# The formula pV=1/3Nm(c_rms)^2 in non cuboids

• GeneralOJB
In summary, the conversation discussed the application of the equation pV = \dfrac 1 3 N m \left(c_{rms}\right)^2 in containers of different shapes. While the derivation typically uses a cuboid container, there is evidence that the shape of the container may not matter. A more sophisticated approach involves considering the molecules reaching a small patch of wall in any container shape. This approach is found in J H Jeans' The Kinetic Theory of Gases.
GeneralOJB
Does $pV = \dfrac 1 3 N m \left(c_{rms}\right)^2$ apply in containers that aren't cuboids? The derivation I have seen uses a cuboid container so I'm not sure if or how this can be generalised.

An amusing 'derivation' uses a spherical container. It is very simple because it doesn't involve x, y and z components. Yet the factor of $\frac{1}{3}$ enters in what seems like a quite different way from the way it enters in the cubical box method. So if you take the cubical and the spherical container methods together, you get quite a strong feeling that the shape of the box doesn't matter! But if you really want to be convinced, you need, imo, a more sophisticated approach...

Consider the molecules reaching a small 'patch', area A of wall in any shape of container. The rate at which they bring momentum normal to the wall up to the area A is given by
$$\frac{\Delta p_x}{\Delta t} = \frac{1}{2} A\ m\ \nu\ \overline{u_x^2}$$
Here x is the direction normal to the wall, $\overline{u_x^2}$ is the mean square velocity component normal to the wall and $\nu$ is the number of molecules per unit volume. It's easy to get from here to the formula you quote – without assuming any particular shape of container.

This derivation is much more satisfying than ones assuming particular shapes of container. It is to be found in J H Jeans: The Kinetic Theory of Gases and no doubt in many other texts. I reproduce a version of it in the thumbnails.

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GeneralOJB Are you any clearer?

## 1. What is the formula pV=1/3Nm(c_rms)^2 in non cuboids?

The formula pV=1/3Nm(c_rms)^2 in non cuboids is used to calculate the average kinetic energy of gas molecules in a non-cuboid container. It takes into account the number of molecules (N), the root mean square speed of the molecules (c_rms), and the volume of the container (V).

## 2. How is this formula different from the ideal gas law?

This formula is different from the ideal gas law (PV=nRT) in that it takes into account the shape of the container. The ideal gas law assumes that the container is a cube, while the formula pV=1/3Nm(c_rms)^2 takes into account the varying dimensions of non-cuboid containers.

## 3. Why is this formula important in scientific research?

This formula is important in scientific research because it allows researchers to accurately calculate the average kinetic energy of gas molecules in non-cuboid containers. This can be useful in various fields, such as physics, chemistry, and engineering.

## 4. Can this formula be applied to all types of non-cuboid containers?

Yes, this formula can be applied to all types of non-cuboid containers as long as the dimensions of the container are known and the gas molecules inside are behaving according to the ideal gas law.

## 5. How is this formula derived?

This formula is derived from the kinetic theory of gases, which states that the average kinetic energy of gas molecules is directly proportional to their temperature. By considering the dimensions of the container and the number of molecules, the formula pV=1/3Nm(c_rms)^2 is derived to calculate this average kinetic energy.

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