Howard 8. Palmer Deportment of Material Sciences The Pennsylvania State University University Park, 16802
A Molecular Model for Adiabatic Reversible Compression and Expansion
The beginning student of physical chemistry sometimes finds himself in conceptual difficultv when he encounters the law of reversible adiabatic exuansion or comuression of an ideal pas. He has accuHtomed himself co thinking that PV =constant for an ideal gas, as long as the temperature is constant. He probably has been exposed to the kinetic-theory derivation of the ideal gas law and has found i t convincing. Now he finds that the thermodynamic treatment of a reversible adiabatic expansion or compression has led to another relation between P and V for an ideal gas, namely PV' = constant. But this time the constant is not related to the temperature; it is simply a number that characterizes the particular adiabatic process in question. The student may follow the thermodynamic derivation and believe the result, hut it may not appeal to his physical intuition. Just as the simple kinetic theory of gases helps in understanding the ideal gas equation of state, i t can contribute also to illuminating processes of compression and expansion. If the student can see what elastic collisions with a moving piston do to the kinetic energy of gas molecules, he may have gained a useful mental picture. If he can carry this model through t o a relation between compression ratio and temperature ratio, he will have a "molecular" understanding of the laws of adiabatic compression. By introducing classical degrees-of-freedom concepts, he can then see why the 7 of a gas enters ae it does. Figure 1 illustrates the first matter, the collision of a molecule with a moving piston. We consider only the x-velocities. In the collision, the velocity u changes to u' and w changes to w'. For an elastic collision, conservation of energy requires
mu2/2
+ MwP/2 = m(u')¶/2+ M(w')'/2
(1)
Conservation of linear momentum requires mu
+ Mw
=
mu'
+ Mw'
Now for the molecule and the piston, let ( m / M j 0. Then
-
That is, the change 'n the absolute value of thex-velocity of the molecule is equal to twice the velocity of the piston. Now we consider a monatomic molecule moving inside a cylinder with a piston (Fig. 2). Let the x-component of the rms average speed he u; i.e., c2 = 3u2, where c is the rms average speed. Thus a typical molecule moves a t speed u in the x direction. The
-
X DIRECTION
Figure 1.
Collision of o molecule with a freely moving piston.
Volume 48, Number 1 1, November 1971
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.PISTON FACE
AREA = A
-x Figure 2.
Cylinder ond piston for somprsrrion or expansion of o gar.
molecule collides with the piston face u/2(Lo - x) times/sec. On each impact, its x-kinetic energy is increased A(KE),/impact
=
A(mua/2) = muAu
But it has been shown (eqn. (4)) that Au
(5)
2w.
=
Thus
A(KE),/inpact = 2muw
(6)
So the rate of change of x-kinetic energy is d(KE),/dt = [u/Z(Lo - x)1.2muw =
d(KE)* = d(KE),,t
(7)
The change in the x-kinetic energy (KE), caused by the piston will be redistributed by collisions into the total kinetic energy (KE),. Thus d(KE), = d(KE),
(8)
However, the equipartitioning of energy requires that W E ) , = (1/3)(KE)t
(9)
for a monatomic gas. Combining eqns. (7), (a), and (9) d(KE)e/(KE)t= (2/3)wdtl(Lo- x) (2/3)dx/(Lo - x)
(10)
Since we are interested in obtaining a relation involving the gas volume, we multiply the right side by (A/A) and obtain d In (KE)t = -(2/3)d In [A(& - x)]
(15)
But now, because there are two rotational degrees of freedom as well as three translational ones (KEL = (1/5)(KE)w
(16)
This equation replaces eqn. (9). Going through the integrations as before, one obtains TII IT^) = ~ V ~ / V ~ ) ~ / I
(17)
Similarly, if one vibrational degree of freedom (two classical square terms in the energy) is added, then ( T j / T o )= (Vo/Vj)'/l
( K E ) ,.2w/(Lo - z )
=
in a real system, the piston velocity does matter, because the adiabaticity of the process depends upon the relative rates of energy addition to the gas and energy loss through the walls. In the limit of very slow piston movement, the compression becomes isothermal.) Molecules with internal degrees of freedom can be treated by this model. Of course only the classical, fully equipartitioned case can be considered easily. For instance, the rigid linear rotator with no vibration will have, as before
(18)
One now sees that the exponents in eqns. (14), (17), and (18) can be identified with the quantity ( y 1) for the gas. Thus the role of the specific heat ratio in the adiabatic, reversible compression process is given rather direct physical interpretation by this model. The entry of y for the gas is seen to be related to the manner by which the energy delivered to a gas molecule by the moving piston distributes itself. If the student is bothered by the artifice in changing eqn. (10) to eqn. (ll), he can imagine a cubical container with three movable walls perpendicular to the three axes. He has to imagine that as the walls move inward, they continuously become smaller in area, so that the container remains cubical and the walls do not interfere with one another. The walls move a t the same velocity, w. I n this case
-
d(KE). = d ( K E ) , = d ( K E ) , = (1/3)d(KE)t
(11)
Integrating, letting the piston move through distance s d ln(KE),,, = -(2/3)ln[A(Lo - s)/ALo] =
-
/
/
v
(12)
where f and o represent final and initial conditions. Having identified kinetic energy as being proportional to absolute temperature, we obtain (TIIT,) = (Vo/Vj)va
(14)
which is one form of the laws of reversible adiabatic compression for a monatomic gas. For expansion, one notes that c, is decreased by collision of a molecule with the piston moving outward. A rather interesting point is that the actual velocity of the piston has no effect on the final result, although it affects the energy changes in the individual collisions. One knows this from the thermodynamic derivation, but the molecular model makes it believable from a mechanistic viewpoint. (It might be pointed out that 756
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Equation (14) follows as before. The treatment presented here cannot be said to be a substitute for the thermodynamic derivation, because it is much less general. In particular, it incorporates the assumption of elastic collisions. Extensions to inelastic collisions would present formidable difficulty that would destroy the usefulness of the treatment as a pedagogical device. The thermodynamic derivation possesses the great virtue that one need not be concerned a t all about the collisions themselves, but only about the initial and final states in the process. Nevertheless, a kinetic theory approach may often be welcomed as an instructive supplement to the thermodynamic treatment.
