Kinetic Models for Adiabatic Reversible Expansion of a Monatomic Ideal Gas On-Kok Chang
University of California-Davis, Davis. CA 95616 A fixed amount of an ideal gas is confined in an adiabatic cylinder and piston device with cross sectional area A. The piston is moved outward slowly with a constant speed u by an external device, with u much smaller than the average velocity of gas molecules. In this process, the gas expands reversibly and adiabatically. The relation between temperature and volume in initial and final states can be derived from the first law of thermodynamics:'
Where y = C&,. For a monatomic ideal gas y = 513, eqn. (1) can be written as
This relation can also be derived based on kinetic models. For a gas sample in a container, the frequency in which a gas molecule collides with a region of the container wall with area A is2
Where NIV is the number density of the gas, and the z axis is chosen to be perpendicular to the container wall. If the wall is moving outward with a velocity u, the expression of collision frequency becomes
Equation (8) is the same as eqn. (1) derived from thermodynamics. The relation in eqn. (8) can also be derived from another simpler kinetics model. Consider an adiabatic expansion process that starts a t time = 0. During the process, as discussed before, every time a gas molecule collides with the piston moving with speed u, its speed decreases by 211, therefore after n collisions, its speed decreases by 2nu. If the initial average speed of gas molecules is Ei, the average speed a t time t is
Where z is the average number of times of collision per molecule. The factor 113 is from the fact that x ,y, and z directions are equivalent, but the moving piston is only in the z direction. Equation (9) is valid only if u is very smallcompared with ui. The length of the cylinder at time t is
The average frequency of collision with the piston for one molecule is just dzldt, which is equal to the average velocity divided by twice the length of the cylinder.
Separating variables and integrating yield
Equation (4) reduces to (3) when u approaches zero. When a gas molecule traveling with a z component velocity u, collides with the piston moving with a velocity u, it bounces back with a z component velocity -(u, - 224). The loss in kinetic energy is
The rate of energy loss of the system is equal to the collision frequency for a small u, interval times the energy loss in each collision integrated over u,:
For the whole systemE = (312) NkT, dE = (312) NkdT. Obviously, the volume at time t is given by V(t) = Vi uAt and therefore dt = dVluA. Substituting the expressions of dE and d t in eqn. (6) gives
+
Canceling terms, separating variables, and integrating,
Let Uf = Ui - (213) zu and Lf = Li
+ ut, eqn. (12) becomes
For gases, the average speed U is proportional to the square root of temperature T. Also for a cylinder of uniform cross section, the length L is proportional to its volume V. Using these relations, eqn. (13) becomes
Which is again the same as eqn. (1). In the derivation based on either kinetic model, the assumption of very small u has to be made, this is expected from a thermodynamic point of view, without the assumption, the orocess is not reversible and eqn. (2) cannot be derived. In the
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Moore. Walter J., "Physical Chemistry," 4th Ed., Prentice-Hall Inc., Englewood Cliffs, NJ, 1972, p. 51. Adamson, Arthur W., "A Textbook of Physical Chemistry," Academic Press, New York. 1973, p. 57. Volume 60 Number 8 August 1983
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extreme case of irreversible process, the piston moves much faster than the average speed of gas, the process becomes afree expansion in which no work is done by the gas and the temperature of the gas does not drop. From the kinetic point of view, if the piston moves very fast from its initial position to
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its final position, gas molecules cannot catch up with the piston and have no chance to collide with the piston when the piston is moving, therefore, there is no decrease in average speed and no decrease in temperature of the gas.
