question_answer 42)

                  Consider an ideal gas with following distribution of speeds.                 Speed (m/s)                       % of molecules                 200                                         10                 400                                         20                 600                                         40                 800                                         20                 1000                                       10                 (i) Calculate \[{{V}_{rms}}\] and hence T. \[(m=3.0\times {{10}^{26}}kg)\]                 (ii)If all the molecules with speed 1000 m/s escape from the system, calculate new \[{{V}_{rms}}\]and hence T.

Answer:

                  (a) \[{{V}_{rms}}=\]             \[=\sqrt{\frac{{{\mu }_{1}}{{({{\upsilon }_{1}})}^{2}}+{{\mu }_{2}}\,{{({{\upsilon }_{2}})}^{2}}+{{\mu }_{3}}{{({{\upsilon }_{3}})}^{2}}+{{\mu }_{4}}{{({{\upsilon }_{4}})}^{4}}+\,{{\mu }_{5}}{{({{\upsilon }_{5}})}^{2}}}{{{\mu }_{1}}+\,{{\mu }_{2}}+\,{{\mu }_{3}}+{{\mu }_{4}}+\,{{\mu }_{5}}}}\]             \[=\sqrt{\frac{10\,{{(200)}^{2}}+\,20\,{{(400)}^{2}}\,+40\,{{(600)}^{2}}+\,20\,{{(800)}^{2}}\,+10{{(1000)}^{2}}}{100}}\]             \[=\sqrt{\frac{4\times \,{{10}^{5}}+32\,\times \,{{10}^{5}}+144\,\times {{10}^{5}}+128\,\times {{10}^{5}}+\,100\times \,{{10}^{5}}}{10}}\]                 \[=\,\sqrt{\frac{408\,\times \,{{10}^{5}}}{10}}\,=-\,638.7\,m{{s}^{-1}}\]                 Now \[\frac{1}{2}\,m\upsilon _{rms}^{2}\,=\frac{3}{2}\,kT\]                 \[\therefore \]\[T\,=\,\frac{m\upsilon _{rms}^{2}}{3k}\,\,=\,\frac{3\times \,{{10}^{-26}}\times \,{{(638.7)}^{2}}}{3\times \,1.38\,\times \,{{10}^{-28}}}\]                 \[=295.5\,K\]                 (b) Since 10% molecules with speed \[1000\text{ }m{{s}^{1}}\] escape, so now the new \[{{\upsilon }_{rms}}\] is given by \[\upsilon {{'}_{rms}}\].             \[=\sqrt{\frac{10\,{{(200)}^{2}}+\,20{{(400)}^{2}}+40\,{{(600)}^{2}}+20\,(800)}{90}}\]                 \[=585\text{ }m\text{ }{{s}^{1}}\]                 and new temperature, \[T\,=\,\frac{m\,{{(\upsilon {{'}_{rms}})}^{2}}}{3k}\]                 \[=\,\frac{3\times \,{{10}^{-26}}\times \,{{(585)}^{2}}}{3\times 1.38\,\times \,{{10}^{-23}}}\]                 \[=247.989\simeq 248\text{ }K.\]