(a) Calculate the total and average kinetic energy of 32 g methane molecules at `27^(@)C (R=8.314 JK^(-1)mol^(-1))`
(b) Also calculate the root mean square speed, average speed and most probable speed of methane molecules at `27^(@)C`.

Explanation

(a) Total K.E.`=(3)/(2)nRT=(3)/(2)xx(32)/(16)molxx8.314" JK"^(-1)xx300 K` (Molar mass of `CH_(4)=16" g mol"^(-1)`)
=7482.6 J
Average K.E., `bar(K.E.)=(3)/(2)kT=(3)/(2)xx(8.314 JK^(-1)mol^(-1))/(6.02x10^(23)"molecules "mol^(-1))xx300 K=6.21xx10^(-21) J molecule^(-1)`
`=(3)/(2)"RT "mol^(-1)=(3)/(2)xx8.314 JK^(-1)mol^(-1)xx300 K =3741.3 J mol^(-1)`
(b) Root mean square speed, `c=sqrt((3RT)/(M))`
Using C.G.S. units, put `R=8.314xx10^(7)" ergs " K^(-1)mol^(-1), T=27+273=300 K`,
M (for `CH_(4)`)=16 g `mol^(-1)`, we get `u_(r.m.s.)=sqrt((3xx8.314xx10^(7)xx300)/(16))=6.839xx10^(4)cm s^(-1)=683.9 m s^(-1)`
`c=sqrt((3xx8.314xx300)/(16xx10^(-3)))=683.9 m s^(-1)`
Similarly, calculate average speed `overset(underline)(c)=sqrt((8 RT)/(pi MT)) " and " c^(**)=sqrt((2RT)/(M))`
or `" " overset(underline)c=0.921xxc=0.921xx683.9 m s^(-1)=629.9 m s^(-1)`
and `" " c^(**)
=0.816xxc=0.816xx683.9 m s^(-1)=558.1 m s^(-1)`.