question_answer

There are N molecules of a gas in a container. If the number of molecules is increased to 2N, what will be (i) pressure of the gas, (ii) total energy of the gas and (ii) r. m. s. speed of the gas?

Answer:

                \[P=\frac{1}{3}mn\overline{{{\upsilon }^{2}}}=\frac{1}{3}\frac{mN}{V}\overline{{{\upsilon }^{2}}}\] \[\left[ n=\frac{N}{V} \right]\] (i) As \[P\propto N,\]so the pressure of the gas is doubled when the number of molecules is increased from N to2N.  (ii) Average K.E. per molecule, \[\frac{1}{2}m\overline{{{\upsilon }^{2}}}=\frac{3}{2}{{k}_{B}}T\]  Total energy of N molecules \[=\frac{1}{2}mN\overline{{{\upsilon }^{2}}}=\frac{3}{2}{{k}_{B}}NT.\] When the number of molecules is increased from N to 2N, total energy of the gas is doubled, though the average K.E. per molecule remains same. (iii) The r. m. s. speed remains same because it depends only upon temperature.