# Calculating the root mean square speed from pressure and density.

## Homework Statement

A tyre contains gas at a pressure of 150 kPa. If the gas has a density of 2.0 kg m-3, find the root mean square speed of the molecules.

## Homework Equations

These are the equations I believe to be relevant:

$$c_{rms} = \frac{\sqrt{<c^2>}}{N}$$

$$pV = \frac{1}{3}Nm<c^2>$$

$$p = \frac{1}{3}ρ<c^2>$$

## The Attempt at a Solution

$$\frac{3p}{ρ} = <c^2>$$
$$\frac{3 * 150000}{2} = <c^2> = 225000ms^{-1}$$

But I'm not sure how to work out N as I don't any volume or temperature. I'm not quite sure how they get their answer in the back of the book which is 474 ms-1 (which is the square root of 225000) meaning that N = 1? How can that be?

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SteamKing
Staff Emeritus
Homework Helper
If c^2 = 225000, then the units can't be m/s

• 1 person
If c^2 = 225000, then the units can't be m/s
Sorry yes <c^2> should be in m^2s^-2 shouldn't it?

BruceW
Homework Helper
Jamesey162 said:
$$c_{rms} = \frac{\sqrt{<c^2>}}{N}$$
Are you sure there should be an ##N## here?

Are you sure there should be an ##N## here?
I'm sure, atleast that's how they have quoted it in my text book: ehild
Homework Helper
How does your book define <c^2>?

ehild

How does your book define <c^2>?

ehild
It defines it as:

$$<c^2> = \frac{{c_1}^2 + {c_2}^2 + {c_3}^2 + ....{c_N}^2}{N} = \frac{{c_i}^2}{N}$$

I'm not exactly sure what the right-hand most part means with the subscript i.

ehild
Homework Helper
It defines it as:

$$<c^2> = \frac{{c_1}^2 + {c_2}^2 + {c_3}^2 + ....{c_N}^2}{N} = \frac{{c_i}^2}{N}$$

I'm not exactly sure what the right-hand most part means with the subscript i.
It should be

$$<c^2> = \frac{{c_1}^2 + {c_2}^2 + {c_3}^2 + ....{c_N}^2}{N} =\frac{\sum{{c_i}^2}}{N}$$

i is the summation index, and √<c2> itself is the rms speed. No need to divide it by N. $$\frac{3p}{ρ} = <c^2>$$

ehild

• 1 person
BruceW
Homework Helper
yeah, I think what ehild said is right. The equation
$$c_{rms} = \frac{\sqrt{<c^2>}}{N}$$
is most likely a mistake in the book. It doesn't make sense, if you think about it. If you had a bunch of atoms all moving at the same speed, then this equation would give an rms value that is smaller for a larger population of these identical atoms. Which doesn't make sense at all.

• 1 person
It should be

$$<c^2> = \frac{{c_1}^2 + {c_2}^2 + {c_3}^2 + ....{c_N}^2}{N} =\frac{\sum{{c_i}^2}}{N}$$

i is the summation index, and √<c2> itself is the rms speed. No need to divide it by N. $$\frac{3p}{ρ} = <c^2>$$

ehild
yeah, I think what ehild said is right. The equation
$$c_{rms} = \frac{\sqrt{<c^2>}}{N}$$
is most likely a mistake in the book. It doesn't make sense, if you think about it. If you had a bunch of atoms all moving at the same speed, then this equation would give an rms value that is smaller for a larger population of these identical atoms. Which doesn't make sense at all.

Okay, I've got it now, thanks for the explanation.