- Ideal gas law
The

**ideal gas law**is theequation of state of a hypotheticalideal gas , first stated byBenoît Paul Émile Clapeyron in 1834.:"The state of an amount of

gas is determined by its pressure, volume, and temperature according to the equation:":$PV\; =\; nRT$where :$P$ is the absolute

pressure of the gas, :$V$ is thevolume of the gas,:$n$ is the number of moles of gas,:$R$ is theuniversal gas constant ,:$T$ is theabsolute 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 hencekinetic energy ) inelastic collision s.Since it neglects both molecular size and intermolecular attractions, the ideal gas law is most accurate for

monoatomic gases 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 thevan der Waals equation ,allow deviations from ideality caused by molecular size and intermolecular forces to be taken into account.**Alternative forms**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 ($n,$) is equal to the mass ($,\; m$) divided by the

molar mass ($,\; M$):: $n\; =\; \{frac\{m\}\{M$By replacing $,\; n$, we get:: $PV\; =\; frac\{m\}\{M\}RT$from where: $P\; =\; ho\; frac\{R\}\{M\}T$.This form of the ideal gas law is particularly useful because it links pressure, density $ho\; =\; m/V$, and temperature in a unique formula independent from the quantity of the considered gas. Instatistical mechanics the following molecular equation is derived from first principles:: $PV\; =\; NkT\; .$Here $,k$ isBoltzmann's constant , and $,N$ is the "actual number" of molecules, in contrast to the other formulation, which uses $,n$, the number of moles. This relation implies that $N,k\; =\; nR$, 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 $mu$ times the

atomic mass constant $m\_mathrm\{u\}$ (i.e., the mass is $mu$ u):$N\; =\; frac\{m\}\{mu\; m\_mathrm\{u$and since $ho\; =\; m/V$, we find that the ideal gas law can be re-written as::$P\; =\; frac\{1\}\{V\}frac\{m\}\{mu\; m\_mathrm\{u\; kT\; =\; frac\{k\}\{mu\; m\_mathrm\{u\; ho\; T\; .$**Calculations**Note_label|A|a|none

**a.**In an isentropic process, system entropy (Q) is constant. Under these conditions, P_{1}V_{1}^{$gamma$}= P_{2}V_{2}^{$gamma$}, where $gamma$ is defined as theheat capacity ratio , which is constant for an ideal gas.**Derivations****Empirical**The ideal gas law can be derived from combining two empirical

gas laws : thecombined gas law andAvogadro's law . The combined gas law states that:$frac\; \{pV\}\{T\}=\; C$

where "C" is a constant which is directly proportional to the amount of gas, "n" (

Avogadro's law ). The proportionality factor is theuniversal gas constant , "R", i.e. $C=nR$.Hence the ideal gas law: $pV\; =\; nRT\; ,$

**Theoretical**The ideal gas law can also be derived from

first principles using thekinetic 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**= ("q_{x}", "q_{y}", "q_{z}") and**p**= ("p_{x}", "p_{y}", "p_{z}") 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:$egin\{align\}langle\; mathbf\{q\}\; cdot\; mathbf\{F\}\; angle\; =\; Bigllangle\; q\_\{x\}\; frac\{dp\_\{x\{dt\}\; Bigr\; angle\; +\; Bigllangle\; q\_\{y\}\; frac\{dp\_\{y\{dt\}\; Bigr\; angle\; +\; Bigllangle\; q\_\{z\}\; frac\{dp\_\{z\{dt\}\; Bigr\; angle\backslash =-Bigllangle\; q\_\{x\}\; frac\{partial\; H\}\{partial\; q\_x\}\; Bigr\; angle\; -Bigllangle\; q\_\{y\}\; frac\{partial\; H\}\{partial\; q\_y\}\; Bigr\; angle\; -\; Bigllangle\; q\_\{z\}\; frac\{partial\; H\}\{partial\; q\_z\}\; Bigr\; angle\; =\; -3k\_\{B\}\; T,end\{align\}$where the first equality isNewton's second law , and the second line usesHamilton's equations and theequipartition theorem . Summing over a system of "N" particles yields:$3Nk\_\{B\}\; T\; =\; -\; iggllangle\; sum\_\{k=1\}^\{N\}\; mathbf\{q\}\_\{k\}\; cdot\; mathbf\{F\}\_\{k\}\; iggr\; angle.$

By

Newton's third law and 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:$-iggllanglesum\_\{k=1\}^\{N\}\; mathbf\{q\}\_\{k\}\; cdot\; mathbf\{F\}\_\{k\}iggr\; angle\; =\; P\; oint\_\{mathrm\{surface\; mathbf\{q\}\; cdot\; mathbf\{dS\},$

where

**dS**is the infinitesimal area element along the walls of the container. Since thedivergence of the position vector**q**is:$oldsymbol\; abla\; cdot\; mathbf\{q\}\; =frac\{partial\; q\_\{x\{partial\; q\_\{x\; +\; frac\{partial\; q\_\{y\{partial\; q\_\{y\; +\; frac\{partial\; q\_\{z\{partial\; q\_\{z\; =\; 3,$

the

divergence theorem implies that:$P\; oint\_\{mathrm\{surface\; mathbf\{q\}\; cdot\; mathbf\{dS\}\; =\; P\; int\_\{mathrm\{volume\; left(\; oldsymbol\; abla\; cdot\; mathbf\{q\}\; ight)\; dV\; =\; 3PV,$

where "dV" is an infinitesimal volume within the container and "V" is the total volume of the container.

Putting these equalities together yields

:$3Nk\_\{B\}\; T\; =\; -iggllangle\; sum\_\{k=1\}^\{N\}\; mathbf\{q\}\_\{k\}\; cdot\; mathbf\{F\}\_\{k\}\; iggr\; angle\; =\; 3PV,$

which immediately implies the

ideal gas law for "N" particles::$PV\; =\; Nk\_\{B\}\; T\; =\; nRT,,$

where "n=N/N

_{A}" is the number of moles of gas and "R=N_{A}k_{B}" is thegas constant .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 theHelmholtz free energy and the partition function, but without using the equipartition theorem, is provided.**ee also***

Combined gas law

*Ideal gas

*Equation of state

*Van der Waals equation

*Boltzmann's constant

*Configuration integral **References**

*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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