A) 1.4 times
B) 4 times
C) 2 times
D) \[\frac{1}{4}\] times
Correct Answer: B
Solution :
Key Idea Ratio of \[{{v}_{av}}/{{v}_{rms}}\] remains constant. Average speed is the arithmetic mean of the speeds of molecules in a gas at a given temperature, ie, \[{{v}_{av}}=({{v}_{1}}+{{v}_{2}}+{{v}_{3}}+.....)/N\] and according to kinetic theory of gases, \[{{v}_{av}}=\sqrt{\frac{8RT}{M\pi }}\] ??(1) Also, rms speed (root mean square speed) is defined as the square root of mean of squares of the speeds of different molecules, ie, \[{{v}_{rms}}=\sqrt{(v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+.....)/N}\] \[=\sqrt{{{(\overline{v})}^{2}}}\] and according to kinetic theory of gases, \[{{v}_{rms}}=\sqrt{\frac{3RT}{M}}\] ??(ii) From Eqs. (i) and (ii), we get \[{{v}_{av}}=\sqrt{\left( \frac{8}{3\pi } \right)}{{v}_{rms}}\] \[=0.92{{v}_{rms}}\] ??..(iii) Therefore, \[\frac{{{v}_{av}}}{{{v}_{rms}}}=\] constant Hence, root mean square velocity also becomes 4 times.
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