Kinetic theory of gases - can someone check this please?

[QueenFisher]

i'm given that p= $$\frac{1}{3}\rho\overline{c^2}$$

without calculation, what happens to rms speed if pressure halved, temperature constant?

firstly, just from rearranging the original equation, you get

$$\overline{c^2}$$ = $$\frac{3p}{\rho}$$

square rooting both sides to get rms speed gives:

rms speed = $$\sqrt{\frac{3p}{\rho}}$$

which rearranges to rms speed = 9 $$\sqrt{\frac{p}{\rho}}$$

then, using original equation, when pressure halved, you get:

$$\overline{c^2}$$ = $$\frac{3p}{2\rho}$$

square rooting both sides to get rms speed gives:

rms speed = $$\sqrt{\frac{3p}{2\rho}}$$

which rearranges to rms speed = $$\frac{9}{4}$$ $$\sqrt{\frac{p}{\rho}}$$

so is it ok to say that the rms speed is quartered?

 

[Astronuc]

What about the density?

What is the relationship between temperature and molecular speed in a gas?

http://hyperphysics.phy-astr.gsu.edu/Hbase/kinetic/kintem.html#c1