Ideal gas law
The ideal gas law is the
equation of stateof a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyronin 1834.
:"The state of an amount of
gasis determined by its pressure, volume, and temperature according to the equation:":
where : is the absolute
pressureof the gas, : is the volumeof the gas,: is the number of moles of gas,: is the universal gas constant,: is the absolute temperature.
The value of the
ideal gas constant, "R", is found to be as follows.
The ideal gas law mathematically follows from a statistical mechanical treatment of primitive identical particles (point particles without internal structure) which do not interact, but exchange
momentum(and hence kinetic energy) in elastic collisions.
Since it neglects both molecular size and intermolecular attractions, the ideal gas law is most accurate for
monoatomicgases at high temperatures and low pressures. The neglect of molecular size becomes less important for larger volumes, i.e., for lower pressures. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy i.e., with increasing temperatures. More sophisticated "equations of state", such as the van der Waals equation,allow deviations from ideality caused by molecular size and intermolecular forces to be taken into account.
As the amount of substance could be given in mass instead of moles, sometimes an alternative form of the ideal gas law is useful. The number of moles () is equal to the mass () divided by the
molar mass():: By replacing , we get:: from where: .This form of the ideal gas law is particularly useful because it links pressure, density , and temperature in a unique formula independent from the quantity of the considered gas. In statistical mechanicsthe following molecular equation is derived from first principles:: Here is Boltzmann's constant, and is the "actual number" of molecules, in contrast to the other formulation, which uses , the number of moles. This relation implies that , and the consistency of this result with experiment is a good check on the principles of statistical mechanics.
From here we can notice that for an average particle mass of times the
atomic mass constant(i.e., the mass is u):and since , we find that the ideal gas law can be re-written as::
Note_label|A|a|none a. In an isentropic process, system entropy (Q) is constant. Under these conditions, P1 V1 = P2 V2, where is defined as the
heat capacity ratio, which is constant for an ideal gas.
The ideal gas law can be derived from combining two empirical
gas laws: the combined gas lawand Avogadro's law. The combined gas law states that
where "C" is a constant which is directly proportional to the amount of gas, "n" (
Avogadro's law). The proportionality factor is the universal gas constant, "R", i.e. .
Hence the ideal gas law:
The ideal gas law can also be derived from
first principlesusing the kinetic theory of gases, in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.
Derivation from the statistical mechanics
Let q = ("qx", "qy", "qz") and p = ("px", "py", "pz") denote the position vector and momentum vector of a particle of an ideal gas,respectively, and let F denote the net force on that particle, then:where the first equality is
Newton's second law, and the second line uses Hamilton's equationsand the equipartition theorem. Summing over a system of "N" particles yields
Newton's third lawand the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure "P" of the gas. Hence
where dS is the infinitesimal area element along the walls of the container. Since the
divergenceof the position vector q is
divergence theoremimplies that
where "dV" is an infinitesimal volume within the container and "V" is the total volume of the container.
Putting these equalities together yields
which immediately implies the
ideal gas lawfor "N" particles:
where "n=N/NA" is the number of moles of gas and "R=NAkB" is the
The readers are referred to the comprehensive article [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 Configuration integral (statistical mechanics)] where an alternative statistical mechanics derivation of the ideal-gas law, using the relationship between the
Helmholtz free energyand the partition function, but without using the equipartition theorem, is provided.
Combined gas law
Equation of state
Van der Waals equation
*Davis and Masten "Principles of Environmental Engineering and Science", McGraw-Hill Companies, Inc. New York (2002) ISBN 0-07-235053-9
* [http://www.gearseds.com/curriculum/learn/lesson.php?id=23&chapterid=5 Website giving credit to
Benoît Paul Émile Clapeyron, (1799-1864) in 1834 ]
* [http://www.stthomas.edu/biol/sciencebear2/respiratory/respiratorycalc/idealgasequation.html Website containing Ideal Gas Law Calculator & a host of other scientific calculators, Rex Njoku & Dr. Anthony Steyermark -University of St.Thomas]
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ideal gas law — idealiųjų dujų dėsnis statusas T sritis fizika atitikmenys: angl. ideal gas law; perfect gas law vok. ideales Gasgesetz, n rus. закон идеального газа, m pranc. loi des gaz idéaux, f … Fizikos terminų žodynas
ideal-gas law — noun : gas law c … Useful english dictionary
ideal gas law — Physics. the law that the product of the pressure and the volume of one gram molecule of an ideal gas is equal to the product of the absolute temperature of the gas and the universal gas constant. Also called gas law. * * * … Universalium
ideal gas law — Physics. the law that the product of the pressure and the volume of one gram molecule of an ideal gas is equal to the product of the absolute temperature of the gas and the universal gas constant. Also called gas law … Useful english dictionary
ideal gas law — the equation of state for an ideal gas: PV = nRT, where P is pressure, V volume, n the number of moles of gas, R the gas constant, and T the absolute temperature. It holds approximately for real gases at low pressures and high temperatures … Medical dictionary
ideal gas law — noun the equation of state of an ideal gas PV = nRT … Wiktionary
ideal gas law — noun → gas laws … Australian English dictionary
Ideal gas — Thermodynamics … Wikipedia
gas law — noun : any of several statements of physics and chemistry relating to the behavior of gases: as a. : boyle s law b. : charles s law c. : a statement that the product of the pressure and volume of one mole of a gas equals the product of the gas… … Useful english dictionary
ideal gas — Physics. a gas composed of molecules on which no forces act except upon collision with one another and with the walls of the container in which the gas is enclosed; a gas that obeys the ideal gas law. Also called perfect gas. [1890 95] * * * … Universalium