A mass M = 15 g of nitrogen is enclosed in a vessel at temperature T = 300 K. What amount of heat has to be transferred to the gas to increase the root-mean-square velocity of molecules 2 times ?

Explanation

We know rms speed of gas molecules is given as
`v_("rms")=sqrt((3RT)/(M))`
As `v_("rms")` is directly proportional to `sqrt(t)`, to double rms speed, temperature must be raised four times to `T_(f)=1200` K.
As the gas is enclosed in a container, it can not expand or it can not loose any energy to external system. Its initial internal energy is given as
`U_(i)=(f)/(2)nRT=(5)/(2)rRT` [As for a diatomic gas f = 5]
`=(5)/(2)xx(15)/(28)xx8.314xx300`
`=3340.45` joule
Finally when gas temperature is raised to 1200 K, its internal energy is given as
`U_(f)=(5)/(2)nRT_(f)`
`=(5)/(2)xx(15)/(28)xx8.314xx1200`
`13361.6` joule
Increase in internal energy or amount of heat supplied is
`DeltaQ=U_(f)-U_(i)=10021.35` joule