M. W. C. Dharmawardhana and K. G. A. Dharmawardane University of Ceylon
Colombo, Ceylon
I
1
Rigorous Derivation of the Kinetic Equation for Ideal Gas Pressure
The simple derivation of the pressure formula PV = l/Jme2 in the kinetic theory of gases proceeds from a consideration of the change in momenta at the wall of the freely moving gas molecules in a cubic container. Although this method leads very simply to the correct gas pressure, it leads to an iucorrect value for the average number of collisions per second per unit area made by molecules of the gas. This limitation is most easily overcome, though with considerable loss of generality, by a consideration of a spherical' container. The other proofs of the pressure formula have the shortcomings of mathematicalz or conceptual3.' difficulty in that it may be difficult to directly form an idea of the physical process considered. The method developed by Clau~ius6.~ for treatment of problems in kinetic theory is especially useful when there are intermolecular forces to be taken into account. The method is so very general and mathematically elegant that its mastery may be considered worthy of its apparent difficulty. In fact, except for the integration of the virial, there are no overbearing mathematical difficulties in it, and it stands well in comparison to some of the other proof^.^ If a simplification of the mathematics is available there is no reason why the Clausius method should not be used with students for the derivation of the pressure formula HULME,R., J. CHEM.EDUC., 34, 459 (1957). PEASE,R. N., J. CHEM.EDITC., 16, 366 (1939). 8 JEANS,J., "Kinet,ie Theory of Gases," Cambridge, 1940, chap. 3. 4 GLASSTONE, S., Text,book of Physical Chemistry, Maemillan, 1940, p. 244. 5 CLAUSIFS, R., Ann. Phl/s., 141, 124 (1870). 6 JEANS,J., IOC. eit., p. 70.
31 8
/
Journal of Chemical Education
of a gas. Its use will help to focus attention on intermolecular forces in gases and their mathematical treatment. We shall give a simplification of the evaluation of the virial of Clausius which was advantageously used with a class of fair mathematical ability. True enough, mathematicians abhor any proofs which involve argument from figures. But, this can hardly be objected to from a teaching point of view. We shall also outline an evaluation of the virial which, we think, is more elegant than that often given,&but we do not claim it to be a very simple proof. I t is giveu to bring out the difference between the mathematician's approach and the simplified teaching approach. Since the motion of any molecule in a gas is governed hy Newtonian laws, we have X
= mi!,
Y
=
mg, Z = m i
(1)
where X, Y, and Z are the components of the total force acting on a gas molecule having the coordinates x, y, z, and mass m. Multiplying (1) by x, y, and z, respectively, we have: zX
+ gY + sZ = ~ ( Z +X yg + nt)
(2)
from (2)
A m dt 6
= 2
where c'
= *?
[ddt
(22
+ y* + r ' ) ] - mcz
+ li' + t'
is the velocity of the molecule.
(3)
+ +
The term (d/dt) (x2 y2 2%)physically corresponds to the rate of change of the (di~tance)~ of the molecule from the origin. As the molecule moves about, this continually fluctuates in value; but there is no tendency to a steady increase or decrease. As such, if we sum over all the molecules of the gas, since the molecules are moving a t random
The term 2dSS = volume of the element ABCD (see diagram) .'.f zPdS, for all dS = P J z d S , = P V (9) where V is the volume of the container. P could he takenout of the integral sign since it is the same throughout, for all elements. Hence, from equations (8) and (9,
Thus, equation (3) becomes on halving,
Equat,ion (5) is the virial equation of Clausius, while the left hand side is called the virial. X, I< Z, are perfectly general forces which act on a molecule a t x, y, z. These will involve intermolecular forces as well a s forces that arise when molecules collide with one another or with the wall of the containing vessel. For an ideal gas, it is assumed by definition, that there are no intermolecular forces. Since we are interested in the pressure exerted by the gas molecules on the wall, we need to consider only forces that arise when molecules collide with the wall of the container. Let the augle between the x axis and the direction of the gas pressure P, acting on the surface element dS be 0 (see diagram). If X8, Ya, Z', he the sum of all the forces acting along the x, y, and z directions respectively, and exerted by the surface element on all the molecnles colliding with it,
where the summation sign and the integral signs go over all the surface elements. Now, Z(XX8
+ yY* + zZ')
where the surnrnat,ion is carried over all the surface elements = Z(xX yY zZ) where the summation is carried out for all the molecules. Hence
+
+
1 Therefore, from equation (5), 3-PI7 = -Zmcz leading to 2 2
which is the required result. This result can be obtained from the virial equation by the following vectorial method. If P is the pressure on the surface element dS, the thrust 011 it due to the pressure is PdS. This would be equal and opposite to the sum of the forces X, Y , Z, acting on the molecules colliding with dS. If, as before, Xs, Y", and Z" are the snm of the forces acting on all the molecules that strike the surface element dS, along the x, y, z, directions respectively, -XI
=
PdS.i,
- Y 8 = PdS.j, -Za = PdS.k
where i, j, k, are the unit vectors aloog the z, y, and z directions respectively. Hence
where
r
= i s
+ jy + kz
Hence, summing up for all surface elements
Now, by Green's theorem J r . d s = f ( v . r ) d V = 3 f d V = RTf
since -xa
=
P cos &dS, ete.,
(6)
the negative sign indicating that the force exerted by the element on the molecules is in the opposite direction to the gas pressnre. If the projection of dS on the yz plane he dS,, since P i s normal to dS, equation (6) becomes, -1'
=
r
Also, as before z(zXs
=
ir
+ j?j + ks,
v.r
=
3
+ y Y 8 + r Z 9 = Z(xX + Y Y + 4
Hence
PdS,
similarly,
- Y'
=
PdS,, - Z 8 = PdS*
.: -(r.YS + yY8 + zZs) = zPdSr + yPdSv + zPdS,
(7) (8)
is the required result. Volume 40, Number 6, June 1963
/
319
