# JEE Main & Advanced Chemistry States of Matter Ideal Gas Equation

## Ideal Gas Equation

Category : JEE Main & Advanced

(1) The simple gas laws relating gas volume to pressure, temperature and amount of gas, respectively, are stated below :

Boyle's law :        $P\propto \frac{1}{V}$ or $V\propto \frac{1}{P}$            (n and T constant)

Charle's law :      $V\propto \text{T}$                 (n and P constant)

Avogadro's law : $V\propto n$                           (T and P constant)

If all the above law's combines, then

$V\propto \frac{nT}{P}$

or               $V=\frac{nRT}{P}$ ($R=$ Ideal gas constant)

or            $PV=nRT$

This is called ideal gas equation. R is called ideal gas constant. This equation is obeyed by isothermal and adiabatic processes.

(2) Nature and values of R : From the ideal gas equation, $R=\frac{PV}{nT}=\frac{\text{Pressure}\times \text{Volume}}{\text{mole}\times \text{Temperature}}$

$=\frac{\frac{\text{Force}}{\text{Area}}\times \text{Volume}}{\text{mole}\times \text{Temperature}}=\frac{\text{Force}\times \text{Length}}{\text{mole}\times \text{Temperature}}$$=\frac{\text{Work or energy}}{\text{mole}\times \text{Temperature}}$.

R is expressed in the unit of work or energy $mo{{l}^{-1}}\,{{K}^{-1}}$.

Since different values of R are summarised below :

$R=0.0821\,L\,atm\,mo{{l}^{-1}}\,{{K}^{-1}}$

$=8.3143\,joule\,mo{{l}^{-1}}\,{{K}^{-1}}$  (S.I. unit)

$=8.3143\,Nm\,mo{{l}^{-1}}\,{{K}^{-1}}$

$=8.3143\,KPa\,d{{m}^{3}}\,mo{{l}^{-1}}\,{{K}^{-1}}$

$=8.3143\,MPa\,c{{m}^{3}}\,mo{{l}^{-1}}\,{{K}^{-1}}$

$=5.189\times {{10}^{19}}\,eV\,mo{{l}^{-1}}\,{{K}^{-1}}$

$=1.99\,cal\,mo{{l}^{-1}}\,{{K}^{-1}}$

(3) Gas constant, R for a single molecule is called Boltzmann constant (k)

$k=\frac{R}{N}=\frac{8.314\times {{10}^{7}}}{6.023\times {{10}^{23}}}ergs\,mol{{e}^{-1}}\,degre{{e}^{-1}}$

$=1.38\times {{10}^{-16}}ergs\,mo{{l}^{-1}}\,degre{{e}^{-1}}$

or $1.38\times {{10}^{-23}}\,joule\,mo{{l}^{-1}}\,degre{{e}^{-1}}$

(4) Calculation of mass, molecular weight and density of the gas by gas equation

$PV=nRT=\frac{m}{M}RT$        $\left( \because n=\frac{\text{mass of the gas (}m\text{)}}{\text{Molecular weight of the gas (}M\text{)}} \right)$

$\therefore$  $M=\frac{mRT}{PV}$

$d=\frac{PM}{RT}$                            $\left( \because d=\frac{m}{V} \right)$

or        $\frac{dT}{P}=\frac{M}{R}$, $\frac{M}{R}=$ Constant

($\because$ M and R are constant for a particular gas)

Thus, $\frac{dT}{P}$ or $\frac{{{d}_{1}}{{T}_{1}}}{{{P}_{1}}}=\frac{{{d}_{2}}{{T}_{2}}}{{{T}_{2}}}$= Constant

(For two or more different temperature and pressure)

(5) Gas densities differ from those of solids and liquids as,

(i)        Gas densities are generally stated in g/L instead of $g/c{{m}^{3}}$.

(ii)       Gas densities are strongly dependent on pressure and temperature as, $d\propto P$$\propto 1/T$

Densities of liquids and solids, do depend somewhat on temperature, but they are far less dependent on pressure.

(iii) The density of a gas is directly proportional to its molar mass. No simple relationship exists between the density and molar mass for liquid and solids.

(iv) Density of a gas at STP $=\frac{\text{molar mass}}{22.4}$

$d({{N}_{2}})$ at STP$=\frac{28}{22.4}=1.25\,g\,{{L}^{-1}}$,

$d({{O}_{2}})$ at STP $=\frac{32}{22.4}=1.43\,g\,{{L}^{-1}}$

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