question_answer 60)

                  If c is r.m.s. speed of molecules in a gas and \[\upsilon \] is the speed of sound waves in the gas, show that the gas, slow \[c/\upsilon \] constant and independent of temperature for all diatomic gases.

Answer:

                  We know r.m.s. speed of a molecule in a gas is given by                 \[c=\sqrt{\frac{3p}{\rho }}\]                 Speed of sound wave in gas is given by                 \[\upsilon =\,\sqrt{\frac{\gamma P}{\rho }}\]                 \[\therefore \] \[\frac{c}{\upsilon }\,=\,\sqrt{\frac{3}{\gamma }}\]                 Now \[\gamma =1+\frac{2}{f}\]. For diatomic gas, \[f=5\]                 \[\therefore \] \[\gamma =\,1+\,\frac{2}{5}=\frac{7}{5}\]                 \[\therefore \] \[\frac{c}{\upsilon }=\,\sqrt{\frac{15}{7}}\,=\,1.46\,(\text{constant})\]