Definitions
Diffusion - The spread of particles through random motion from regions of higher concentration to regions of lower concentration.
Effusion - The process in which individual molecules flow, through a hole without collisions between molecules, into a vacuum.
Graham's Laws
Graham's law, also known as Graham's law of effusion, was
formulated by Scottish physical chemist Thomas Graham.
Graham found experimentally that the rate of effusion of a gas is inversely
proportional to the square root of the mass of its particles. This formula can
be written as:
where:
Rate1 is the rate of effusion of the first gas (volume
or number of moles per unit time).
Rate2 is the rate of effusion for the second gas.
M1 is the molar mass of gas 1
M2 is the molar mass of gas 2.
Graham's law states that the rate of effusion of a gas is inversely
proportional to the square root of its molecular weight. Thus, if the molecular
weight of one gas is four times that of another, it would diffuse through a
porous plug or escape through a small pinhole in a vessel at half the rate of
the other. A complete theoretical explanation of Graham's law was provided years
later by the kinetic theory of gases.
Graham's law provides a basis for separating isotopes by diffusion — a
method that came to play a crucial role in the development of the atomic
bomb.
Graham's law is most accurate for molecular effusion which involves the
movement of one gas at a time through a hole. It is only approximate for diffusion of
one gas in another or in air, as these processes involve the movement of more
than one gas.
Graham's law states that the rate at which gas molecules diffuse is inversely proportional to the square root of its density. Combined with Avogadro's law (i.e. since equal volumes have equal number of molecules) this is the same as being inversely proportional to the root of the molecular weight.
History
Graham's research on the diffusion of gases was triggered by his reading
about the observation of German chemist Johann Döbereiner
that hydrogen gas diffused out of a small crack in a glass bottle faster than
the surrounding air diffused in to replace it. Graham measured the rate of
diffusion of gases through plaster plugs, through very fine tubes, and through
small orifices. In this way he slowed down the process so that it could be
studied quantitatively. He first stated the law as we know it today in 1831.
Graham went on to study the diffusion of substances in solution and in the
process made the discovery that some apparent solutions actually are suspensions of
particles too large to pass through a parchment filter. He termed these
materials colloids, a term that has come
to denote an important class of finely divided materials.
At the time Graham did his work the concept of molecular weight was being
established, in large part through measurements of gases. Italian physicist Amadeo Avogadro had
suggested in 1811 that equal volumes of different gases contain equal numbers of
molecules. Thus, the relative molecular weights of two gases are equal to the
ratio of weights of equal volumes of the gases. Avogadro's insight together with
other studies of gas behaviour provided a basis for later theoretical work by
Scottish physicist James Clerk Maxwell
to explain the properties of gases as collections of small particles moving
through largely empty space.
Perhaps the greatest success of the kinetic theory of gases, as it came to be
called, was the discovery that for gases, the temperature as measured on the Kelvin (absolute)
temperature scale is directly proportional to the average kinetic energy of the
gas molecules. The kinetic energy of any object is equal to one-half its mass
times the square of its velocity. Thus, to have equal kinetic energies, the
velocities of two different molecules would have to be in inverse proportion to
the square roots of their masses. The rate of effusion is determined by the
number of molecules entering an aperture per unit time, and hence by the average
molecular velocity. Graham's law for diffusion could thus be understood as a
consequence of the molecular kinetic energies being equal at the same
temperature.
Example
Let gas 1 be H2 and gas 2 be O2.
Therefore, hydrogen molecules effuse four times faster than those of
oxygen.
Graham's Law can also be used to find the approximate molecular weight of a
gas if one gas is a known species, and if there is a specific ratio between the
rates of two gases (such as in the previous example). The equation can be solved
for either one of the molecular weights provided the subscripts are
consistent.
Graham's law was the basis for separating 235U from
238U found in natural uraninite (uranium ore) during
the Manhattan project to
build the first atomic bomb. The United States government built a gaseous
diffusion plant at the then phenomenal cost of $100 million in Clinton,
Tennessee. In this plant, uranium from uranium ore was
first converted to uranium
hexafluoride and then forced repeatedly to diffuse through porous barriers,
each time becoming a little more enriched in the slightly lighter
235U isotope.
For more information: https://en.wikipedia.org/wiki/Graham's_law
Source: Wikipedia (All text is available under the terms of the GNU Free Documentation License and Creative Commons Attribution-ShareAlike License.)
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