4.4 Graham’s Law

Conservation of momentum for collisions between gas molecules and solid particles for an isobaric gas as a whole means that, on average, momentum transferred to particles by species A must be equal in magnitude and opposite in direction to that transferred by species B. The rate of momentum exchanged with the solids for each species is proportional to N_{i}^{D}m_{i}\bar{v}_{i}, i = A, B, wherein mi is the molecular mass and \bar{v}_{i} is the mean molecular speed. Therefore, the momentum balance is expressed by Equation 18.

\displaystyle N_{A}^{D}m_{A}\bar{v}_{A}+N_{B}^{D}m_{B}\bar{v}_{B}=0 (18)

From kinetic theory, the average values of molecular speed are inversely proportional to the square root of their respective molecular masses. Hence, Equations 17 and 18, together, provide the important result given by Equation 19.

\displaystyle \frac{D_{B}^{K}}{D_{A}^{K}}=\left ( \frac{M_{A}}{M_{B}} \right )^{0.5}=\left ( M_{AB} \right )^{0.5} (19)


MA = molecular weight of A (mass / mol)
MB = molecular weight of B (mass / mol)
MAB = ratio of molecular weights of A to B, MA/MB, (dimensionless)

In Equation 19, molecular weights Mii = A, B, are used in place of molecular masses and MAB denotes the ratio MA/MB. Prescription of the isobaric condition in our development is a sufficient condition for Equation 19, but not a necessary one. The more rigorous Dusty Gas Model development shows the Knudsen diffusion coefficients are inversely proportional to the square root of the respective molecular masses and that Equation 19 holds for both constant and variable pressure (Cunningham and Williams, 1980).

A rearrangement of Equation 16, together with the use of Equation 19 and the ideal gas law, results in Equation 20.

\displaystyle N_{B}^{D}=-D_{B}^{K}\frac{dC}{dl}-\left ( M_{AB} \right )^{0.5}N_{A}^{D} (20)

This shows in compact form the coupling that exists between the total diffusive fluxes when the pressure is not uniform. Even under isobaric conditions (i.e., constant C), these fluxes remain coupled, but by the simpler expression of Equation 21.

\displaystyle N_{B}^{D}=-\left ( M_{AB} \right )^{0.5}N_{A}^{D} (21)

Equation 21 is known as Graham’s law of diffusion. In what could be the earliest scientific investigation of diffusion, Thomas Graham (1833) studied steady counter-current diffusion of the components of binary gases through a porous plug under uniform temperature and pressure. Uniform pressure was achieved by frequent adjustment of the pressure on the face of the porous plug so as to negate the spontaneous pressure gradient that was otherwise engendered. He determined the ratio of the magnitudes of both component fluxes and noted the fluxes pointed in opposite directions. In the context of the foregoing equations, Graham measured the magnitude and direction of N_{i}^{D}i = AB, for 10 gas pairs. His experiments foretold Equation 21, a result we have seen to arise from the momentum balance for the gas as a whole under isobaric conditions. We emphasize that Graham’s law holds only under the isobaric condition but Equation 19 holds for both variable and constant pressure. Graham’s law has been experimentally verified many times (e.g., Evans III et al., 1962; Gunn and King, 1969) since Graham’s pioneering investigations.


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