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Graham's law of effusion (also called Graham's law of
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
) was formulated by Scottish physical
chemist A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin ''alchemist'') is a graduated scientist trained in the study of chemistry, or an officially enrolled student in the field. Chemists study the composition of ...
Thomas Graham in 1848. Keith J. Laidler and John M. Meiser, ''Physical Chemistry'' (Benjamin/Cummings 1982), pp. 18–19 Graham found experimentally that the rate of
effusion In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules. Such a hole is often described as a ''pinhole'' and the escape ...
of a gas is inversely proportional to the square root of the
molar mass In chemistry, the molar mass () (sometimes called molecular weight or formula weight, but see related quantities for usage) of a chemical substance ( element or compound) is defined as the ratio between the mass () and the amount of substance ...
of its
particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
s. This formula is stated as: :=\sqrt, where: :Rate1 is the rate of effusion for the first gas. (
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
or number of moles per unit time). :Rate2 is the rate of effusion for the second gas. :''M1'' is the
molar mass In chemistry, the molar mass () (sometimes called molecular weight or formula weight, but see related quantities for usage) of a chemical substance ( element or compound) is defined as the ratio between the mass () and the amount of substance ...
of gas 1 :''M2'' is the molar mass of gas 2. Graham's law states that the rate of diffusion or 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 (heavier gases diffuse more slowly). 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 Isotopes are distinct nuclear species (or ''nuclides'') of the same chemical element. They have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), but ...
by diffusion—a method that came to play a crucial role in the development of the atomic bomb.R.H. Petrucci, W.S. Harwood and F.G. Herring, ''General Chemistry'' (8th ed., Prentice-Hall 2002) pp. 206–08 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. In the same conditions of temperature and pressure, the molar mass is proportional to the
mass density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter rho), although the Latin letter ''D'' (or ''d'') ...
. Therefore, the rates of diffusion of different gases are inversely proportional to the square roots of their mass densities: : \propto where: :''ρ'' is the mass density.


Examples

First Example: Let gas 1 be H2 and gas 2 be O2. (This example is solving for the ratio between the rates of the two gases) : =\sqrt

\sqrt = 4
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 the unknown molecular weight. := Graham's law was the basis for separating
uranium-235 Uranium-235 ( or U-235) is an isotope of uranium making up about 0.72% of natural uranium. Unlike the predominant isotope uranium-238, it is fissile, i.e., it can sustain a nuclear chain reaction. It is the only fissile isotope that exists in nat ...
from uranium-238 found in natural uraninite (uranium ore) during the
Manhattan Project The Manhattan Project was a research and development program undertaken during World War II to produce the first nuclear weapons. It was led by the United States in collaboration with the United Kingdom and Canada. From 1942 to 1946, the ...
to build the first atomic bomb. The United States government built a gaseous diffusion plant at the Clinton Engineer Works in Oak Ridge, Tennessee, at the cost of $479 million (equivalent to $ in ). In this plant,
uranium Uranium is a chemical element; it has chemical symbol, symbol U and atomic number 92. It is a silvery-grey metal in the actinide series of the periodic table. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons. Ura ...
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 uranium-235 isotope. Second Example: An unknown gas diffuses 0.25 times as fast as He. What is the molar mass of the unknown gas? Using the formula of gaseous diffusion, we can set up this equation. :\frac = \frac Which is the same as the following because the problem states that the rate of diffusion of the unknown gas relative to the helium gas is 0.25. :0.25 = \frac Rearranging the equation results in :M = (\frac)^2 = \frac


History

Graham's research on the diffusion of gases was triggered by his reading about the observations 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 in 1831 that the rate of effusion of a gas is inversely proportional to the square root of its density, and later in 1848 showed that this rate is inversely proportional to the square root of the molar mass. 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
colloid A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others exte ...
s, a term that has come to denote an important class of finely divided materials. Around the time Graham did his work, the concept of molecular weight was being established largely through the measurements of gases.
Daniel Bernoulli Daniel Bernoulli ( ; ; – 27 March 1782) was a Swiss people, Swiss-France, French mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applicati ...
suggested in 1738 in his book ''
Hydrodynamica ''Hydrodynamica, sive de Viribus et Motibus Fluidorum Commentarii'' (Latin for ''Hydrodynamics, or commentaries on the forces and motions of fluids'') is a book published by Daniel Bernoulli in 1738. The title of this book eventually christened ...
'' that heat increases in proportion to the velocity, and thus kinetic energy, of gas particles. Italian physicist Amedeo Avogadro also 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 James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
to explain the properties of gases as collections of small particles moving through largely empty space.See: * Maxwell, J.C. (1860
"Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres,"
''Philosophical Magazine'', 4th series, 19 : 19–32. * Maxwell, J.C. (1860
"Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another,"
''Philosophical Magazine'', 4th series, 20 : 21–37.
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 The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
(absolute) temperature scale is directly proportional to the average kinetic energy of the gas molecules. Graham's law for diffusion could thus be understood as a consequence of the molecular kinetic energies being equal at the same temperature. The rationale of the above can be summed up as follows: Kinetic energy of each type of particle (in this example, Hydrogen and Oxygen, as above) within the system is equal, as defined by
thermodynamic temperature Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion. Thermodynamic temperature is typically expres ...
: : \fracm_v^_=\fracm_v^_ Which can be simplified and rearranged to: : \frac = \frac or: : \frac = \sqrt Ergo, when constraining the system to the passage of particles through an area, Graham's law appears as written at the start of this article.


See also

* Sieverts' law * Henry's law *
Gas laws The laws describing the behaviour of gases under fixed pressure, volume, amount of gas, and absolute temperature conditions are called gas laws. The basic gas laws were discovered by the end of the 18th century when scientists found out that re ...
* Scientific laws named after people *
Viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
*
Drag (physics) In fluid dynamics, drag, sometimes referred to as fluid resistance, is a force acting opposite to the direction of motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers, two solid surfaces, or b ...
* Vapour Density


References

{{DEFAULTSORT:Graham's Law Eponymous laws of physics Gas laws