KINETIC THEORY OF GASES. PART I ROBERT N. PEASE Princeton University, Princeton, New Jersey
I
N HOMOGENEOUS gas reactions activation is assumed to be a product of the energy exchanges resulting from molecular collisions. Consequently we want to be able to calculate collision rates, and steady-state energy distributions. Information on these subjects is supplied by the kinetic theory. The development will be rather more complete than is absolutely essential because kinetic theory is of fundamental interest to students of physical chemistry. We begin our discussion with a consideration of the properties of an ensemble of perfectly elastic, non&acting spheres of fixed diameter, in rapid and random motion. This is the model of a monatomic ~ e r f e c gas, t if we add the restriction that the averane distanc'between the spheres is very large as compargd to their diameters. Individual sphereslet us call them molecules from now on-are continually changing their velocities as a result of collisions with other molecules and with the walls of the container. Nevertheless, a t constant energy (that is, a t constant temperature) the distribution of velocities among all molecules in the ensemble is assumed to hold close to a b e d pattern-the steady-state distribution. Our first problem is to obtain an expression for this steadystate distribution. We will follow Maxwell's original derivation, although it has been subject to criticism for a reason which will be discussed later. Maxwell's Distribution Law.*-The translational velocity, c, of a given molecule may be expressed in terms of its rectangular components, u, er, w, cZ = 3
+ + w2 V=
we assume--and
this is the assumption that has laid the derivation open to criticism-that for a picked a t random the values of u, W, and w are entirely independent of one another. Thus, a large value of the component, u, implies neither large nor small values of w and w. If this is true, the chance that the u component lies between the limits u and u + du is a function of u alone. If this chance is w,, we may write
W. = f(u)du (The magnitude of W. obviously is proportional to the magnitude, du, of the range between u and u du. Over this infinitesimal range, j(u) is constant.) Similarly, for er and w,
These functions will obviously all be of the same form since there is nothing to choose as between the three components. (If some force, such as that of gravity, acted along one component, this would of course no longer be true.) The form of the functions, f(u), f(er), j(w), is obtained as follows. The chance, WUm,that a molecule chosen a t random has simultaneously a value of u between u and u of u between u and u and of w between w and w
S e e HIN~EELWOOD, "Kinetics of chemical change in gaseous systems," 3rd ed., Oxford University Press, Oxford, England, 1933, pp. 6 seq.; JEANS. "Dynamical theory of gases," 4th ed., Cambridge University Press, Cambridge, England, 1925, pp. 55 sep.; JELLINEK. "Lehrbuch der Physikalischen Chemie,"F.Enke. Stuttgart, Germany, 1936, Vol. I, pp. 139 seq.
+ + dv. dm,
is the product of the independent probabilities,
w.,
=
w..w,.w.
= f(u)du~f(u)dv~f(w)dw = f(u)f(u)f(w)dududw
This is also the fraction,
% , N
of all molecules with
components in the prescribed ranges,
dN "" N
=
dN...
= Nf(u)f(~)j(w)dudwdw
W,,
= f(u)f(v)f(w)dududw,
or At this point it is helpful to give the problem a graphical representation. We think of a rectangular coordinate system with axes, u, w, w. At any instant we assume to make an observation of u, u, and w for every molecule. These observations are plotted. The result is a distribution of points throughout the coordinate system, each point representing a molecule In terms of its velocity components. The quantity, dud~dw,represe~tsan infinitesimal volume-element. In such a volume-element a t any position u, er, w in the System there will be the number, dN,,, of points representing molecules having the corresponding ranges of represents the density of u, nt, and w. Thus, dN,, d a w
points a t that spot. This is given by
+
-
+ du,
dN"." dudvdw
= Nf(ir)f(o)j(w)
Now we can make an important statement about this distribution of points in our coordinate system. It is obvious that each point corresponds notonlyto values of u, u, and but also to a value of c, the velocity, since
242
c' = uz
+ v2 + wa
The quantity, c, is the distance from the origiu to the corresponding point. The direction of the point with respect to the origin corresponds to some particular direction of the path of the molecule in space. Now since all such directions are equally probable, it follows that the distribution of points in the coijrdinate system will be symmetrical with respect to the origin. Hence a t a distance, c, from the origin the poiut-density, and hence f(u)f(u)f(w), will be the same irrespective of direction and of the individual values of u, u, and w. These considerations lead to a solution for the form of the distribution function. Suppose we vary u, u, and w infinitesimally by the amounts 6% 60, 6w keeping c constant. This amounts to altering infinitesimally the direction of c without altering its magnitude. For the variation, 6c, and therefore 6cz, is zero 6ca = 0
+ uf + wz 6(ua + o' + wa) = u6u + o8u + w6w = 0 P = u2
(1)
The variation also leaves f(u)f(u)f(w) unaltered a[ f ( u ) f ( ~ w I = ) 0
The quantity, A, was chosen to satisfy a particular value of u. It now appears in relations involving some other values of w and w. Evidently i t must be a constant for all these relations to hold true. The form of the functions f(u), f(v), and f(w) follows a t once. Remembering that
we have
and similarly for u and w. (Note here that the exponent must have a negative sign as otherwise the distribution function would increase without limit as u increases. This would lead to probabilities greater than unity. Likewise i t would lead to infinite gas pressures.) It is convenient to set
whence bf(u) and so forth. Write f'(u) for bu
f(u) = ' ~ c - . - / - =
A may now be expressed in terms of a as follows. The chance that a molecule has a value of u between u and u du is
+
f(u)du = Ac-uV='du
dN. This also expresses the fraction, -, of all molecules
N
Equations (1) and (2) are solved as follows. Multiply equation (1) by a quantity, X, such that
in this range of u values.
Thus,
dN. = NA~-u'/=Idu
Now obviously Then, Au6u
+ Au6v 4- Awdw
Now add (2) and (3)
[f%
+AU]
6,
+ [f%
+ A"]
6,
+
=
0
(3)
[fs; + +w
= 0
The first term is obviously zero by choice of A, whence
(Values of u taken from the origin may be either positive or negative.) Then
+
m
N A f e-='/ai du = N
-m
But 6u and 6w may have any infinitesimal values whatsoever, the correction necessary to keep c constant being effected by a corresponding alteration in 6u. Equation (4) can then only be true in general if
and
The value of the integral is ru&
whence
JEANS, ''Dynamical theory of gases," 4th ed., Cambridge University Press. Cambridge, England, 1925.
and j(u)du =
1 e-ui/=l
du
a = c,, the most probable velocity. It is customary to express a in terms of the root The Distribution Lawfor the Vehcity, 6.-The chance mean square velocity rather than of the most probable that a given molecule has values of u, v, win the ranges
+ du u + dv
u and u u and
velocity. Let c, = the root mean square velocity. This is defined by the equation
c , ~is the sum of all the squared velocities divided by
dNUW This is also the fraction - of molecules with comN ponents in the stated ranges. In a u,v, w coordinate system dN., is the number of points representing molecules in the volume du, dv, dw located in the region defined by u, v, w . The quantity
-
the total number of molecules. Substituting for dN,
era ' =
~
f
4
ca p r e - W = 2
c2Jc
0
The value of the integral is a/s&a5, whence represents the density of points a t u m . The points corresponding to values of the velocity between c and c dc will lie in a spherical shell of radius c and thickness dc. The volume of this shell is 4rrc2dc. The density of points in this shell will he everywhere the same since all directions for the velocity are equally probable. The density a t the position defined by u, v, w was given above. Multiplying this by the volume of the shell, and substituting
+
we obtain for the number dN, of molecules with values of the velocity between c and c dc
+
Substituting this in the original equation for dN,
The advantage of using c, comes from the simple relation between this and the temperature. This relation (which we shall later obtain) is mc,? = '/. kT where m = mass of the molecule k = gas constant per molecule
We may now identify a. From the above equation
Thus,
dN, against c, we obtain a curve starting dc from the origin, rising to a maximum and falling to zero when c becomes infinite. The maximum corresponds to the velocity about which most molecules are grouped, and is called the most probable velocity, c,. Its value is obtained by taking the second derivative and equating to zero (corresponding to the maxiIf we plot
,a,
mum in the plot of
alv
dc
against c)
We may also write JN,
N
=
--
I/lmut
Aoerage Velocity.-We have incidentally defined two characteristic velocities-the most probable, and
the root-mean-square. We may as well introduce This is obtained here a third-the average velocity, 6. from the relation Scd
C. =
N.
0 -
N
The value of the integral is cr4/2, whence
with respect to the line of centers will involve more than one of the components of velocity on the uvur coordinate system. Thus, the exchanging of momenta in the line of centers involves an alteration of more than one velocity component. More than one velocity component will thus appear in the equation for this exchange. Hence the velocity components are related. This criticism of the Maxwell method is perfectly valid. Nevertheless, i t is to be pointed out that the method is applied not to the mechanical process by which the velocity distribution is achieved, but to the steady-state distribution which results. It is comforting, moreover, to find that a more exhaustive analysis of the problem leads to the same result. The Energy Distribution Law.-Maxwell's law of distribution of molecular velocities provides a basis for a distribution law for molecular energies. A molecule with a translational velocity, c, has a translational kinetic energy, mc2. For the probability, W,, that a molecule has a translational velocity between c and c dc, we can write
+
We may add for completeness where These three characteristic velocities are not very different numerically. In terms of To obtain the corresponding probability, We,, that the translational energy lies between r, and e, dh, we substitute
+
e, = I/, mc2
whence M
= molecular weight.
Just as the energy, '/zmcZ, corresponds to the Criticism of Above Derivation.-The form of the distribution law which we have obtained by Maxwell's velocity, c, so to each component of c there corresponds original method is the same as that obtained by all a component of the translational kinetic energy. Since methods, and may therefore be taken to be correct. ct = u9 + o2 + w z The derivation was stated to be open to question in '/s mca = muz + I/% mv2 + mwa that i t assumes the distributions of the velocity components to be independent of one another. This Thus, the distribution law for c is also a distribution assumption has been criticised because (as Jeans puts law for the sum of the components of the translational it), "The velocities do not enter independently into the kinetic energy. Wecis in effect the chance that this dynamical equations of collisions between molecules, sum has a value within the prescribed limits (ec to so that until the contrary has been proved, we should E, de,) without regard to how it is distributed among expect to find correlation between these velocities." the three components. The meaning of this may be made clear by the following. We will want to use a similar expression for the total When two molecules collide, they exchange their thermal energy of polyatomic molecules. This is to components of momentum along the line of centers, include terms not only for translational energy but retaining their components a t right angles to the line also for rotational and vibrational energy. In arriving of centers. Since the line of centers may lie a t any a t this expression we should properly take account of angle whatsoever with respect to the coordinate the quantum restrictions on rotational and vibrational system along which the components of velocity are energy. This, however, would call for information laid off, it is evident that the components of momentum about moments of inertia and fundamental frequencies
+
which is not in general available. In consequence we fall back on the classical treatment based on the theorem of equipartition of energy. According to this, each "square-term" in the energy contributes on the average '/n kT to the total. In the case of translational energy, these square-terms are simply ' / z mu2 where u is one component of the translational velocity. For rotation, the terms contain the squares of the components of angular velocity multiplied by a constant of such magnitude that again 1/2mu,2is on the average equal to kT. There will be two such terms for all diatomic molecules and for "linear" polyatomic molecules (e. g., Oa); while for other polyatomic molecules there are three such terms. For each vibrational degree of freedom there will be two such t e r m s o n e for kinetic and one for potential energy. The first involves a velocity squared and the second a displacement squared. The energy per molecule can then be written Since the terms (ul. . . .u,) are all equivalent, we may apply to all the distribution law already found for translational terms
For the distribution among n such "square-terms"
and
Also
as previously obtained.) We shall be interested in the distribution with respect to the energy, en, rather than with respect to the quantity, 7.. To obtain this we substitute for 7, in the above expression. We have
Let us write rnp=
+ .. .. + u 2
ulZ
We want now the distribution law for r. This has already been obtained for the case that n = 3 in the discussion of translational velocity. We have dr.
This was a three-dimensional problem. It introduces the volume-element 4 d d r ~which is the volume of a spherical shell in three-dimensional space. The corresponding case of n-dimensions is amved a t as follows. The volume of a hyper-sphere in n-dimensional space is given by
=
[LlL/* m
=
[Zme.l-'/~d h
..d
l/*h-L/l
We have already found
" -- r(%n + 1 ) *l/'",""
Then
Making the substitutions dlrn
dv, = r(lll.cl) nr2-1 dr"
Since r(m r(%n
Thus
+ 1) = mr(m),
+ 1) = %nr(%n) We shall want to calculate the fraction of molecules with energy, e,,, greater than a given minimum G. This will be given by the expression
the value at the upper limit(%= +, =
-
J
xme.dz
= -
e.
m(rn
m)
is zero, leaving
- I)%'-' - .. . .
)/I
Then
4 O menis greater than unity, this series is approxi-
kT
mated by the first term giving
This result is exact for n = 2. In this case
This simple expression has been widely used in work on reaction kinetics.
