1.

For a gas sample with N_(0) number of molecules, function N(v) is given by : N(v)=(dN)/(dv)=((3N_(0))/(v_(0)^(3)))v^(2)" for "0ltvltv_(0)andN(v)=0" for "vgtv_(0), where dN is number of molecules in speed range v to v+dv. Find the rms speed of the molecules.

Answer»

`sqrt((5)/(3))v_(0)`
`sqrt((3)/(5))v_(0)`
`sqrt((5)/(3v_(0)))`
`sqrt((3)/(5v_(0)))`

Solution :`v_(RMS)^(2)=ltv^(2)GT =(v_(1)^(2)+v_(2)^(2)+v_(3)^(2)+....)/(N)=(intv^(2)dN)/(intdN)`
Here `(dN)/(dv)=N(v)`,
So, `v_(rms)^(2)=(1)/(N)underset(0)OVERSET(oo)intN(v)v^(2)dv=(1)/(N)underset(0)overset(v_(0))int((3N)/(v_(0)^(3)).v^(2))v^(2)dv=(3)/(5)v_(0)^(2)`
`implies v_(rms)=sqrt((3)/(5))v_(0)`


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