KINETIC THEORY of GASES. PART I1 ROBERT N. PEASE Princeton University, Princeton, New Jersey
LAW^ UR first problem is to derive the expression for the pressure exerted on the walls of the containing vessel by the impacts of molecules, taking the account Of the fact that a may wall from any direction a t any velocity. For this purpose we first determine the distribution of velocities with respect to direction a t any point 0 in the interior of the gas. Erect a rectangular coordinate system about the point 0. Construct a circle of unit radius in the YZ
CD
BOYLE'S
0
=
-CG .EF= OE
sinEd+
Area men = B G C D = sin 0 a d +
Since all directions are equally probable, the chance that the molecule is to pass through the area ABCD is the ratio of this area to the total area of the sphere (of unit radius)
We, = sin 0 dedd 4r
This is also the fraction of molecules starting from 0 which would pass through ABCD. We next calculate the number of molecules which will strike unit surface of the container in unit-time coming from a particular direction defined by 8 and 4, with velocities between c and c dc. Construct a prism of unit base inclined a t an angle 8 to the vertical, and turned through an angle 4. Its side length is c. The base is supposed to rest on the wall of the container. The volume of this prism is c cos 8 (the altitude times the base, which is unity). All molecules in this prism with velocities between c and c dc directed a t angles between 8 and 9 dB, and 4 and 4 d4 will strike the unit surface in unit-time. (Strictly we should say, "all molecules or their substitutes arisingfrom collision.") If
+
r
z
+
+
+
N = total number of molecules V = total volume
FIGURE1
there will be c cos 0 plane with 0 as center. Construct semicircles (supN -V posed to be infinitesimally separated) of unit radius terminating on the X-axis and cutting the circle a t E in the prism, of which the number and F, respectively. Connect these semicircles by c "0s a arcs A B and CD drawn in planes parallel to the YZ dNGT plane. We wish to know the chance that a molecule at 0 is so directed that it will pass through the area have velocities between c and c dc. Of these, the ABCD (see Figure 1). fraction Let sin - - 8 d0dd L YOE = 4 4r = EF (in radians, the radius being LEOF = d+ unity) has velocities directed as required. The number of molecules of velocities between c and c dc from the LGOC = 0 required directions striking unit surface in unit-time is LBOC = dB = .BC then c cos e .-sin a dad+
+
+
dN, 7
' JELLINEK,
"LehTbuch der Physikalischen Chemie." F.Enke, Stuttgart, Germany, 1936, Vol. I, p. 153.
366
4r
The resulting pressure is obtained
follows. For
any molecule, the component of velocity normal ta the wall is C COS
8
Heat CaNcity of a Mmzatomic Gas.-If the change in energy of a monatomic gas with temperature is due solely to the change in translational energy, the heat capacity a t constant volume should be simply
To this corresponds a momentum change on collision with the surface of
CY
1/2NomcP =--
=/a
2mc cos 8
Multiplying this by the number of molecules of the required kind striking unit surface in unit-time, we obtain for the contribution of these to the pressure dp.86
=
c cos 8 sin 8 dBd6 2mc cos DdNc v 4r
=
m dN.. cP sin 8 cos' 8d8d6 2rV
as is known to be the case. (No = 6.06 X loz3; R = 1.9885 cals./mole/degree C.) Rate of Collision of Molecules with a Surface.-The total number of molecules striking unit surface in unittime is found as follows. We had for the number of molecules of velocity bedc directed a t angles between 0 and tween c and c B dB, and q5 and q5 d 4
+
+
mc2 e--/z
dp. = d N . - f 2nV # = o
m=2-
f
sin 8 cos' 8d8d+
+=o
With respect to q5
These molecules will strike unit surface in unittime. To obtain the total number we first integrate with respect to B and q5 n=n/2
2~
f
c cos 8 sin 8 d8dQ = dNe v 4"
d"*
Integrating for all directions
+
+=2
dv. = d N . - f 4"V s - 0
d+ = 2 s
0
f
+=o
sin 8 cos 8 dad+
With respect to q5 With respect to B With respect to 0
d cos 8 = -sin 8d9
8 = 7/2
sin 8 cos' 8d8 = - c a s V d cos 8 cos 8 =
fcos"d
f sin
'Ir cos"
8 = 0,
8 = s / 2 , cos 8 = 0 ; 8 =Or
c a s 8 = 1 , cos"
f sin 8
cos'
sin 8 = 0 , '/nsinP 8 = 0
8 = s / 2 , sin 8 = 1,
= 1
sin' 8 =
cos 8d8 = d sin 8
O(8 = r / 2 )
8=r/2
0 cas 8d8 =
n=o
= +'/s
@dB =
8-0
Whence Thus
pV = '/~Nmc,'
Since by observation pV = n R T '/sNmc,' = n R T ' / m ~= , ~kT
(k
= gas constant per molecule)
when R is expressed in ergs per degree ( R = 8.316 X lo7 ergs per degree)
This result has already been used in introducing T into the expression for velocity distribution.
This quantity, v , is also the number of molecules per second crossing unit area of any imaginary boundary within the body of the gas in one direction.
368
JOURNAL OF
Collisions beheen Molecu1es.l-Our next problem is to determine the total number of collisions occurring between the molecules of a gas, taking into account the Maxwellian distribution of velocities. consider two molecules of fixed diameter a in collision. At the instant of collision the center of either
CHEMICAL EDUCATION
in time dt, the center of the one molecule must lie in a volume whose base is the ring of area 2 u d sin E ~ E
and whose thickness is r cos 0dt. This volume is 2 ~ sin d 0 d E ~cos edt
If there are a total of N molecules, the volume be2NuoPsin 8 a . r cos Bdt
If the total volume is V, the chance that the molecule actually does lie in the above volume is the ratio 2
FIGURE5
sin 8 a . r cos Bdt
FIGURE 2
or if N/V = n, the number of molecules per unit molecule must lie somewhere on the surface of a sphere volume, the chance is of radius a drawn around the center of the other 2nmSsin Bd8.r cos edt molecule (see Figure 2). ~~tus think now of collisions in termsof the direction This is the chance that the one molecule collides with of approach of the two molecules with respectto the S@me0th- in the time d t when the relative velocity is at r. The number of collisions in unit-time is obtained line joining their collision, ~~t the angle by summing up these individual chances over the between relative velocity and line of centers be 0. By way of illustration we may represent two limiting cases. If the angle is zero, we have a "head-on" collision (see Figure 3). If the angle is 90' we have a "grazing" collision (see Figure 4). For intermediate values (see Figure 5) suppose the angle lies FIGURE3 between 0 and 0 dB. Then the center of the one FIGURE6 molecule meets the surface of a sphere of radius a drawn around the center of the Other corresponding period, that is, by integrating the above where on a ring (see Figure 6). The radius of this ring is a sin 0. Its circumference expression with respect to t between 0 and 1. Whence the number of collisions is is 2na sin 0. Its breadth is ad0. Its area is therefore
+
2 m sin
= 2 m s sin BdE
Let r be the velocity of the one molecule relative to the other. The rate of approach of the centers is determined by the component of this relative velocity parallel to the line of centers. This is r cos 0 (see Figure 7). Now if the one molecule is to collide with the other ' TOLMAN,"Statistical meFIGURE 4
chanics," Chemical Catalog Co., New York City, 1927, p. 68.
2nr.9 sin Bd8.r cos
e
Or
2nuoS sin 8 cos
We next need to know the fraction of all collisions in which the relative velocity r lies between r and r -I-dr. r is a function of the velocity components of the two molecules concerned, ul ol wl and uz va w2. Let
- UI = B - VI = y wg - w, = ux uz
6
1
-e
YI~+U,~+IUI.+",~+O~~+W~.~
-
du,du,dw,du$~$w.
CI*
a'+
In place of [dul dw, dw,] and [duz d n dwz] we may
7 FIGURE
wish to consider all directions for the relative velocity and for the velocity of the center of gravity, we substitute 4arzdr for dB dy de and 4?rgzdg for dX dp dv. Then substituting ' / % r 2 2g2 in the exponent, we obtain
+
1 -e
(See Figure 8.) X, fi, and v are the velocity components of the center of gravity of the two molecules. If g is the velocity of the center of gravity
aC3
1/.+
+ 2gx
''
4ddv4ugzdg
This is the chance that for a pair of molecules r lies between r and r dr, and g lies between g and g dg. We are not interested in the velocity, g, of the center of gravity and may a t once eliminate it by integrating over all values3
+
+
The chance that the two molecules have simultaneously velocity components in the ranges ul and u l
+ whence we obtain
for the chance that the relative velocity lies between r and r dr. Multiplying the above by the number of collisions when the relative velocity is r, we obtain
+
du,, and so forth, and ua and u2
+
duo, and so forth, is the product of the independent probabilities
"Dynamical theory of gases," 4th ed., Cambridge JEANS, University Press, Cambridge, England, 1925, Appendix A.
The total, Z12,of such collisions per unit-volume per unit-time is
+
for the number of collisions when r is between r and r dr and B is between Band B dB. Integrating with respect to B between 0 and a/2
+
rcos ode = d sin
el
e = ,/2
f
Collisions with Relative Velocity i n the Line of Centers Minimum.-The simplest collision mechanism for bimolecular reactions assumes that reaction follows a collision in which the velocity of one molecule with respect to the other along the line of centers exceeds some critical value. The wrresponding collision-rate is obtained as follow^.^ We had for the number of collisions in which the relative velocity lies between r and r d r , and the angle between the direction of the relative velocity and the line of centers lies between B and B dB Greater Than a &en
(go0)
sin 8 cos @dl?5
8=0
8
-J
,./2
e=o
sin
sin
s
1
=
I/.
sin*s
1
s
=
,/2
6 =0
=
'12
+
+
whence
Now integrating with respect to r
whence
where 21, is the number of collisions one molecule makes with other like molecules in unit-time. From the above may be obtained the total number of collisions, Z between like molecules per nnitvolume per unit-time. If each molecule suffers 211 collisions per second and there are n molecules per unitvolume, then all molecules will suffer nzll collisions. This, however, counts each collision twice--once for each molecule participating. The number of collisions is therefore
Z,,= 'l2nzll
=
2 4 ; an%+ per unit-volume per unit-time
Since
Let R be the component of the relative velocity in the line of centers (see Figure 9). R
I COS
F
e
Our condition is that R shall exceed some minimum value, Ro. This obviously requires that r shall also exceed Ro, and thus gives 7 = Ro as the lower limit of integration with respect to r. As to 8, we note that if R is to exceed Ro it is necessary that r cos 0 > RO cos
e >
Ro r
-
This will correspond to the upper limit for 8. The lower limit for 0 is B = O(RO= r), for which cos B = 1. Thus
With respect to cos B eos
e- Ra/r
f
sin 0 cos Ed0 = -
eos e
eos e
eon 8 = 1
-
J
R,/r
---- 1
cos Ed cos 0
whence In a similar manner, one may calculate the number of collisions a molecule of one kind enters into with molecules of a second kind. The result is
" ,/+ dE2 [ f - X"-radr - R# f 2 a
ncl
= 2
ff
where
-
,=& r=
r=
r=Ra
r=Ra
I
I/,,'
-~
r d r ]
I1
8 TOLMAN, "Statistical mechanics," Chemical Catalog Co., New York City, 1927, p. 70.
plate which is at rest. The result is a uniform gradient of velocity over the moving layers from the fixed to the moving plate. The force which must be applied to the moving plate to maintain its motion may be defined in terms of this velocity gradient, du/dx, the area, a, of the plate, and a proportionality constant characteristic of the fluid and termed its coefficient of viscosity, q . Therefore
Integral I becomes
Integral I1 becomes ~/*Ro~ -LI%
0%
Therefore Force
-
=?a.
Since
If, E = '14MRo2, M = molecular weight
For the total, Z,,, of such collisions per cubic centimeter per second
(Here R = 8.316 X 10' ergs per degree under the radical. The value 1.989 will be used in the exponent if E is expressed in calories.) I n a similar way it can be shown that for unlike molecules
-A,,
a.
- I
dz du
The coefficientof viscosity, T , is thus equal to the force per unit area of the moving plate per unit velocity gradient. In liquids, this force is presumably to be thought of as opposed to the so-called van der Waals force of attraction between molecules. In gases, i t is a consequence of molecular bombardment of the moving plate. Molecules colliding with the plate acquire an additional component of momentum in the direction of motion. Subsequent collisions with other molecules result in a transport of this momentum to portions of the gas at a distance from the plate. Viscosity is one of three so-called transport phenomena, the other two being thermal conductivity, which is transport of energy, and diiusion, which is transport of mass. A general treatment of all three can be given. We will make our application specifically to visco~ity.~ Let a gas be contained between the two parallel plates one of which is a t rest and the other in uniform motion parallel to its plane. Consider the rate at which molecules cross an imaginary plane in the gas parallel to the plates. This will be the same as the rate a t which molecules would collide with a wall a t this point. For the moment we restrict the problem to molecules of velocities between c and c dc and dB and directions defined by the angles 0 to 0 4 to 4 d4. In the section on gas pressure (Boyle's Law) we obtained for the rate a t which such molecules struck unit surface in unit-time the quantity (page 366)
+ +
+
where
Viscosity of Gases.-In order to calculate collision frequencies, we need a value of a, the molecular diameter. This is best obtained from measurements of the viscosity of the gas, for which reason we next treat this property. Viscositv mav be defined as the resistanre of a fluid .7 to a shearing force. SupFIGURE 10 pose a fluid is contained between 'two parallel plates, one of which is a t rest and the other in uniform motionparallel to its plane. ~h~ moving plate drags with it the adjacent layer of fluid. This layer in turn imparts a drag to layers farther removed, and to the
v
c cos 0 sin 0 dOd4 4" '
dN. y
or if dnc
- = dn.,
v
dn,.c 4r
-sin 8 cos 8 dad+
Integrating with respect to 4 ( 2 . 4
= 2~
we obtain for the number of molecules of velocities
,TOLMAN, .,Statistical mechanics,M Chemical N ~ W~
~ City, r k 1927, pp. 222-4.
Catalog Co,,
+
between c and c dc which will cross unit area of the plane from one side in unit-time, a t an angle to the normal of the plane between 0 and 0 d0, the quantity
+
that is, the distance the average molecule travels per unit-time, by the total number of collisions which the average molecule suffers per second. The latter is given by
'/&. .c sin 0 cos 0 d8 (See Figure 10.) Let X, be the mean free path for molecules of velocity c. This is the average distance a molecule of velocity c passes over between collisions with other molecules. Thus a molecule which bas traversed a distance X, before arriving a t the plane will on the average collide with a molecule on arriving a t the plane. Its last collision will have been a t a distance A. cos 8
measured perpendicular to the plane. Let the fixed plate be beneath the plane, and the moving plate be above the plane. In virtue of the motion, there is a uniform gradient of velocity, du/dx, along the normal, x , to the plane. If mua is the average value of the added momentum in the plane due to the "mass motion," then a t a distance, A, cos 0, above the plane the average value will be muo
+ m i , cos 0 du -.d z
andhelow the plane mu.
u - mA, cos 8 ddx
If we assume A, independent of c and substitute X , we obtain for the net rate of transfer of momentum across unit area per second
This is also the rate of transfer of momentum from the moving plate to the gas, and is therefore the retarding force exerted on unit area of the plate. (Momentum has the dimensions M L / T . Rate of transfer of momentum has the dimensions M L / T 2 which are likewise the dimensions of force.) For unit velocity gradient (du/dx = I), this force is the coefficient of viscosity, q. Thus
We now assume that molecules coming from positions defined by X, cos 0 have on the average the corresponding momentum and, on colliding with a molecule a t the plane, acquire the momentum characteristic of the This is the result when X is taken independent of the plane. The net transfer of momentum to molecules velocity. Taking account of the distribution of h with the velocity introduces about a five per cent. in the plane is then the diierence correction. A further correction is necessitated by du consideration of the so-called persistence of velocities.' muo - m i c cos 8 We have assumed that "molecules coming from posid u tions defined by X, cos 0 have on the average the corre2mX. cos 0 d x sponding momentum and, on colliding with a molecule For molecules of velocities c to c dc and directions in the plane, acquire the momentum characteristic of 0 to 0 do, this becomes per unit area per unit-time the plane." This is equivalent to assuming that a single collision is sufficient to effect complete adjustdu ment in the value of the momentum. This is incor2mA, cos 0 - .'/zdn,.c sin 0 cos 0 dB, dx rect, and consideration of the problem further alters or the coefficient in the expression for q. The finally du n L . c , d n , - .sin 0 cos' 0d0 accepted equation is dx
+
+
Integrating with respect to 0
we obtain du '/amA.cdn, dz
The next step is to integrate with respect to c from This requires that X, be expressed as a funcm. tion of c, which is a matter of some difficulty. The value of ' 9 the mean free path for is obtained very simply by dividing the average velocity,
0 to
"'
These two corrections have thus increased q by fifty per cent. It will be noted that the above expression does not contain the concentration. Thus, q should be independent of concentration (or pressure), and this is found to be the case. Temperature should enter in only through c,, the average velocity. Since 8 "JJynamical theory of gases," 4th ed,, Cambridge University Press. Cambridge, England, 1925. pp. 274-6.
d ~ .
7 would be expected to vary directly as Actually q is found to increase somewhat more rapidly with the
is a limiting diameter corresponding to T,. It is this limiting diameter which we shall want to use in calculations of collision frequency, since it corresponds to the closest possible approach of two molecules. Values of q a t some temperature, and values of C may be found in any collection of physico-chemical data8 (e. g., in Landolt-Biirnstein's Tabellen). o2 is calculated from the relation
temperature. The discrepancy is a consequence of the assumption that molecules are solid elastic spheres of fixed diameter. This can only be approximately true. A better assumption is that the molecule is surrounded by a field of force which is attractive a t long distances and repulsive a t short distances. The amount of interaction between two approaching molecules depends on the parts of the fields interpenetrating, and also on the time of interaction. The latter varies inversely as the relative velocity. At low velocities of approach the interaction is large; distant parts of the field exert an appreciable effect, and the molecules behave as if the diameter were large. At high velocities only the intense field close in plays any considerR k = 6.06 able part, and the effective diameter is small. High average velocities are associated with high temperatures; hence the average effective diameters will decrease with rising temperatures. Since the diameter, s, appears in the denominator of the expression for q, this decrease of effective diameter with rising temperature will lead to the higher temperature coefficient Then which is observed. The variation of diameter with temperature may be expressed by Sutherland's equation,'
"2
=
M 6.06 X 1 P a
X = 8.316 6,06 c2
=
10'
ergs per degree per molecule
X 0.499d8.316 X 10' rV*.6.06X lozS
2
2.697 X lo-"-
~n T T
dTT 9
.'= .,c
(1
+ $)
=, .
=-
C
where C is known as Sutherland's constant, and o, 'JEANS, "Dynamical theory of gases." 4th ed., Cambridge University Press, Cambridge, England, 1925, p. 284.
=-
sZT
T+ C LANDOLT-BORNSTEIN, "Physikalisch-Chemische Tabellen,'
5th ed., Verlag Julius Springer, Berlin, 1923, Vol. I, pp. 171-81.
