Graham's law

Last updated October 23, 2023

Graham's law of effusion (also called Graham's law of diffusion ) was formulated by Scottish physical chemist Thomas Graham in 1848.^[1] Graham found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the molar mass of its particles.^[1] This formula is stated as:

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 by diffusion—a method that came to play a crucial role in the development of the atomic bomb.^[2]

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.^[2]

In the same conditions of temperature and pressure, the molar mass is proportional to the mass density. Therefore, the rates of diffusion of different gases are inversely proportional to the square roots of their mass densities:

{\mbox{r}}\propto {{\mbox{1}} \over {\sqrt {\rho }}}

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)

{{\mbox{Rate H}}_{2} \over {\mbox{Rate O}}_{2}}={\sqrt {M(O_{2}) \over M(H_{2})}}={{\sqrt {32}} \over {\sqrt {2}}}={\sqrt {16}}=4

Therefore, hydrogen molecules effuse four times faster than those of oxygen.^[1]

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.

{M_{2}}={M_{1}{\mbox{Rate}}_{1}^{2} \over {\mbox{Rate}}_{2}^{2}}

Graham's law was the basis for separating uranium-235 from uranium-238 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 Clinton Engineer Works in Oak Ridge, Tennessee, at the cost of $479 million (equivalent to $5.8 billion in 2021). 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 uranium-235 isotope.^[2]

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 {\mathrm {Rate} _{\mathrm {unknown} }}{\mathrm {Rate} _{\mathrm {He} }}}={\frac {\sqrt {4}}{\sqrt {M_{2}}}}

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 {\sqrt {4}}{\sqrt {M_{2}}}}

Rearranging the equation results in

M=({\frac {\sqrt {4}}{0.25}})^{2}={\frac {\mathrm {64g} }{\mathrm {mol} }}

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.^[1] 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.^[3]

Around the time Graham did his work, the concept of molecular weight was being established largely through the measurements of gases. Daniel Bernoulli suggested in 1738 in his book Hydrodynamica 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 to explain the properties of gases as collections of small particles moving through largely empty space.^[4]

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. Graham's law for diffusion could thus be understood as a consequence of the molecular kinetic energies being equal at the same temperature.^[5]

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:

{\frac {1}{2}}m_{\rm {H_{2}}}v_{\rm {H_{2}}}^{2}={\frac {1}{2}}m_{\rm {O_{2}}}v_{\rm {O_{2}}}^{2}

Which can be simplified and rearranged to:

{\frac {v_{\rm {H_{2}}}^{2}}{v_{\rm {O_{2}}}^{2}}}={\frac {m_{\rm {O_{2}}}}{m_{\rm {H_{2}}}}}

or:

{\frac {v_{\mathrm {H} _{2}}}{v_{\mathrm {O} _{2}}}}={\sqrt {\frac {m_{\mathrm {O} _{2}}}{m_{\mathrm {H} _{2}}}}}

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.

Related Research Articles

Brownian motion is the random motion of particles suspended in a medium.

<span class="mw-page-title-main">Molecular diffusion</span> Thermal motion of liquid or gas particles at temperatures above absolute zero

Molecular diffusion, often simply called diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration. Once the concentrations are equal the molecules continue to move, but since there is no concentration gradient the process of molecular diffusion has ceased and is instead governed by the process of self-diffusion, originating from the random motion of the molecules. The result of diffusion is a gradual mixing of material such that the distribution of molecules is uniform. Since the molecules are still in motion, but an equilibrium has been established, the result of molecular diffusion is called a "dynamic equilibrium". In a phase with uniform temperature, absent external net forces acting on the particles, the diffusion process will eventually result in complete mixing.

<span class="mw-page-title-main">Fick's laws of diffusion</span> Mathematical descriptions of molecular diffusion

Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, $D$ . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation.

In physics, the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.

Stoichiometry is the relationship between the weights of reactants and products before, during, and following chemical reactions.

The molar gas constant is denoted by the symbol $R$ or $R$ . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount of substance, rather than energy per temperature increment per particle. The constant is also a combination of the constants from Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. It is a physical constant that is featured in many fundamental equations in the physical sciences, such as the ideal gas law, the Arrhenius equation, and the Nernst equation.

<span class="mw-page-title-main">Kinetic theory of gases</span> Historic physical model of gases

The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles, all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles.

<span class="mw-page-title-main">Effusion</span> Process of a gas escaping through a small hole

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 the gas is due to the pressure difference between the container and the exterior. Under these conditions, essentially all molecules which arrive at the hole continue and pass through the hole, since collisions between molecules in the region of the hole are negligible. Conversely, when the diameter is larger than the mean free path of the gas, flow obeys the Sampson flow law.

In physical chemistry, Henry's law is a gas law that states that the amount of dissolved gas in a liquid is directly proportional to its partial pressure above the liquid. The proportionality factor is called Henry's law constant. It was formulated by the English chemist William Henry, who studied the topic in the early 19th century.

<span class="mw-page-title-main">Reaction rate</span> Speed at which a chemical reaction takes place

The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit time. Reaction rates can vary dramatically. For example, the oxidative rusting of iron under Earth's atmosphere is a slow reaction that can take many years, but the combustion of cellulose in a fire is a reaction that takes place in fractions of a second. For most reactions, the rate decreases as the reaction proceeds. A reaction's rate can be determined by measuring the changes in concentration over time.

The Sherwood number (Sh) is a dimensionless number used in mass-transfer operation. It represents the ratio of the convective mass transfer to the rate of diffusive mass transport, and is named in honor of Thomas Kilgore Sherwood.

The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. The number is named after Danish physicist Martin Knudsen (1871–1949).

In chemistry, colligative properties are those properties of solutions that depend on the ratio of the number of solute particles to the number of solvent particles in a solution, and not on the nature of the chemical species present. The number ratio can be related to the various units for concentration of a solution such as molarity, molality, normality (chemistry), etc. The assumption that solution properties are independent of nature of solute particles is exact only for ideal solutions, which are solutions that exhibit thermodynamic properties analogous to those of an ideal gas, and is approximate for dilute real solutions. In other words, colligative properties are a set of solution properties that can be reasonably approximated by the assumption that the solution is ideal.

<span class="mw-page-title-main">Equipartition theorem</span> Theorem in classical statistical mechanics

In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in rotational motion.

The Lawson criterion is a figure of merit used in nuclear fusion research. It compares the rate of energy being generated by fusion reactions within the fusion fuel to the rate of energy losses to the environment. When the rate of production is higher than the rate of loss, the system will produce net energy. If enough of that energy is captured by the fuel, the system will become self-sustaining and is said to be ignited.

Gaseous diffusion is a technology that was used to produce enriched uranium by forcing gaseous uranium hexafluoride (UF₆) through microporous membranes. This produces a slight separation (enrichment factor 1.0043) between the molecules containing uranium-235 (²³⁵U) and uranium-238 (²³⁸U). By use of a large cascade of many stages, high separations can be achieved. It was the first process to be developed that was capable of producing enriched uranium in industrially useful quantities, but is nowadays considered obsolete, having been superseded by the more-efficient gas centrifuge process (enrichment factor 1.05 to 1.2).

<span class="mw-page-title-main">Collision theory</span> Chemistry principle

Collision theory is a principle of chemistry used to predict the rates of chemical reactions. It states that when suitable particles of the reactant hit each other with the correct orientation, only a certain amount of collisions result in a perceptible or notable change; these successful changes are called successful collisions. The successful collisions must have enough energy, also known as activation energy, at the moment of impact to break the pre-existing bonds and form all new bonds. This results in the products of the reaction. The activation energy is often predicted using the Transition state theory. Increasing the concentration of the reactant brings about more collisions and hence more successful collisions. Increasing the temperature increases the average kinetic energy of the molecules in a solution, increasing the number of collisions that have enough energy. Collision theory was proposed independently by Max Trautz in 1916 and William Lewis in 1918.

In physics, Knudsen diffusion, named after Martin Knudsen, is a means of diffusion that occurs when the scale length of a system is comparable to or smaller than the mean free path of the particles involved. An example of this is in a long pore with a narrow diameter (2–50 nm) because molecules frequently collide with the pore wall. As another example, consider the diffusion of gas molecules through very small capillary pores. If the pore diameter is smaller than the mean free path of the diffusing gas molecules, and the density of the gas is low, the gas molecules collide with the pore walls more frequently than with each other, leading to Knudsen diffusion.

Diffusivity, mass diffusivity or diffusion coefficient is usually written as the proportionality constant between the molar flux due to molecular diffusion and the negative value of the gradient in the concentration of the species. More accurately, the diffusion coefficient times the local concentration is the proportionality constant between the negative value of the mole fraction gradient and the molar flux. This distinction is especially significant in gaseous systems with strong temperature gradients. Diffusivity derives its definition from Fick's law and plays a role in numerous other equations of physical chemistry.

<span class="mw-page-title-main">Diffusion</span> Transport of dissolved species from the highest to the lowest concentration region

Diffusion is the net movement of anything 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 potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as statistics, probability theory, information theory, neural networks, finance, and marketing.

References

1 2 3 4 Keith J. Laidler and John M. Meiser, Physical Chemistry (Benjamin/Cummings 1982), pp. 18–19
1 2 3 R.H. Petrucci, W.S. Harwood and F.G. Herring, General Chemistry (8th ed., Prentice-Hall 2002) pp. 206–08 ISBN 0-13-014329-4
↑ Laidler and Meiser p.795
↑
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.
↑ "The Kinetic Molecular Theory". Chemed.chem.purdue.edu. Retrieved 2017-07-20.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[LM-1] 1 2 3 4 Keith J. Laidler and John M. Meiser, Physical Chemistry (Benjamin/Cummings 1982), pp. 18–19

[Petrucci-2] 1 2 3 R.H. Petrucci, W.S. Harwood and F.G. Herring, General Chemistry (8th ed., Prentice-Hall 2002) pp. 206–08 ISBN 0-13-014329-4

[3] Laidler and Meiser p.795

[Maxwell-4] 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.

[mwvA] 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.

[mwwQ] 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.

[5] "The Kinetic Molecular Theory". Chemed.chem.purdue.edu. Retrieved 2017-07-20.

[1]

[2]

[3]

[4]

[5]

Graham's law

Contents

Examples

History

See also

Related Research Articles

References