What is the approximate size of a Helium atom at a temperature of 10^(-9)K?

[mmh37]

Can anyone help me to find an approximation to the size of a He atom at very low temperature (10^(-9)K)?

My attempt:

At this very low temperature all electrons will be in the ground state, thus using the equipartition theorem:

$$1/2kT = h*c/lambda$$

now, for the zero point energy, lambda is 1/2 the diameter of the Helium atom.
However, this approach does not yield the correct answer.

I have had a look at several textbooks, but couldn't find any hints in there.
Thanks a lot for your help!

 

[Andrew Mason]

Can anyone help me to find an approximation to the size of a He atom at very low temperature (10^(-9)K)?

My attempt:

At this very low temperature all electrons will be in the ground state, thus using the equipartition theorem:

$$1/2kT = h*c/lambda$$

now, for the zero point energy, lambda is 1/2 the diameter of the Helium atom.
However, this approach does not yield the correct answer.

I have had a look at several textbooks, but couldn't find any hints in there.
Thanks a lot for your help!

Temperature, being a statistical concept, is not defined for an atom. I don't see how temperature would affect the size of any atom.

But, assuming this question is asking what the uncertainty of a He atom's position would be if it had a kinetic energy in the range of He atoms in He gas at 10^-9 K, you would have to apply the Heisenberg uncertainty principle:

$$\Delta x \Delta p = \hbar/2$$

You would have to work out [itex]\Delta p[/tex] from the energy range - which is roughly 0 < E < kT. Using $E = p^2/2m$ would give $\Delta p = \sqrt{2mE} = \sqrt{2mkT}$<br /> <br /> So<br /> <br /> $$\Delta x = \frac{\hbar}{\sqrt{8mkT}}$$AM[/itex]

 

[Gokul43201]

I believe the question may be referring to the de Broglie wavelength (this might be the preamble to a discussion of BECs), which is of the same order as the number calculated by AM.

 

[mmh37]

it is indeed, as the question then asks to compare this with BEC.

However, the solution manual says that d = 60 um, but with the uncertainty principle I got 2.45 *10^(-7)m ?

 

[Gokul43201]

I get a number that's pretty close to 60um (I get p ~ 10^{-29} Ns). You must have made a numerical error. If you still can't find the error, post your calculation here, and someone will point it out.