# How fast did hydrogen atoms travel when they were first formed in the early universe?

I can't seem to find any data on this, is it a known value?

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What you will find is information on the temperature of the universe at various times –  dmckee Apr 17 '13 at 16:16
Wikipedia says that hydrogen atoms were formed hundreds of thousands of years after the Big Bang, when it was cool enough for electrons and protons to combine to form atoms. And the temperature at this time (around 3000K), combined with the weight of a hydrogen atom, will tell you how fast the atoms are moving. –  Peter Shor Apr 17 '13 at 16:17
I got around 8000m/s using v=sqrt[3kt/m], is that along the right lines? Thanks. –  Theo Windebank Apr 17 '13 at 16:34

## 1 Answer

Re your comment, the speed is related to the temperature by the Maxwell-Boltzmann distribution:

$$f(v) = \sqrt{\left( \frac{M}{2\pi R T} \right)^3} \space 4 \pi v^2 \space exp \left( -\frac{Mv^2}{2RT} \right)$$

where M is the molar mass i.e. 0.001kg for atomic hydrogen. I did a quick graph of this for $T$ = 3000K and got:

So it looks as if your estimate of 8,000 m/s is pretty close!

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Wikipedia says 4000K, another (maybe more reliable) source says 3000K. The exact temperature depends on what fraction of the protons you require to be in hydrogen atoms. But 8000m/s is a reasonably good estimate. –  Peter Shor Apr 17 '13 at 17:46
That's great, thanks! I don't need it to be hugely accurate so as long as its close that's good. –  Theo Windebank Apr 17 '13 at 18:08