Readings for this section

Petrucci: Section 6-7

Kinetic Molecular Theory

To understand a gas and it's properties, we need to lay out a few concepts. In the kinetic molecular theory, there are several postulates upon which we base our explanations.

  1. a gas is composed of molecules that are far apart from each other in comparison to their own dimensions (most volume occupied by a gas is empty space).
  2. gas molecules are in constant random motion, each molecule continues to move in a straight line unless it collides with another molecule or with a wall of the container
  3. the molecules exert no force on each other or on the wall unless they collide. These collisions are elastic (the total translational kinetic remains unchanged).
  4. the average kinetic energy of the molecules is proportional to the absolute temperature.

The kinetic energy of a single molecule is

KE = e = 1/2 mv2 where m is the mass of the particle and v is the speed.

Let's consider a single molecule of mass m traveling with speed v in a box where all sides (x,y, and z) have length L. The average time it takes for the molecule to strike a wall, bounce around and return to a wall to strike it again depends on the component of speed perpendicular to the wall and on L.

Thus, for the wall perpendicular to the x direction, the number of times per second a molecule strikes one wall is . (it must travel length L twice between each collision with a given wall.

Since the collisions are elastic, the change in momentum is simply mvx – (–mvx). or 2mvx. Thus, the change in momentum per second of one wall is

.

Since force equals the rate of change of momentum, we've just calculated the force exerted on the wall. We know that Pressure is measured as Force divided by area so we can now calculate the pressure exerted by a single molecule in the x direction.

since L × A = Volume (V). similarly, we can measure force and pressure in y and z directions. The overall pressure caused by one molecule will be the average of the three component pressures

NOTE: Note that when talking about the speed of the molecule in 3-d, the text uses the variable u. I will continue to use v for the speed in any dimension.

We can extrapolate this idea to a gas sample with N molecules in it by using,  instead of the speed v, the average of the square of the speeds. This is done to allow greater influence of the outlier speeds in determining the average value. This is the mean-squared-speed.

NOTE: Angle brackets <v> means average and may also be written as v with a bar over it. Since that particular typesetting is difficult to do in a web page, I'll stick with the equally valid alternate means (pun) of indicating average.

Thus,

PV = 1/3 Nm< v2 > .

The average translational kinetic energy

< e > =1/2 m < v2 >

can now be incorporated into this equation.

Remember that PV has units of energy:
P º Nm-2 and V º m2, so
PV º Nm º Joules

Thus, for one mole (n=1) of an ideal gas we can write

Total translational kinetic energy per mole

KE = 3/2 PV = 3/2 RT

We have seen that P comes from collisions with the walls. If the volume is reduced then the length of time it takes a molecule to travel back and forth between the walls is reduced and the number of collisions goes up, thus the pressure goes up.

Now let's look at the effects these have on our relationships we've seen earlier:

Boyle's Law says that, for a sample of gas, PV is constant at any given temperature.

We can see that the Kinetic molecular theory predicts that PV is dependent only on the number of molecules and on the temperature. If we reduce the volume of a container of gas by 1/2 (at constant T) then the pressure should rise by 2.

Charles' Law says that the volume to temperature ratio of a gas is constant. Again, we can see that this exactly what theory predicts.

Notice that I did not specify the molar mass of the molecules when I stated that kinetic energy ? was proportional to T. Thus whether the molecules are heavy or light or if there is a mixture of molecules they all have the same (average) kinetic energy at a given T.

This is called the Principle of Equipartition of Energy.

We can look at how molecular motion varies with molar mass using a few simple
relationships. Let's consider one mole of an ideal gas. There are NA molecules in
one mole.

We can measure this effect quite handily. Heavy molecules move slower than lighter molecules. Thus, if we were to release two different scents into the air at one time, the one coming from the lighter molecule would be smelled (assuming no wind currents) first by an observer as some distance from the two samples. This effect is also seen when we have helium balloons The helium is a light molecule and quickly escapes through the pores in the latex rubber of the balloon. Balloons inflated by breath (with heavier air molecules) stay inflated much longer.

One other consequence of the Kinetic Molecular Theory is the fact that we've not actually been able to measure the speed of any given molecule, just their average speeds. There is an equation called the Maxwell-Boltzman distribution, which predicts the proportion of molecules in a gas sample to have any given speed (or kinetic energy). We can see this in action from the following diagram plotting the fraction of molecules versus the speed of the molecules for a sample of molecules at several temperatures. An Excel spreadsheet has been kindly supplied by Prof. Horton that allows you to experiment with these settings. You can download it from here. Try inserting different molar masses or temperatures into the data page and see what happens to the graph.