Level: introductory (ex. 1), advanced (ex. 2)
Reference: Rudolf Clausius, "On the nature of the motion which we call heat," Annalen der Physik 100, 353-380 (1857); translation published in Philosophical Magazine 14, 108-127 (1857)
Notes: Rudolf Clausius (1822-1888) made fundamental contributions to thermodynamics, including coining the term entropy. He also provided a molecular explanation for heat, one that included a kinetic model for gases. The paper on which these exercises are based includes an expression for the root mean square speed of a gas (although it does not use that term). Clausius' derivation is based on the mechanics of momentum transfer of billiard-ball-like gas molecules to the walls of a container and on consideration of the kinetic energy (or, in Clausius' terms, "vis viva") of the gas. Clausius said that the speed he derived would give the gas its proper "vis viva," but he recognized that it is possible that the actual velocities of the several molecules differ materially from their mean value.
Exercise 1 is simply a matter of plugging numbers into a textbook formula. It does not use Clausius' results so much as show that he could estimate molecular speeds back in 1857. Exercise 2 is an exercise in mathematical manipulation or derivation rather than in chemistry. The difference between Clausius' formula and a modern textbook formula is only a matter of emphasis. Clausius' formula is in the form of a typical value (485 m/s) times a factor that won't differ drastically from unity; the factor differs from unity to the extent that the temperature differs from 0°C and to the extent that the gas density differs from that of air. The modern formula is given in terms of more fundamental quantities, perhaps, but it contains no obvious clue of a typical value.
Solutions: To download solutions, go to:
http://web.lemoyne.edu/giunta/classicalcs/clausius.doc
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