IDEAL GAS LAW

The 'ideal gas law' is the equation of state of a hypothetical ideal gas, first stated by Benoît 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 [Pa],
:$V$ is the volume [m³] of the vessel containing $n,$ moles of gas,
:$n$ is the amount of substance of gas [mol],
:$R$ is the gas constant [8.314 472 m³·Pa·K⁻¹·mol⁻¹],
:$T$ is the temperature in kelvin [K].
The ideal gas constant (''R'') depends on the units used in the formula. The value given above, 8.314472, is for the SI units of pascal cubic meters per mole per kelvin, which is equal to joule per mole per kelvin (J mol^-1 K^-1). Another value for ''R'' is 0.082057 L·atm·mol⁻¹·K⁻¹)
"R" has a different value for each different unit of pressure used. The values are...
R = 8.314472 (pascals/kPa)
R = .0821 (atms)
R = 62.4 (torr/mmHg)
R = 1.2 (psi)
The ideal gas law is the most accurate for monoatomic gases at high temperatures and low pressures. This follows because the law neglects the size of the gas molecules and the intermolecular attractions. Obviously 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 3''kT''/2, i.e., with increasing temperatures. The more accurate Van der Waals equation takes into consideration
molecular size and attraction. The ideal gas law mathematically follows from statistical mechanics of primitive identical particles (point particles without internal structure) which do not interact, but exchange momentum (and hence kinetic energy) in elastic collisions.

Contents

Alternative forms

Proof

Empirical

Theoretical

Derivation from the statistical mechanics

See also

References

Alternative forms

Considering that 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 ($,\; M$):
:
Then, replacing $,\; n$ gives:
:
from where
: $p\; =$
ho rac{R}{M}T .
This form of the ideal gas law is particularly useful because it links pressure, density $$
ho = m/V, and temperature in a unique formula independent from the quantity of the considered gas.
In statistical mechanics the following molecular equation is derived from first principles:
: $pV\; =\; NkT\; .$
Here $,k$ is Boltzmann's constant, and $,N$ is the ''actual number'' of molecules, in contrast to the other formulation, which uses $,n$, the number of moles. This relation implies that $N,k\; =\; nR$, and the consistency of this result with experiment is a good check on the principles of statistical mechanics.
From here we can notice that for an average particle mass of $mu$ times the
atomic mass constant $m\_mathrm\{u\}$ (i.e., the mass is $mu$ u)
:
and since $$
ho = m/V , we find that the ideal gas law can be re-written as:
:
ho T .

Proof

Empirical

The ideal gas law can be proved using Boyle's law, Charles's law, and Gay-Lussac's law.
Consider one mole of gas. Let its initial state be defined as:
: volume = $v\_0$
: pressure = $p\_0$
: temperature = $t\_0$
If this gas now undergoes an isobaric process, its state will change:
:volume:
:pressure $p\text{'}\; =\; p\_0\; ,$
:temperature .
If it then undergoes an isothermal process:
: $pv\; =\; p\_0v\text{'}\; ,$
where
:p = final pressure
:v = final volume
:T = final temperature (= t')
So:
: ;
where
:, termed $R$, is the universal gas constant.
Using this notation we get:
: $pv\; =\; RT\; ,$
And multiplying both sides of the equation by ''n'' (numbers of moles):
: $pnv\; =\; nRT\; ,$
Using the symbol $,V$ as a shorthand for $,nv$ (volume of $,n$ moles) we get:
: $pV\; =\; nRT\; ,$

Theoretical

The ideal gas law can also be derived from first principles using 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 monatomic 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} rac{dp_{x}}{dt} Bigr
angle +
Bigllangle q_{y} rac{dp_{y}}{dt} Bigr
angle +
Bigllangle q_{z} rac{dp_{z}}{dt} Bigr
angle\
&=-Bigllangle q_{x} rac{partial H}{partial q_x} Bigr
angle -
Bigllangle q_{y} rac{partial H}{partial q_y} Bigr
angle -
Bigllangle q_{z} rac{partial H}{partial q_z} Bigr
angle = -3k_{B} T,
end{align}

where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition 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 the divergence of the position vector 'q' is
:$$
oldsymbol
abla cdot mathbf{q} =
rac{partial q_{x}}{partial q_{x}} +
rac{partial q_{y}}{partial q_{y}} +
rac{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_Ak_B'' is the gas constant.

See also

★ Ideal gas

★ Equation of state

★ Thermodynamics

★ Van der Waals equation

★ Boyle's law

★ Charles's law

★ Dalton's law

★ Amagat's law

References

★ Davis and Masten ''Principles of Environmental Engineering and Science'', McGraw-Hill Companies, Inc. New York (2002) ISBN 0-07-235053-9

★ Website giving credit to Benoît Paul Émile Clapeyron, (1799-1864) in 1834

★ Website containing Ideal Gas Law Calculator & a host of other scientific calculators, Rex Njoku & Dr. Anthony Steyermark -University of St.Thomas