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

 

[kyle8921]

I would say that your reasoning and calculations are correct. When pressure is halved and temperature is constant, the rms speed of the gas molecules will indeed decrease by a factor of 1/4. This can be seen from the equations you provided, as well as from the Kinetic Theory of Gases, which states that the rms speed of gas molecules is directly proportional to the square root of temperature and inversely proportional to the square root of pressure. Therefore, when pressure is halved, the rms speed will decrease accordingly.