JEE Main & Advanced Physics Kinetic Theory of Gases Gaseous Mixture

If two non-reactive gases are enclosed in a vessel of volume V. In the mixture \[{{\mu }_{1}}\] moles of one gas are mixed with \[{{\mu }_{2}}\] moles of another gas. If \[{{N}_{A}}\] is Avogadro's number then

Number of molecules of first gas \[{{N}_{1}}={{\mu }_{1}}\,{{N}_{A}}\]

and number of molecules of second gas \[{{N}_{2}}={{\mu }_{2}}{{N}_{A}}\]

(1) Total mole fraction \[\mu =({{\mu }_{1}}+{{\mu }_{2}})\].

(2) If \[{{M}_{1}}\] is the molecular weight of first gas and \[{{M}_{2}}\] that of second gas.

Then molecular weight of mixture \[M=\frac{{{\mu }_{1}}{{M}_{1}}+{{\mu }_{2}}{{M}_{2}}}{{{\mu }_{1}}+{{\mu }_{2}}}\]

(3) Specific heat of the mixture at constant volume will be  

\[{{C}_{{{V}_{mix}}}}=\frac{{{\mu }_{1}}{{C}_{{{V}_{1}}}}+{{\mu }_{2}}{{C}_{{{V}_{2}}}}}{{{\mu }_{1}}+{{\mu }_{2}}}\]\[=\frac{\frac{{{m}_{1}}}{{{M}_{1}}}{{C}_{{{V}_{1}}}}+\frac{{{m}_{2}}}{{{M}_{2}}}{{C}_{{{V}_{2}}}}}{\frac{{{m}_{1}}}{{{M}_{1}}}+\frac{{{m}_{2}}}{{{M}_{2}}}}\]

(4) Specific heat of the mixture at constant pressure will be

\[{{C}_{{{P}_{mix}}}}=\frac{{{\mu }_{1}}{{C}_{{{P}_{1}}}}+{{\mu }_{2}}{{C}_{{{P}_{2}}}}}{{{\mu }_{1}}+{{\mu }_{2}}}\]\[=\frac{{{\mu }_{1}}\left( \frac{{{\gamma }_{1}}}{{{\gamma }_{1}}-1} \right)R+{{\mu }_{2}}\left( \frac{{{\gamma }_{2}}}{{{\gamma }_{2}}-1} \right)R}{{{\mu }_{1}}+{{\mu }_{2}}}\]

\[=\frac{R}{{{\mu }_{1}}+{{\mu }_{2}}}\left[ {{\mu }_{1}}\left( \frac{{{\gamma }_{1}}}{{{\gamma }_{1}}-1} \right)+{{\mu }_{2}}\left( \frac{{{\gamma }_{2}}}{{{\gamma }_{2}}-1} \right) \right]\]

\[=\frac{R}{\frac{{{m}_{1}}}{{{M}_{1}}}+\frac{{{m}_{2}}}{{{M}_{2}}}}\left[ \frac{{{m}_{1}}}{{{M}_{1}}}\left( \frac{{{\gamma }_{1}}}{{{\gamma }_{1}}-1} \right)+\frac{{{m}_{2}}}{{{M}_{2}}}\left( \frac{{{\gamma }_{2}}}{{{\gamma }_{2}}-1} \right) \right]\]

(5) \[{{\gamma }_{\text{mixture}}}=\frac{{{C}_{{{P}_{mix}}}}}{{{C}_{{{V}_{mix}}}}}=\frac{\frac{({{\mu }_{1}}{{C}_{{{P}_{1}}}}+{{\mu }_{2}}{{C}_{{{P}_{2}}}})}{{{\mu }_{1}}+{{\mu }_{2}}}}{\frac{({{\mu }_{1}}{{C}_{{{V}_{1}}}}+{{\mu }_{2}}{{C}_{{{V}_{2}}}})}{{{\mu }_{1}}+{{\mu }_{2}}}}\]

\[=\frac{{{\mu }_{1}}{{C}_{{{P}_{1}}}}+{{\mu }_{2}}{{C}_{{{P}_{2}}}}}{{{\mu }_{1}}{{C}_{{{V}_{1}}}}+{{\mu }_{2}}{{C}_{{{V}_{2}}}}}\]\[=\frac{\left\{ {{\mu }_{1}}\left( \frac{{{\gamma }_{1}}}{{{\gamma }_{1}}-1} \right)R+{{\mu }_{2}}\left( \frac{{{\gamma }_{2}}}{{{\gamma }_{2}}-1} \right)R \right\}}{\left\{ {{\mu }_{1}}\left( \frac{R}{{{\gamma }_{1}}-1} \right)+{{\mu }_{2}}\left( \frac{R}{{{\gamma }_{2}}-1} \right) \right\}}\]

\[\therefore \]\[{{\gamma }_{\text{mixture}}}=\frac{\frac{{{\mu }_{1}}{{\gamma }_{1}}}{{{\gamma }_{1}}-1}+\frac{{{\mu }_{2}}{{\gamma }_{2}}}{{{\gamma }_{2}}-1}}{\frac{{{\mu }_{1}}}{{{\gamma }_{1}}-1}+\frac{{{\mu }_{2}}}{{{\gamma }_{2}}-1}}=\frac{{{\mu }_{1}}{{\gamma }_{1}}({{\gamma }_{2}}-1)+{{\mu }_{2}}{{\gamma }_{2}}({{\gamma }_{1}}-1)}{{{\mu }_{1}}({{\gamma }_{2}}-1)+{{\mu }_{2}}({{\gamma }_{1}}-1)}\]