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 :




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








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




(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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