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at what temperature the r.m.s speed of N_{2} will be tripled and K.E will be 4 times .

${v}_{rms}=\sqrt{\frac{3RT}{M}}andK.E.=\frac{3}{2}RT$

For calculating temperature at which rms speed is tripled , we assume that T

_{1}is the initial temp at which rms speed is v

_{1}and T

_{2}is the temp. at which rms speed is tripled .

$\frac{{v}_{1}}{3{v}_{1}}=\sqrt{\frac{3R{T}_{1}}{M}}\times \sqrt{\frac{M}{3R{T}_{2}}}\phantom{\rule{0ex}{0ex}}\frac{1}{3}=\sqrt{\frac{{T}_{1}}{{T}_{2}}}\phantom{\rule{0ex}{0ex}}{T}_{2}=9{T}_{1}\phantom{\rule{0ex}{0ex}}$

So, the temp. will be 9 times the initial temp. to triple the rms speed.

Now, to increase the K.E. 4 times of initial ; similarly

$\frac{K{E}_{1}}{4K{E}_{1}}=\frac{3R{T}_{1}}{2}\times \frac{2}{3R{T}_{2}}\phantom{\rule{0ex}{0ex}}\frac{1}{4}=\frac{{T}_{1}}{{T}_{2}}\phantom{\rule{0ex}{0ex}}{T}_{2}=4{T}_{1}$

Thus to increase K.E. 4 times temperature should also be the 4 times of initial temperature.

Regards

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