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Effusion

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Process of a gas escaping through a small hole

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Not to be confused withAffusion.

For other uses, seeEffusion (disambiguation).

The image on the left shows effusion, whereas the image on the right showsdiffusion. Effusion occurs through an orifice smaller than the mean free path of the particles in motion, whereas diffusion occurs through an opening in which multiple particles can flow through simultaneously.

In physics and chemistry,effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than themean free path of the molecules.^[1] Such a hole is often described as apinhole 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 themean free path of the gas, flow obeys theSampson flow law.

In medical terminology, aneffusion refers to accumulation of fluid in ananatomic space, usually withoutloculation. Specific examples includesubdural,mastoid,pericardial andpleural effusions.

Etymology

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The word effusion derives from theLatin word, effundo, which means "shed", "pour forth", "pour out", "utter", "lavish", "waste".

Into a vacuum

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Effusion from an equilibrated container into outside vacuum can be calculated based onkinetic theory.^[2] The number of atomic or molecular collisions with a wall of a container per unit area per unit time (impingement rate) is given by: $J_{\text{impingement}}={\frac {P}{\sqrt {2\pi mk_{\text{B}}T}}}.$ assuming mean free path is much greater than pinhole diameter and the gas can be treated as anideal gas.^[3]

If a small area $A {\displaystyle A}$ on the container is punched to become a small hole, the effusive flow rate will be ${\begin{aligned}Q_{\text{effusion}}&=J_{\text{impingement}}\times A\\&={\frac {PA}{\sqrt {2\pi mk_{\text{B}}T}}}\\&={\frac {PAN_{\text{A}}}{\sqrt {2\pi MRT}}}\end{aligned}}$ where $M {\displaystyle M}$ is themolar mass, $N_{\text{A}}$ is theAvogadro constant, and $R=N_{\text{A}}k_{\text{B}}$ is themolar gas constant.

The average velocity of effused particles is ${\begin{aligned}{\overline {v_{x}}}&={\overline {v_{y}}}=0\\{\overline {v_{z}}}&={\sqrt {\frac {\pi k_{\text{B}}T}{2m}}}.\end{aligned}}$

Combined with the effusive flow rate, the recoil/thrust force on the system itself is $F=m{\overline {v_{z}}}{\times }Q_{\text{effusion}}={\frac {PA}{2}}.$

An example is the recoil force on a balloon with a small hole flying in vacuum.

Measures of flow rate

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According to thekinetic theory of gases, thekinetic energy for a gas at a temperature $T {\displaystyle T}$ is

{\frac {1}{2}}mv_{\rm {rms}}^{2}={\frac {3}{2}}k_{\rm {B}}T

where $m {\displaystyle m}$ is the mass of one molecule, $v_{\rm {rms}}$ is theroot-mean-square speed of the molecules, and $k_{\rm {B}}$ is theBoltzmann constant. The average molecular speed can be calculated from theMaxwell speed distributionas ${\textstyle v_{\rm {avg}}={\sqrt {8/3\pi }}\ v_{\rm {rms}}\approx 0.921\ v_{\rm {rms}}}$ (or, equivalently, ${\textstyle v_{\rm {rms}}={\sqrt {3\pi /8}}\ v_{\rm {avg}}\approx 1.085\ v_{\rm {avg}}}$ ). The rate $\Phi _{N}$ at which a gas ofmolar mass $M {\displaystyle M}$ effuses (typically expressed as thenumber of molecules passing through the hole per second) is then^[4]

\Phi _{N}={\frac {\Delta PAN_{A}}{\sqrt {2\pi MRT}}}.

Here $\Delta P$ is the gas pressure difference across the barrier, $A {\displaystyle A}$ is the area of the hole, $N_{\text{A}}$ is theAvogadro constant, $R {\displaystyle R}$ is thegas constant and $T {\displaystyle T}$ is theabsolute temperature. Assuming the pressure difference between the two sides of the barrier is much smaller than $P_{\rm {avg}}$ , the average absolute pressure in the system (i.e. $\Delta P\ll P_{\rm {avg}}$ ), it is possible to express effusion flow as a volumetric flow rate as follows:

\Phi _{V}={\frac {\Delta Pd^{2}}{P_{\rm {avg}}}}{\sqrt {\frac {\pi k_{\text{B}}T}{32m}}}

\Phi _{V}={\frac {\Delta Pd^{2}}{P_{\rm {avg}}}}{\sqrt {\frac {\pi RT}{32M}}}

where $\Phi _{V}$ is the volumetric flow rate of the gas, $P_{\rm {avg}}$ is the average pressure on either side of the orifice, and $d {\displaystyle d}$ is the hole diameter.

Effect of molecular weight

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At constant pressure and temperature, the root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight. Gases with a lower molecular weight effuse more rapidly than gases with a higher molecular weight, so that thenumber of lighter molecules passing through the hole per unit time is greater.

Graham's law

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Main article:Graham's law

Scottish chemistThomas Graham (1805–1869) found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles.^[5] In other words, the ratio of the rates of effusion of two gases at the same temperature and pressure is given by the inverse ratio of the square roots of the masses of the gas particles.

{{\mbox{Rate of effusion of gas}}_{1} \over {\mbox{Rate of effusion of gas}}_{2}}={\sqrt {M_{2} \over M_{1}}}

where $M_{1}$ and $M_{2}$ represent the molar masses of the gases. This equation is known asGraham's law of effusion.

The effusion rate for a gas depends directly on the average velocity of its particles. Thus, the faster the gas particles are moving, the more likely they are to pass through the effusion orifice.

Knudsen cell

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TheKnudsen cell is used to measure thevapor pressures of a solid with very low vapor pressure. Such a solid forms a vapor at low pressure bysublimation. The vapor slowly effuses through a pinhole, and the loss of mass is proportional to the vapor pressure and can be used to determine this pressure.^[4] Theheat of sublimation can also be determined by measuring the vapor pressure as a function of temperature, using theClausius–Clapeyron relation.^[6]

References

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^K.J. Laidler and J.H. Meiser, Physical Chemistry, Benjamin/Cummings 1982, p.18.ISBN 0-8053-5682-7
^"5.62 Physical Chemistry II"(PDF).MIT OpenCourseWare.
^"Low-Pressure Effusion of Gases".www.chem.hope.edu. Hope College. Retrieved6 April 2021.
^^a ^bPeter Atkins and Julio de Paula,Physical Chemistry (8th ed., W.H.Freeman 2006) p.756ISBN 0-7167-8759-8
^Zumdahl, Steven S. (2008).Chemical Principles. Boston: Houghton Mifflin Harcourt Publishing Company. p. 164.ISBN 978-0-547-19626-8.
^Drago, R.S.Physical Methods in Chemistry (W.B.Saunders 1977) p.563ISBN 0-7216-3184-3

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