JEE Main & Advanced Physics Kinetic Theory of Gases Equation of State or Ideal Gas Equation

The equation which relates the pressure (P) volume (V) and temperature (T) of the given state of an ideal gas is known as ideal gas equation or equation of state.

For 1 mole of gas \[\frac{PV}{T}=R\] (constant)  \[\Rightarrow \] \[PV=RT\]

where R = universal gas constant.

Different forms of gas equation

(1) Universal gas constant (R) : Universal gas constant signifies the work done by (or on) a gas per mole per kelvin.

\[R=\frac{PV}{\mu T}=\frac{\text{Pressure }\times \text{Volume}}{\mu \times \text{ Temperature}}\]\[=\frac{\text{Work done}}{\mu \text{ }\times \text{ Temperature}}\]

(i) At S.T.P. the value of universal gas constant is same for all gases \[R=8.31\frac{J}{mole\times kelvin}=1.98\frac{cal}{mole\times kelvin}\] \[\tilde{-}\,2\frac{cal}{mol\,\times kelvin}\] \[=0.8221\frac{\,litre\times atm}{mole\times kelvin}\].

(ii) Dimension : \[[M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]\]

(2) Boltzman's constant (k) : It is represented by per mole gas constant i.e., \[k=\frac{R}{N}=\frac{8.31}{6.023\times {{10}^{23}}}\] \[=1.38\times {{10}^{-23}}\ J/K\]

It's dimension : \[[M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]\]

(3) Specific gas constant (r) : It is represented by per gram gas constant i.e., \[r=\frac{R}{M}\]. It's unit is \[\frac{Joule}{gm\times kelvin}\] and dimension \[[{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]\]

Since the value of M is different for different gases. Hence the value of r is different for different gases. e.g. It is maximum for hydrogen \[{{r}_{{{H}_{2}}}}=\frac{R}{2}\]