Calculation of the Reversibility Factor for an Irreversible Adiabatic Process of an Ideal Gas On-Kok Chang University of California-Davis, Davis, CA 95616 Adiabatic expansion processes of an ideal gas are discussed in most physical chemistry texts. Most often only two extreme cases are discussed, namely reversible expansion and free expansion. For the completeness of the discussion on ideal gas processes, it is necessary to study general cases in between the two extremes. The temneratule-volume relationship of an ideal gas system undergoing an adiabatic expansion o; compression process (reversible or irreversible) can be given by where i and f refer to initial and final states, and r is the "reversibility factor." The assumption of constant Cv has also been made. From thermodynamics considerations, the T-V relationship for a reversible process is1 (Tr/Ti)C"/R= VJVr
When the piston moves outward (away from the center of the cylinder) its velocity u is positive, and the process is an expansion process. When the piston moves inward, u is negative and the process is a compression process. When a molecule with a perpendicular velocity u collides with the piston moving with velocity u, it bounces back with a perpendicular velocity -(u - 2u). The change in kinetic energy in one collision can be written as The rate a t which the internal energy of the system changes is equal to the collision frequency times the energy change per collision followed by integration over u ,
(2)
Equation (2) can be considered a special case of eqn. (1)in which r = 1.In this article, the reversibility factor r for an irreversible process will be calculated from a model based on kinetic theory of gases. Consider a fixed amount of an ideal gas confined in an adiabatic cylinder-piston device with cross-sectionalarea A. The frequency a t which molecules collide with the piston surface is2
Where NIV is the number of molecules per unit volume and u is the perpendicular component of the velocity of the molecules (perpendicular to the piston surface). When a molecule moves toward the piston, itsvelocity is chosen t o be positive. If the piston is moving with a constant velocity u, the collision frequency from eqn. (3)becomes
' Moore. Walter J.. "Physical Chemistry." 4th ed., Prentice-Hall.
E lewood Cliffs. NJ, 1972, p. 51. nqAdamson,Arthur W.. "A Textbook of Physical Chemistry,'' A c a demic Press, New York. 1973, p. 57.
let us define dimensionless "reduced" velocities u, and u,,
where u,, is the root-mean-square velocity of the gas molecules. Substituting eqn. (7) into eqn. (6)gives
For a system of an ideal gas,
where N Ais the Avogadro number. The volume of the cylinder a t any time t is and therefore
Table 1. m e Reverslblllty Factor r as a Fundlon ot the Reduced Plston Velocity u, Range of u,
Range of r
*=+=
r=o
O
