A Kinetic Theory Dilemma? consider one mole of a perfect monatomic gas a t one atmosphere and 2 5 T . What fraction of these molecules has a velocity equal to taro a t s, given instant? Maxwell stated,' "The velocities of the molecules have values ranging from zero to infinity. ." Boltzmann said,' "In the course of further collisions it would soon happen, if the number of molecules were large enough, that all possible velocities would occur, from zero up to a, velocity muoh larger than the original common velocity of all the molecules. . ." According to the well-known Maxwell-Boltemann distribution law,' the fraction of molecules dnln that have velocities between c and c +&is given by
.
According to this equation the fraction of molecules having a zero velocity a t 25'C is zero. However, for the "head-on" elastic collision of two molecules of equal mass, there must be an instant at which a t least one of the two molecules must stop (i.e., have zero velocity). How can this contradiction be resolved? The question i. asked in terms of the ideas of Maxwell and Boltzmann and not in terms of quantum mechanics. Resolutia: The number of collisions is proportional to the square of the collision diameter of the molecule. Strictly speaking, a perfect gas consists of molecules which are dimensionless point masses (collision diameter equal to zero) with no attractive forces between them. Therefore, in a perfect gas there will be no molecular collisions (and, of coume, no "head-on" collisions during which a t least one molecule would stand still). Chollage: Consider one mole of a real monatomic gas (e.g., Ar with a collision diameter of 2.86 Angstrom units) a t one atmosphere and 35'C. Assuming that the molecules are rigid, noninteracting spheres, what fraction of the molecules a t a given instant has a velocity equal to zm7 S c o n L. KITTSLEY MARQWETTEUNIVERSITY MILWAUKEE, WISCONSIN
' MAXWELL,JAMES CLBRK,"The Scientific Papers of James Clerk Maxwell," (Editor: NIVEN, W. D.), Dover Publications, New York, 1952, Vol. 2, p. 428. BOLTZMANN, LUDWIG, "Lectures on Gas Theory," (Translatm: BRUSH,STEPREND.), Univ. of California Press, Berkeley and Los Angeles, 1964, p. 36. BARROW, GORDON M., "Physical Chemistry," McGraw-Hill, New York, 1961, p. 37.
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