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
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
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
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
of the gas, flow obeys the
Sampson flow Sampson flow is defined as fluid flow through an infinitely thin orifice in the viscous flow regime for low Reynolds number. It is derived from an analytical solution to the Navier-Stokes equations. The below equation can be used to calculate the ...
law.
In medical terminology, an effusion refers to accumulation of fluid in an
anatomic space {{set index article
In anatomy, a spatium or anatomic space is a space (cavity or gap). Anatomic spaces are often landmarks to find other important structures. When they fill with gases (such as air) or liquids (such as blood) in pathological ways, ...
, usually without
loculation. Specific examples include
subdural,
mastoid
The mastoid part of the temporal bone is the posterior (back) part of the temporal bone, one of the bones of the skull. Its rough surface gives attachment to various muscles (via tendons) and it has openings for blood vessels. From its borders, ...
,
pericardial and
pleural effusion
A pleural effusion is accumulation of excessive fluid in the pleural space, the potential space that surrounds each lung.
Under normal conditions, pleural fluid is secreted by the parietal pleural capillaries at a rate of 0.6 millilitre per k ...
s.
Etymology
The word effusion derives from the
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
word, effundo, which means "shed, pour forth, pour out, utter, lavish, waste."
Effusion into vacuum
Effusion from an equilibrated container into outside vacuum can be calculated based on
kinetic theory
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to its motion
Art and ente ...
.
The number of atomic or molecular collisions with a wall of a container per unit area per unit time (
impingement rate) is given by:
assuming mean free path is much greater than pinhole diameter and the gas can be treated as an
ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
.
If a small area
on the container is punched to become a small hole, the effusive flow rate will be
where
is the
molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
,
is the
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
, and
is the
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
.
The average velocity of effused particles is
Combined with the effusive flow rate, the recoil/thrust force on the system itself is
An example is the recoil force on a balloon with a small hole flying in vacuum.
Measures of flow rate
According to the
kinetic theory of gases, the
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
for a gas at a temperature
is
:
where
is the mass of one molecule,
is the
root-mean-square speed of the molecules, and
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
. The average molecular speed can be calculated from the
Maxwell speed distribution
Maxwell may refer to:
People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of ...
as
(or, equivalently,
). The rate
at which a gas of
molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
effuses (typically expressed as the ''number'' of molecules passing through the hole per second) is then
Peter Atkins
Peter William Atkins (born 10 August 1940) is an English chemist and a Fellow of Lincoln College at the University of Oxford. He retired in 2007. He is a prolific writer of popular chemistry textbooks, including ''Physical Chemistry'', ''Ino ...
and Julio de Paula, ''Physical Chemistry'' (8th ed., W.H.Freeman 2006) p.756
:
Here
is the gas pressure difference across the barrier,
is the area of the hole,
is the
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
,
is the
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
and
is the
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic w ...
. Assuming the pressure difference between the two sides of the barrier is much smaller than
, the average absolute pressure in the system (''i.e.''
), it is possible to express effusion flow as a volumetric flow rate as follows:
:
or
:
where
is the volumetric flow rate of the gas,
is the average pressure on either side of the orifice, and
is the hole diameter.
Effect of molecular weight
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 the ''number'' of lighter molecules passing through the hole per unit time is greater.
Graham's law
Scottish chemist
Thomas 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. 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.
:
where
and
represent the molar masses of the gases.
This equation is known as
Graham'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 effusion cell
The
Knudsen effusion cell In crystal growth, a Knudsen cell is an effusion evaporator source for relatively low partial pressure elementary sources (e.g. Ga, Al, Hg, As). Because it is easy to control the temperature of the evaporating material in Knudsen cells, they are c ...
is used to measure the
vapor pressure
Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phas ...
s of a solid with very low vapor pressure. Such a solid forms a vapor at low pressure by
sublimation. 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.
[ The ]heat of sublimation
In thermodynamics, the enthalpy of sublimation, or heat of sublimation, is the heat required to sublimate (change from solid to gas) one mole of a substance at a given combination of temperature and pressure, usually standard temperature and pr ...
can also be determined by measuring the vapor pressure as a function of temperature, using the Clausius–Clapeyron relation.[Drago, R.S. ''Physical Methods in Chemistry'' (W.B.Saunders 1977) p.563 ]
References
{{reflist
Physical chemistry
Gases