Now we must treat the motion of a molecule in three dimensions.
We know that the gas is isotropic (all directions are equivalent) so that
the velocity probability distribution function must be the same in all
directions. This means that the distribution functions, *f*(*v _{x}*),

Define the three dimensional velocity probability distribution
function, *F*(*v _{x}*,

Since the three directions are independent of each other this probability must be written as the product of the individual probabilities in the three directions. That is,F(v,_{x}v,_{y}v)_{z}dv= the probability that a molecule has_{x}dv_{y}dv_{z}

anx-component of velocity betweenvand_{x}v+_{x}dvAND_{x}

ay-component of velocity betweenvand_{y}v+_{y}dvAND_{y}

az-component of velocity betweenvand_{z}v+_{z}dv._{z}

All the differentials cancel so that we conclude thatF(v,_{x}v,_{y}v)_{z}dv=_{x}dv_{y}dv_{z}f(v)_{x}dv(_{x}fv)_{y}dv(_{y}fv)_{z}dv. (1)_{z}

We are now going to use this three-dimensional velocity probability distribution function to determine the form of the one-dimensional distribution function,F(v,_{x}v,_{y}v) =_{z}f(v)_{x}f(v)_{y}f(v). (2)_{z}

Following the derivation of James Clerk Maxwell - a giant of science - we take the natural logarithm of Equation 2,

(3)Now take the derivative of this with respect to

(4)We have already defined the molecular "speed,"

(5)The function

With this understanding, let's rewrite the left-hand derivative
as a derivative with respect to the speed, *v*,

. (6)(One might ask why Equation 6 does not include derivatives with respect to the angles which determine the direction of the velocity. Since we know that the gas is isotropic, none of its properties can depend on direction so that the angular derivatives of ln

Now, we know (or can figure out easily from Equation 5) that,

(7)Placing this result in Equation 6 we get

(8)and then inserting this back into Equation 4 we find that,

(9)Rearranging to place everything involving

(10)We could just as easily have carried out this derivation with

(11)In Equation 11

(12)or

(13a, b)We can easily integrate Equation 13b to obtain (calling the constant of integration ln

(14)or, taking the exponential of both sides,

(15)Equation 15 gives the functional form of

The function 15 contains two unknown constants, but we have enough information to determine what these constants are. First, the function must be normalized,

(16)and, second,

(17)Let's look at the normalization first,

(18)This integral is a member of a class of integrals called "Gaussian integrals" and can be found in any standard table of integrals or on this web site,

(19)The normalization integral becomes,

(20)from which we see that

(21)The function,

(22)We determine the constant,

(23)which becomes, after inserting our form for

(24)This integral is another one of the Gaussian integrals and can be found in standard integral tables or at the reference given above,

(25)which gives,

(26)or

(27)We can now put together the complete form of

(28)This equation is one of the fundamental equations of kinetic molecular theory. It will provide the basis for all of our other probability distribution functions. Notice that the exponent is a kinetic energy (or a mechanical energy) divided by a thermal energy,

**Applications of f(v_{x})**

Our first application of our new-found one-dimensional velocity probability distribution function will be to calculate

(29a, b)This integral you can do without an integral table. Change the variable, let

(30)then

(31a, b)When we change variables in the integral we have to change the limits also. When

(32a, b, c)The integral,

(33)so

(34)You will see why we made this last change on the right-hand-side in a little while.

**Three-dimensional Speed Distribution**

(Recall that velocity is a vector quantity and has three components, but speed is the magnitude of the velocity and contains no directional information. We will be very careful to preserve this distinction in all of our discussions.)

Let's go back to our original three-dimensional velocity
distribution function. Recall that we defined the three dimensional probability
distribution function, *F*(*v _{x}*,

Recall also that we can writeF(v,_{x}v,_{y}v)_{z}dv= the probability that a molecule has an_{x}dv_{y}dv_{z}

x-component of velocity betweenvand_{x}v+_{x}dvAND_{x}

ay-component of velocity betweenvand_{y}v+_{y}dvAND_{y}

az-component of velocity betweenvand_{z}v+_{z}dv._{z}

. (35)

. (36)The same is true when we transform from a Cartesian velocity space to a polar velocity space,

.) (37)So,

(38)We already know that the function,

So let's see whatF(v,_{x}v,_{y}v) =_{x}f(v)_{x}f(v)_{y}f(v). (39)_{z}

(40a, b, c, d)We can just as well define the velocity probability in terms of speed,

This probability is, then,F(v,θ,φ)v^{2}sinθdθ dφ dv= the probability that a molecule has

a speed betweenvandv+dvAND

a polar angle of the velocity vector betweenθandθ+dθAND

an azimuthal angle of the velocity vector betweenφandφ+dφ.

(41)If we sum (integrate) this probability over all angles (

(42a, b, c)We conclude that the "speed probability distribution function" is,

(43)We can use this speed probability distribution function to calculate two new things. First, note that the range of

(44)We will leave it to the reader to show that this is true so that the function,

**Applications of the Speed Distribution Function**

With our speed distribution function we can calculate the average speed as follows,

(45a, b)From the integral tables, or by working it out, we find that

(46)so our integral

(47)Then

(48)Notice that,

(49)We now have two different measures of the average velocity, the rms velocity,

(50)and the average speed,

(51)These two averages are very close to each other (√ (8/π ) = 1.60 and √ 3 = 1.73) even though they describe slightly different aspects of molecular average velocities.

**Most probable Speed**

One other application of the molecular speed distribution function is to find the "most probable speed." If we look carefully at this distribution function,

(52)we see that it is zero at

(53)Note that there is no need to keep the constants in because the position of the maximum does not depend on the values of the constants. So we can write,

(54)The exponential cancels out and we solve

(55a, b, c)which gives

(56)We now have three measures of various aspects of molecular speeds,

(57)and(58)

(59)all of which are close to each other in magnitude. Each of theses quantities gives us an estimate of how fast molecules are moving in a gas at some temperature,

WRS

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