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Question

What is the ratio of the number of molecules having speeds in the range of 2ump and 2ump+du to the number of molecules having speeds in the range of ump and ump+du?
(Answer should be closest integer to the ratio)

Solution
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Let dN1 represents the number of molecules in the speed range ump to ump+du .
dN2 represents the number of molecules in the speed range 2ump to 2ump
According to maxwell distribution,
dN1N=4π(M2πRT)1.5u2mpexp(Mu2mp/2RT)du and

dN2N=4π(M2πRT)1.52u2mpexp(4Mu2mp/2RT)du

Hence, dN2dN1=4exp(4Mu2mp/2RT)exp(Mu2mp/2RT)=4exp(3Mu2mp/2RT)

But, u2mp=2RTM

Hence, dN2dN1=4e3=0.199.
Hence answer is 0.

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