(1)Kinetic molecular theory, KMT for short, is very different from thermodynamics although it deals with some of the same variables, such as pressure, temperature, volume, and density.
Thermodynamics does not care whether molecules exist or not. Essentially all the results of thermodynamics can be worked out without any assumptions concerning the particle nature of matter. KMT, however, is a molecular theory (or a molecular model) of matter. (Molecular theories of matter are sometimes called "microscopic" models as opposed to the "macroscopic" model of bulk material. KMT is a microscopic model of a gas.)
Thermodynamics assumes equilibrium (infinite time), KMT
assumes molecules are in motion and includes time explicitly, for example
in molecular velocities.
The Model
Assume that the gas consists of:
N molecules (N is large, on the order of Avogadro's number, NA = 6.023 × 1023 molecules/mol)("Isotropic" means that the properties of the gas are the same in every direction. Crystals, for example are not isotropic because some of their properties depend on your direction relative to the crystal axes. The statement that the gas is isotropic and chaotic also includes the notion that motion in any one of the three coordinate directions is independent of the motion in the other two dimensions.)
The molecules have mass, m,
The molecules are contained in a volume, V,
The gas is at some temperature, T (which is the Kelvin temperature).
The dimensions of the molecules are small compared to the average distance between molecules and compared to the size of the container,
There is no potential energy of interaction between the molecules. (Potential energy can be added in later if desired.)
The molecular motion is random, or chaotic, so that the gas is isotropic.
We will use the "number density," N/V, the number of molecules per unit volume, instead of the usual mass density.
Since we are dealing with individual molecules we will write the gas constant on a molecular basis instead of a molar basis, and we will give it a new name and symbol, k or kB. that is,
kB (or k) is called "Boltzmann's constant." However it is really not a new constant, it is just the gas constant per molecule instead of per mole. You can easily find, using Equation 1, that kB = k = 1.380658 × 10−23 J/K. (We will most often symbolize Boltzmann's constant simply as k in our discussions.)
We will assume that the ideal gas law holds. That is, pV = nRT = NkT.
If that last term is a problem, note that,
(We have made use of the fact that Avogadro's number times the number of moles in the system gives the number of molecules, N, in the system.)(2)
Consider one molecule.
The molecule moves in three spatial dimensions, x, y, z, so it has a velocity which is a vector quantity. Write,
(Note that we are using(3)
, 
,
and 
for
unit vectors directed along the three Cartesian coordinate axes instead
of 
,
,
and 
). The
velocity is a vector quantity and has both a direction and a magnitude. Occasionally we will want to
refer just to the magnitude of the velocity. We will refer to the magnitude
of the velocity, independent of direction, as the "speed" of the molecule.
The molecule has a kinetic energy which we will write as E, defined as usual,
The internal energy of the gas in this model is the sum of the kinetic energies of all the molecules. That is,(4)
where the summation is over all the molecules in the gas.(5)
It is impossible to track the motion of 1023 molecules so we must deal with averages. We will write the average kinetic energy of one molecule as,
where the angle brackets indicate that we are taking the average of the quantity enclosed within them. Since we can't add up all the individual kinetic energies of all the molecules we will obtain the internal energy of the gas by just multiplying the average energy by the number of molecules,(6)
Since(7)
and we know that the average of a sum of terms is the same as the sum of the averages of the individual terms, it is easy to see that(8)
Since the motion is random, and hence isotropic, there is no preferred direction for the motion. The average velocities (and the average of the velocities squared) must be the same in every direction. That is,(9)
hence(10)
or(11)
(We could have just as well used vy or vz here, it makes no difference.)(12)
A little later on we will need to talk about the momentum of a molecule. The momentum of a molecule is defined as usual:
(13)
or, in terms of components,
etc.(14)
Velocity Probability Distribution Functions
Probably the simplest way to deal with random motion is
to use probabilities (or statistics). In order to use probabilities or
statistics we need to define and use the velocity probability distribution
function for motion in one dimension, 
.
(Sometimes we will just call this the velocity distribution function, but
it really is a probability distribution function.) This function is defined
so that (and this is important)
f(vx) dvx = the probability that a molecule has x-component of velocity between vx and vx + dvx.
(Probability distribution functions are very important and very useful in chemistry and in many branches of science. We will define two more velocity-related probability distribution functions in a similar manner in this course and you will see two or three more probability distribution functions in the introduction to quantum mechanics and in statistical mechanics.)
Since the molecule must have some velocity the sum of all these probabilities must be equal to unity. The way we sum all these probabilities is, of course, by integration. That is,
Equation 15 says that our probability distribution function is "normalized." If the integral in Equation 15 does not equal unity the function is not normalized and would not be a valid probability distribution function. We can normalize an unnormalized distribution function by multiplying it by a suitable constant which will make it satisfy equation 15.(15)
(Our KMT is a nonrelativistic theory. The molecules never get close to the speed of light so we don't have to worry about the fact that material particles can't go faster than the speed of light. This is reflected in kinetic molecular theory in that the probability function for molecules being near the velocity of light becomes vanishingly small.)
We don't know the form of f yet, but we will figure out what it is later.
There are several properties of f that we can tell right away. We will assume that our sample of gas is not going anywhere so the bulk velocity is zero in every direction. This means that
In words, this means that the properties of the velocity distribution function must be the same in the negative x-direction as they are in the positive x-direction. Mathematically we would say that that f is an even function of vx.(16)
From the fact that the gas is isotropic we conclude that f (vx), f (vy), and f (vz) all have the same functional form so that if we can find the form of one of them we will know them all.
We can use these velocity probability distribution functions
to find averages of quantities that depend on velocity. (As already stated,
we will denote averages by angle brackets, 
.
We have already done this with average kinetic energy and average of vx2.)
Find averages as follows: Take the value of your molecular property that you want to average and evaluate it at some velocity, vx. Multiply that value by the probability that the molecule has the velocity, vx, and then sum over all velocities. It's easier to write this as an equation than to say it in words,
where you place the quantity you want averaged within the parentheses. For example, if you wanted the average of the kinetic energy in the x-direction you would write,(17)
Other examples:(18)
and so on.(19)
(20)
(21)
(22)
Let's take a look at the average of vx.
In these equations we have made use of several things we know about integrals and about the function f(vx). You can break an integral from −infinity to +infinity into the sum of two integrals, one from −infinity to 0 and the other from 0 to +infinity (Equation 23a). Also, when you interchange the upper and lower limits of integration the integral changes sign (going from Equation 23c to Equation 23d). We have also used the fact that f is an even function of vx going from Equation 23b to 23c).(23,a,b,c,d,e)
There are several reasons why we would expect the average of the x−component of velocity to be zero. One, vx is an odd function so that vxf (vx) is an odd function. The integral over the entire x−axis of an odd function is zero. Second, we have assumed that the gas sample is standing still. That is, it is not moving in any direction. Therefore, the average of any of the three components of velocity should be zero.
However,
We can calculate(24)
without knowing the detailed functional form of f by considering
the pressure of a gas against a wall. The pressure of a gas against a wall
is caused by collisions of gas molecules with the wall. When a molecule
collides with a wall it exerts a pulse of force against the wall. Since
there are so many molecules colliding with the wall the pulses average
out to give a pressure that seems constant. We know that
but according to Newton's second law(25)
so that(26)
Let's consider a portion of the wall of area, A, and a molecule with x-component of velocity, vx. Think about what happens in a small time interval Δt. In the time Δt the molecule will travel a distance vxΔt in the x-direction. (Let vx be positive so that the molecule is traveling in the +x direction.) The molecule will hit the wall in a time Δt if it is within a distance vxΔt of the wall. This distance and the area, A, create a small volume A vxΔt. Let the number density be N/V. Then this small volume contains(27)
molecules. The number of these molecules which have velocity, vx, is(28)
(That is, the number of molecules with velocity vx in our small volume is equal to the number of molecules in that volume times the probability that a molecule has velocity vx.)(29)
Consider, now, what happens when the molecule hits the wall. The molecule has initial momentum, mvx directed toward the wall. Assuming that the collision is elastic the molecule will bounce off the wall with momentum, −mvx away from the wall. The change of momentum for the molecule is
final momentum − initial momentum = − mvx − (+mvx) = − 2mvx. (30)According to the laws of physics, momentum is conserved, so this momentum had to be transferred to the wall. The momentum transferred to the wall is +2mvx. Then the momentum transferred to the wall by all the molecules with this velocity is
We obtain the total momentum transferred to the wall in time Δt by integrating this expression from 0 to ∞ . (We integrate only over the positive values of vx because the molecules with negative vx are going the other way and will not hit the wall.) The total momentum transferred to the wall in the time interval Δt is then,(31)
(The integral from zero to infinity is just half of the integral from minus infinity to infinity because the integrand is symmetric.)(32a, b)
The rate of change of momentum is this quantity divided by Δt,
Since the rate of change of momentum is, in fact, the force on the section of wall, the pressure is then,(33a, b, c)
So, if we combine this with the ideal gas equation of state we get(34a, b, c)
or(35)
or(36)
But we know that,(37)
so(38)
from which we get(39)
We call this average vrms for "root-mean-square" average velocity and it is defined by Equation 40. (Root-mean-square, or rms, averages are a common way to describe the magnitudes of quantities which average to zero. For example, common house current (electricity ) is usually around 110 volts. However, household electricity is "alternating current" so that the voltage alternates between positive and negative values such that the average voltage is zero. If you place the probes of a direct current volt meter in the terminals of a wall electrical outlet you will get a reading of zero - if it doesn't burn your volt meter out. But you can easily determine that there is a voltage there by shorting the terminals out with your fingers.)(40)
Equation 40 is the first of three quantities that we will
define and use to describe the velocities of molecules in a gas. We will
see that all three of the velocity measures will have the same order of
magnitude but will differ slightly in their exact value.
Example
As an example, let's calculate the rms velocity of a nitrogen molecule at 25oC. Sometimes it is easier to work with a mole of molecules instead of individual molecules. It is easy to see that.
where FW is the formula weight of the gas (in kg). For N2 the formula weight is 0.028013 kg/mol. R is 8.3145 J/K mol so the rms velocity of a nitrogen molecule is,(41)
(42)
We will leave it to the reader to show that the units
really do work out to give m/s. (515.25 m/s is about 1153 miles per
hour.)
WRS
From here you can:
Copyright 2004, W. R. SalzmanReturn to the local Table of Contents,
Return to the Table of Contents for the Dynamic Text, or
Return to the WRS Home Page.