It is agreed that the calculation of work in processes such as the movement of a piston dividing two chambers of gas a t different pressures is tricky in that various different assumptions may be used in evaluating the path for the compression work in the system of interest where the pressure is non-constant.
To the Editor: The difficulty expressed by Professor Chesick in understanding "the pressure drop across the system boundary" points directly to the source of confusion. The system is the gas and there is no discontinuity of pressure a t or near the boundary of the gas, unless there is a shock wave within the gas. Straightforward application of Newton's third law of motion shows that in any system, static or dynamic, there is a balance of forces when inertial effects are properly considered. Thus it can make no difference whether the driving force or opposing force is employed in calculating work done, provided these are properly chosen. The preference indicated for looking a t the opposing force is based on the conviction that it is more often easier to recognize the opposing force correctly than the driving force. The new experiment suggested, in which a mass M rests on a frictionless, weightless piston, which moves between stops, is illustrative and may he considered in somewhat greater detail. I n the first place, if there were no gas in the container the mass, M, would be in free fall and would exert no force on the piston, or the vacuum, beneath it. It exerts a force only when it is prevented from free fall, and the force exerted is just equal to the reaction force provided by the gas. Therefore, as the piston falls the work done is JPdV where P is the restraining pressure exerted by the gas on the piston. The difference between the gravitational force exerted on the mass M and the restraining force exerted by the gas goes into increasing the kinetic energy of the mass with its associated piston. At the end of the fall, this kinetic energy will be dissipated by collision with the lower set of stops. If we retain the original definition of the system as the gas only, then the collision with the lower stops is irrelevant and the total work done is JPdV. This pressure will be the pressure of the gas if, and only if, the gas has a well-defined, uniform pressure a t each instant, which will be true for small piston speeds, and which in turn requires a short fall of the piston in the vertical cylinder (or eounterwcights) or a large mass for the piston in a horizontal cylinder. If the pressure within the gas is not uniform the problem cannot be solved by ordinary thermodynamics; one can then only say that the work will be somewhat greater than for a reversible compression hut certainly less than P.,tAV. Apart from trivial or non-trivial variations in the experimental arrangement, the important point is quite general and straightforward. If the pressure of a gas is
uniform during any expansion or compression, the process will be thermodynamically reversible with respect to the gas (although there may be frictional forces or heat transfers across temperature differentials that can make the total process irreversible). The requirement for a uniform pressure is normally that any motions be slow compared to the relaxation time of the gas and that there should be no internal churning of the gas, as with a rotating paddle. Experimentally, the requirement of constant temperature also requires that the process be slow when work is done. When the pressure is non-uniform it may still be possible to calculate the work done if the pressure is known at each point in the gas at all times. If this information is not available, the work cannot be calculated. It is extremely important, therefore, that examples be chosen with care. I hope we may all agree not to give such poorly stated problems as the one I had selected as a bad example. ROBERTBAUMAN POLYTECHNIC INSTITUTE OF BROOKLYN BROOKLYN, NEWYORK
To ths Editor: A recent article in THIS JOURXAL~ reaches the couclusion that when an ideal gas is compressed by an external pressure greater than the internal pressure, the work done is independent of the external pressure. I n this treatment, work and heat are assessed by effects in the system. Historically, however, heat and work are assessed by events outside the system; i.e., in the surroundings.2 The power of thermodynamics stems largely from these traditional ground rules, which make the over-all effects independent of transient events within the system. Nevertheless, the peek inside the system does not automatically invalidate the analysis; it just makes erroneous conclusions more likely. Thus, it is not surprising to find that the analysis in ref. 1 is meretricious. Let us consider a specific case of an irreversible compression of an ideal gas. Assume that initially the ideal gas is confined in an upright cylinder with a frictionless piston of mass m uppermost. At equilibrium, the pressure of the gas is equal to the downward pressure of the mass nz. Mechanistically, the gas pressure arises from elastic collisions with the piston. Before and after a collision the momentum of a gas molecule is mu, and -mucx, respectively; hence, each collision involves a momentum change of -2mu, and this is the origin of the gas pressure. I n this static situation 3, = 3" = Dz = 0. Now let us add a mass M to the piston. The piston moves and hence acquires kinetic energy. Consider, however, the effect of the moving piston on a colliding 'Textbook Error, 49, BAUMAN, R. P., J. CHEM.ED., 41, 102 (1964). 1 PLANCK, M., "Treatise on Thermodynamics," Longmans, Green and Co., Ltd., London, 1927, p. 42.
Volume 41, Number 12, December 1964
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