Portion problem including Vrms

[brycenrg]

Homework Statement

The rms speed of molecules in a gas at 20.0C is to be increased by 2 percent. To what temperature must it be raised?

Homework Equations

Vrms = Squareroot((3kt)/m)

The Attempt at a Solution

vrms.02 + vrms = squareroot((3kt)/m)
vrms(1.02) = squareroot((3kt)/m)

Too many unknowns.

In the solution manual it does
(vrms1/vrms2) = 1.02 = (squareroot t1)/(squareroot t2)
t2 1.02^2 = t1 then plug in 20 plus 275 for t2 to get t1

My miss understanding is how does the solution manual know its 1.02 as a portion?
They use proportionality or something. I can't see the logical steps.

Thank you

 

[haruspex]

vrms.02 + vrms = squareroot((3kt)/m)

You should have two equations here, one relating the unincreased vrms to the unraised temperature, and one relating the increased vrms to the new temperature.

 

[NascentOxygen]

vrms(1.02) = squareroot((3kt)/m)

Too many unknowns.

Once you've written an equation for each of the two different cases, in both of these equations the k and m have unchanged values for your gas sample (even though you may not know those values).

 

[rude man]

Equating k.e. to molecular temperature energy we get
mv²/2 = 3kT/2 (kT/2 in each of the three directions)
So we see that v² ∝ T
and so v_RMS ∝ √T.
Of course T is in Kelvin.