Effect of density on rms speed in ideal gas eqn

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1. Apr 2, 2016

toforfiltum

1. The problem statement, all variables and given/known data

2. Relevant equations
1) PV = nRT
2 )$P = ⅓ ρ<c^2>$
3) KE ∝ T

3. The attempt at a solution
According to the second equation above, density is inversely proportional to root mean square speed at constant pressure, but the answer states that the root mean square speed depends only on the temperature so the answer is no effect.

I don't see how when the second equation suggests otherwise. Can someone help me out?

2. Apr 2, 2016

haruspex

Yes, but the question states constant temperature, not constant pressure.

3. Apr 2, 2016

toforfiltum

Oops, how could I misread that. Thanks.
So to prove this, it is $\frac {1} {2} Nm<c^2> = \frac{3} {2} NkT$ ? Which is KE ∝ T?
Since ρ is not in the above equation, it is proven that density has no effect on root mean square speed?

Thanks!

4. Apr 2, 2016

Ok.