Effect of density on rms speed in ideal gas eqn

[toforfiltum]

Homework Statement

Homework Equations

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

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?

 

[haruspex]

According to the second equation above, density is inversely proportional to root mean square speed at constant pressure

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

 

[toforfiltum]

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

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!

 

[haruspex]

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!

Ok.