Graham's Law: Defining Gas Velocities

Tom Kenney Montgomery College, Rockville. MD 20850

Freshman chemistry texts traditionally present the deri- vation of Graham's law from the kinetic molecular theory. The explanation usually involves the equating of the average kinetic energies of two gases and rearranging to isolate the velocities, u, from the masses, m, or:

and since mass is directly dependent on gram molar mass, M, and ondensity,d, the right-handside ofthe equationmay he written in thiequivalent forms:

The left-hand side of the equation is usually just restated as a "rate", r:

However, rate may he defined or measured avariety of ways: 1. One might actually consider the velocity in terms of

distance, I , per time, t, so that the ratio of the rates would read:

It is this expression that is used for the popular demonstra- tion of HCl(g) diffusing through an air-filled glass tube to form a smoke ring when it meets NH3(g) diffusing in the opposite direction. In this case tl = t2, and the equation becomes:

I t is also possible to word a problem: "Gas # 2 travels the same distance in the tube as gas #1 in (fill in time da- ta) . . . ." In this case, since the distances are equal, but the times vary, the subscripts invert to become:

2. One might also imagine a molecular-sized observer sit- ting a t a point in space (usually a pinhole) and counting the number of passing particles as a function of time. Given the validity of Avogadro's hypothesis, this is most conveniently done by measuring volume, V, as a function of time. The ratio of the rates has the same general symmetry as before:

I t is this expression that was demonstrated in the CHEM- study film on gases. Time was held constant and volumes were measured, so that:

Again, word problems may be written: "A fixed volume of gas #1 takes (time data) as long to diffuse as the same volume of gas #2. . . ." As'hefore, the subscripts invert to become:

3. At least one popular text (Kotz and Purcell) describes velocity in terms of mass as a function of time. This is the equivalent of our molecular observer weighing the particles going past rather than just counting them. Exercises of this type may read: "Equal masses of two gases (time data). . . ." or "Gas #I diffuses a t x mg/h and gas # 2 diffuses a t y mgl h. . . ." In these cases, the velocities must be changed from volumes to mass, m, through the density, d.

Since V = mld, then the ratio of the rates is: .. - = r, (md(d,)( tJ

A problem that refers to equal masses would use Graham's equation as

which, it will be noted, is inverted compared to the previous two timeldensity situations. Further, if time is held con- stant, the law reads:

In this case, the "look" of the equation might suggest to the careless reader that anumber, viz., aratio of masses, is equal to its square root.

The referenced text does an excellent job of leading the student through the reasoning involved in placing data in the proper places in equations, but unfortunately, many of our students prefer a "plug in the numbers" algebraic ap- proach. Possibly, in the interests of being kinder, gentler teachers of beginning students, we should defer defining velocity in terms of mass per time until later courses.

Volume 67 Number 10 October 1990 87 1