

 

substituting for ρV into the Kinetic Theory equation (i ,

 

kinetic theory equation #4                  (ii            

 

With some simple arithmetic and a more detailed description of M, this equation can be amended into a more useful form:

 

one third is  two thirds multiplied by a half

 

If N is the total number of molecules and m is the mass of one molecule:

 

mass M  in terms of N and m

 

Now, substituting for 1/3 and M into equation (ii ,

 

molecular KE equation #7

 

The Ideal Gas Equation is,

 

ideal gas equation

 

where,

 

n is number moles of gas
R is the Universal Gas Constant
T is the temperature in kelvin

 

 

Elimenating pV between the last two equations,

 

molecular KE equation #8

 

Making molecular KEthe subject of the equation,

 

molecular KE equation #10                 (iii            

 

The Avagadro Number NA is by definition the number of molecules per mole. It is obtained by dividing the total number of molecules by the number of moles of matter:

 

Avogadro Number in terms of numbers of molecules and moles

 

We can now modify equation (iii to include NA by substituting for n/N ,

 

molecular KE equation #12

 

By definition the Boltzmann's constant k is given by:

 

Boltzmann's constant

 

So the final form of the equation is:

 

molecular KE equation #14

 

 

average translational KE of a molecule is called the average translational KE of a molecule

 

 

So the average kinetic energy of gas molecules is proportional to the temperature.

 

This can also be said in the converse:

 

temperature is a measure of the average kinetic energy of gas molecules

 

 

 

 

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