# Is it possible to find the number of gas atoms/molecules in a box when the number is small?

Given very low number of particles in a system (e.g. in the 100s), is there a way to accurately measure the number of particles in the system? Assume temperature, pressure and volume is constant and we are using state of the art pressure and temperature sensors. Is it possible to do it from observing the speed distribution of particles?

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Any reason you can't just take the temperature, pressure, and volume of the gas and use the ideal gas law? If one or more of those quantities are unknown, could you edit the question to say so? – David Z Nov 1 '12 at 17:52

This greatly depends on the volume of the box. While the ideal gas law given by John probably holds, for any macroscopic box (say, $V = 1\textrm{ m}^3$), $n = 100$ is close to a perfect vacuum (in fact, better than any vacuums we can achieve nowadays).

Wikipedia mentions a minimum of $10^{-12} \textrm{ Torr}$ for the cold cathode method of pressure measurement. This is equivalent to $1.3\cdot10^{-12} \textrm{ hPa}$. However, using the ideal gas law for $V = 1\textrm{ m}^3$, $T = 300\textrm{ K}$ and $n = 100$ yields $p = 4\cdot 10^{-21}\textrm{ hPa}$, that is, 9 orders of magnitude less.

I would therefore be very surprised if it were possible to measure the number of particles in this system correctly. However, the result will change if the volume or temperature changes. For example, for $V = 1 \textrm{ mm}^3$, we get $p = 4\cdot 10^{-12}\textrm{ hPa}$, which is at least in the same range as the number given by Wikipedia.

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Thank you Claudius, I was working on something that gave me the answer using statistical means (speed measurement of particles), but it was shot down by the journals, so I wanted to ask the question to see if it is that obvious that it wasn't getting the attention. From your reply it seems that we have still a long way to go when dealing with low n value. I guess i'll post the research work online now and see how things go. Thanks much. – Shahid Nov 1 '12 at 19:18

Yes, the number of moles of (ideal) gas is given by:

$$n = \frac{PV}{RT}$$

Then multiply by $6.023 \times 10^{23}$ to get the number of atoms.

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Thanks for the reply John, apologies my first post wasn't very clear. – Shahid Nov 1 '12 at 19:19