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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