question_answer 30)

                  The molecules of a given mass of a gas have root mean square speeds of \[100\,m\,{{s}^{-1}}\]at \[27{}^\circ C\] and 1.00 atmospheric pressure. What will be the roots means square speeds of the molecules of the gas at \[127{}^\circ C\] and 2.0 atmospheric pressure?

Answer:

                  \[P=\,\frac{1}{2}\,\rho {{c}^{2}}\,=\,\frac{1}{2}\,\frac{M}{V}\,{{c}^{2}}\]                 or \[c=\,\sqrt{\frac{2PV}{M}}\]                 \[\therefore \]\[\sqrt{\frac{{{c}_{2}}}{{{c}_{1}}}}=\sqrt{\frac{{{P}_{x}}{{V}_{2}}}{{{P}_{1}}{{V}_{1}}}}\]              ?. (1)                 Now \[{{P}_{1}}{{V}_{1}}=R{{T}_{1}}\] and \[{{P}_{2}}{{V}_{2}}=R{{T}_{2}}\]                 \[\therefore \] \[\frac{{{P}_{2}}V{{ & }_{2}}}{{{P}_{1}}{{V}_{1}}}\,=\,\frac{{{T}_{2}}}{{{T}_{1}}}\,=\,\frac{400}{300}\,=\,\frac{4}{3}\]                 Hence \[\frac{{{c}_{2}}}{{{c}_{1}}}\,=\frac{2}{\sqrt{3}}\] or \[{{c}_{2}}\,-\frac{2{{c}_{1}}}{\sqrt{3}}\,=\frac{200}{\sqrt{3}}\,m\,{{s}^{-1}}\]