Question

at what temperature will the root mean square velocity of sulphur dioxide be same as that of methane at 27c?

Answer 1

SO2 vs Methane

Methane: symettrical molecule, one degree of freedom (translational)
SO2: linear molecule, two degrees of freedom (translational, vibration).

One of the fundamental postulates in statistical thermodynamics, is that energy <i>almost</i> divides equally in each degree of freedom.

So 1/2 of the energy in SO2 goes to translational energy, and

KE then = 1/2 energy available at a temp, or
energy available= 2*KE
(rmsVelocity)^2 proportional to KE, so
rmsVelocity proportional to sqrt 1/2*KE available
rmsVelocity proportional to sqrt (.5* total energy) or
rmsVelocity proportional to .707 sqrttotal energy

finally, total energy proportional to abs temp.
so rmsVelocity to be the same is .707*sqrt(temp)/sqrt(300)
or temp=2*300K or 600K or 600-273 C

check my thinking. I am a bit surprised at this question, so I assume you are studying statistical thermodynamics.

check my work carefully.