Graham's_law - Wikiparency

# Graham's law **Graham's law**, or Graham's law of effusion, states that the rate at which a gas escapes through a tiny hole into a vacuum is inversely proportional to the square root of its molar mass, so that for two gases at the same temperature and pressure `r1/r2 = sqrt(M2/M1)`.[^averill-945] The law is named for the Scottish chemist Thomas Graham, who measured the passage of gases through porous plugs and fine apertures in the 1830s and 1840s and found the square-root rule before the [[Kinetic_theory_of_gases|kinetic theory of gases]] existed to explain it.[^graham1833][^graham1846] Kinetic theory later supplied the reason: at a given temperature every gas has the same average molecular kinetic energy, so lighter molecules move faster, `v_rms = sqrt(3·R·T/M)`, and reach the hole more often.[^averill-950] [[Effusion]] through a hole smaller than the [[Mean_free_path|mean free path]] follows the law closely; [[Diffusion|diffusion]] of one gas through another only approximately, since collisions intervene.[^averill-945] On the [[Chemistry]] flagship the law is the second section of Part XI, *Chemical laws*, under the heading *Graham's law: effusion and enrichment*, following [[Dalton's_law]] and preceding the optical law treated on [[Beer–Lambert_law]]. Its spine neighbours are [[Gaseous_diffusion]], [[Isotope_separation]] and [[Enriched_uranium]], the last of which carries the same microsim as a variant on the Energy and Physics flagships. In the microsim below the reader runs an effusion cascade. For the two [[Uranium|uranium]] hexafluorides, ²³⁵UF₆ at 349.03 g/mol and ²³⁸UF₆ at 352.04 g/mol, the law gives a per-stage enrichment factor of sqrt(352.04/349.03) = 1.0043, and the reader slides the number of stages *n* from 0 to 1,200 while the ²³⁵U fraction `x_n = x0 · 1.0043^n` climbs a logarithmic axis from the natural 0.720 % past the reactor band near 3 % toward the weapons band at 70 % and above.[^averill-948][^averill-948-949][^ball-757] The plot marks the book's result of 1.15 × 10³ stages to reach 99 %, labelled as the book's fraction method, beside the ratio method's figure of about 2.2 × 10³ stages, and a preset switches the cascade to helium-3, where a per-stage factor of 1.152 needs only 96 stages.[^averill-949] ## Examples The law in its working form compares two gases through their molar masses alone. Because the mean molecular speed at a fixed temperature scales as `1/sqrt(M)`, the ratio of root-mean-square speeds of two gases is `v_rms,2/v_rms,1 = sqrt(M1/M2)`, and the rate at which molecules of each gas strike a small opening, and so leak through it, follows the same ratio when the two gases are at the same number density.[^averill-950][^averill-945] The Portal Book *General Chemistry* states the law as `r1/r2 = sqrt(M2/M1)`, with *r* the rate of effusion, and warns that although effusion closely approximates diffusion the two are different phenomena: effusion is escape through a hole into a region of lower pressure, diffusion is the spreading of one gas through another by countless collisions.[^averill-945] The examples below run from the balloon on a string to the cascade that made the first nuclear fuel, and every one of them is the same square root. #### Helium against air Air has an average molar mass of about 29 g/mol and [[Helium|helium]] 4.00 g/mol, so helium effuses sqrt(29/4.00) = 2.7 times as fast as air.[^averill-946-947] That ratio is why a helium [[Balloon|balloon]] sags overnight while an air-filled one does not: helium atoms find the pores of the rubber faster than the nitrogen and oxygen molecules outside find their way in. [[Hydrogen|Hydrogen]], at 2.016 g/mol, would leak sqrt(4.00/2.016) = 1.41 times faster still (derived), which is one of several reasons it lost the airship trade to helium despite its slightly greater lift. The [[Lifting_gas|lifting-gas]] article on the flagship carries the buoyancy side of that comparison; this page carries the leak. The same ratio runs the other way for heavy gases: sulfur hexafluoride, at about 146 g/mol, effuses at sqrt(29/146) = 0.45 of the rate of air (derived).[^ball-122-124] Rates, not amounts, are what the law compares; how much gas a balloon loses also depends on the area of the pores and the pressure difference across them, which the law leaves out. #### Ammonia and hydrogen chloride The classroom demonstration of the law places cotton soaked in concentrated [[Ammonia|ammonia]] solution at one end of a glass tube and cotton soaked in hydrochloric acid at the other; where the two vapours meet, a white ring of ammonium chloride forms. With molar masses of 17.0 g/mol for NH₃ and 36.5 g/mol for HCl the law predicts a rate ratio of sqrt(36.5/17.0) = 1.47, so the ring should form about 1.47 times farther from the ammonia end. The Portal Book reports an observed ratio of 15.5 cm to 9.2 cm, or 1.7.[^averill-946] The discrepancy is the diffusion caveat made visible: the vapours are moving through air, not into a vacuum, and the heavier hydrogen chloride is slowed more by collisions than the square root alone allows. The demonstration confirms the direction of the law and the rough size of the effect, and it is a fair warning that the cascade calculations below assume true effusion. #### The uranium cascade Natural uranium is 0.720 % ²³⁵U, the fissile isotope, and the rest is almost entirely ²³⁸U.[^averill-948] Converted to uranium hexafluoride, the only compound of uranium that is a gas at practical temperatures, the two isotopic species differ in molar mass by 3.01 g/mol out of about 350, so a single pass through a porous barrier enriches the lighter one by the factor sqrt(352.04/349.03) = 1.0043, taking the fraction from 0.720 % to 0.723 %.[^averill-948] The book's model applies that factor to the fraction at every stage: to reach 99 % requires `0.990/0.00720 = 138 = 1.0043^n`, so `n = ln 138/ln 1.0043 ≈ 1.15 × 10³` stages.[^averill-948-949] The microsim draws this curve on a log axis with two shaded bands taken from the Portal Book *Introductory Chemistry*: light-water reactor fuel at roughly 3 % and weapons material at 70 % and above.[^ball-757] Reaching the reactor band takes about 330 stages by the same arithmetic and the weapons band about 1,070 (derived; ILLUSTRATIVE). The multiplication of the fraction is a simplification that holds only while the fraction is small; the exact treatment multiplies the isotope *ratio* x/(1 − x) by 1.0043 per stage, which gives about 2.2 × 10³ stages to 99 % and about 1,350 to 70 % (derived; ILLUSTRATIVE), and the fraction method's curve would pass 100 % just after 1,150 stages if the microsim did not clamp it. The sim shows the book's number with that label and clamps the fraction at 1. The first industrial enrichment plants were built on this principle, the wartime K-25 gaseous-diffusion plant at Oak Ridge, Tennessee among them, and ran thousands of stages in series.[citation needed] #### The helium-3 crossing The same cascade with a lighter pair shows how much the isotopic mass difference matters. For [[Helium-3]] against ordinary [[Helium-4|helium-4]], the per-stage factor is sqrt(4.00260/3.01603) = 1.15200, and one stage takes the natural abundance of helium-3 in atmospheric helium from 0.000134 % to 0.000154 %.[^averill-949] Reaching 99.0 % helium-3 by the book's fraction method needs only 96 stages, against more than a thousand for uranium; the ratio method gives about 128 stages (derived; ILLUSTRATIVE).[^averill-949] The microsim's helium preset sets α = 1.152 and a stage range of 0 to 120 so the reader sees the curve climb four orders of magnitude on the same axis that the uranium curve barely tilts; the two presets side by side are the clearest statement the article can make that the mass ratio, not the mass, is what the square root cares about. In practice helium-3 is not made by effusion; it is collected from the decay of [[Tritium|tritium]], whose half-life of about 12.3 years converts stored tritium to helium-3 at a useful rate.[^ball-ch15] ## History Thomas Graham published his first account of the law in 1833 in the *Philosophical Magazine*, under the title "On the law of the diffusion of gases", reporting that the rates at which different gases passed through a plug of plaster of Paris were inversely as the square roots of their densities.[^graham1833] In 1846 he extended the work to passage through a fine aperture into a vacuum, the case now called effusion, in a long paper to the Royal Society, "On the motion of gases", which separated the three regimes of effusion, transpiration through capillaries and diffusion through porous plates.[^graham1846] The square-root rule was empirical: Graham measured it, tabulated it and used it, but had no molecular theory to derive it from. He went on to apply the same idea of unequal rates of passage to liquids, and his 1861 paper on liquid diffusion introduced the words *colloid* and *dialysis* for substances that diffuse slowly through membranes and for the method of separating them.[^graham1861] The gas law, the [[Colloid|colloid]] and the dialysis membrane are one habit of mind: measure what leaks through and read the size of the particle from the rate. #### From Graham to kinetic theory The explanation came in 1860, when [[James_Clerk_Maxwell|James Clerk Maxwell]] derived the distribution of molecular speeds in a gas and showed that the mean speed at a given temperature varies as the inverse square root of the molecular mass.[^maxwell1860] With the [[Maxwell–Boltzmann_distribution|Maxwell–Boltzmann distribution]] in hand, Graham's law became a corollary: the number of molecules striking a hole per unit area per unit time is proportional to the number density times the mean speed, so at equal density the lighter gas leaks faster by exactly the factor Graham had measured. The mean speed and the root-mean-square speed differ by a fixed numerical factor for every gas, so either one gives the same ratio and the law can be written with whichever is convenient. The Portal Books teach the law in that order, first as `v_rms = sqrt(3·R·T/M)` with *M* in kilograms per mole, then as the ratio of rates.[^averill-950][^openstax-ch8] The [[Kinetic_energy|kinetic energy]] shared equally among gases at one temperature is the physical content; the square root is its arithmetic. #### From the law to the cascade Graham's plaster plug became, a century later, the porous barrier of the gaseous-diffusion plant, and his square root became the enrichment factor of 1.0043 per stage that fixed the scale of the first uranium enrichment plants.[^averill-948-949] A real barrier does not achieve even that ideal factor, because part of the gas passes by bulk flow rather than by effusion and because the enriched and depleted streams mix at the barrier face, so working cascades needed more stages than the textbook count and were engineered as countercurrent arrangements that recycle the depleted stream.[^benedict1981] The same arithmetic set the size of every diffusion plant built afterward, and it is the reason the method was eventually displaced by centrifuges, whose per-stage factor is far larger; the pages on [[Gaseous_diffusion]] and [[Isotope_separation]] follow that story, and [[Enriched_uranium]] carries the microsim as a variant with the reactor and weapons bands foregrounded. In the [[Nuclear_fuel_cycle|nuclear fuel cycle]] the 3 % band is where the cascade hands off to fuel fabrication. ## See also - [[Gaseous_diffusion]] - [[Isotope_separation]] - [[Effusion]] - [[Enriched_uranium]] - [[Dalton's_law]] - [[Kinetic_theory_of_gases]] - [[Maxwell–Boltzmann_distribution]] - [[Helium-3]] ## References [^averill-945]: Averill, B.; Eldredge, P. (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", §10.7, p. 945 (statement of Graham's law; effusion against diffusion). https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^averill-946]: Averill, B.; Eldredge, P. (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", p. 946 (the ammonia–hydrogen chloride tube: predicted 1.47, observed 15.5/9.2 = 1.7). https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^averill-946-947]: Averill, B.; Eldredge, P. (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", pp. 946–947 (air at about 29 g/mol; helium against air, sqrt(29/4.00) = 2.7). https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^averill-948]: Averill, B.; Eldredge, P. (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", p. 948 (²³⁵UF₆ 349.03 and ²³⁸UF₆ 352.04 g/mol; natural abundance 0.720 %; one stage to 0.723 %). https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^averill-948-949]: Averill, B.; Eldredge, P. (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", pp. 948–949 (the cascade calculation: 0.990/0.00720 = 138 = 1.0043ⁿ, n = 1.15 × 10³). https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^averill-949]: Averill, B.; Eldredge, P. (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", p. 949 (helium-3: factor 1.15200; 0.000134 % → 0.000154 %; 96 stages to 99.0 %). https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^averill-950]: Averill, B.; Eldredge, P. (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", p. 950 (v_rms = sqrt(3RT/M) with M in kg/mol; the ratio of rms speeds). https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^ball-757]: Ball, D. W. (2011). *Introductory Chemistry*. Chapter 15, "Nuclear Chemistry", p. 757 (enrichment levels: natural 0.7 %, reactor fuel about 3 %, weapons 70 % and above). https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry [^ball-ch15]: Ball, D. W. (2011). *Introductory Chemistry*. Chapter 15, "Nuclear Chemistry", pp. 720–764 (tritium decay to helium-3 and its half-life; page to pin). https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry [^openstax-ch8]: Flowers, P.; Neth, E.; Robinson, W.; et al. (2019). *Chemistry: Atoms First*, 2nd ed. OpenStax. Chapter 8, "Gases", pp. 363–420 (effusion and diffusion of gases; Graham's law; page to pin). https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first [^ball-122-124]: Ball, D. W. (2011). *Introductory Chemistry*. Chapter 3, §3.3 "Masses of Atoms and Molecules", pp. 122–124 (molecular masses, including SF₆ at about 146 u). https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry [^graham1833]: Graham, T. (1833). "On the law of the diffusion of gases." *Philosophical Magazine*, 3rd series, vol. 2. [^graham1846]: Graham, T. (1846). "On the motion of gases." *Philosophical Transactions of the Royal Society of London*, vol. 136. [^graham1861]: Graham, T. (1861). "Liquid diffusion applied to analysis." *Philosophical Transactions of the Royal Society of London*, vol. 151. [^maxwell1860]: Maxwell, J. C. (1860). "Illustrations of the dynamical theory of gases." *Philosophical Magazine*, 4th series, vols. 19–20. [^benedict1981]: Benedict, M.; Pigford, T. H.; Levi, H. W. (1981). *Nuclear Chemical Engineering*, 2nd ed. New York: McGraw-Hill (gaseous-diffusion barriers and cascade design; chapter and page to pin).  **Microsim — three.js (Wikitube framework):** *Graham's law* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Graham's_law.html" data-title="Graham's law"></div> *Built from `MICROSIM_GUIDE/specs/sims/Graham's_law.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Graham's_law) : [Wikitube](https://en.wikitube.io/wiki/Graham's_law) · pinned revision [1370697843](https://en.wikipedia.org/w/index.php?oldid=1370697843) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Chemistry]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Chemistry row K52 · sim pending (matter/Graham's_law).*