Estimating Atom Size using the Uncertainty Principle

[Quantum_man]

The typical energy needed to ionise an atom is around 5 eV. Use the Uncertainty principle to estimate the size of an atom.

Homework Equations

E=mc^2

E = p^2/2m

Δx.Δp ≥ h/4π

The Attempt at a Solution

So I got the mass rearranging E = mc^2

m = 5*1.6*10^-19 / (3*10^8)^2

m = 8.88*10^-36 kg

then Δp = √(2mE)

which = 3.77*10^-27 kg m s^-1

Finally

Δx ≥ h/(4*pi*3.77*10^-27)

= 14nm

My only problem is that this nucleus is greater than the hydrogen atom. So shouldn't its ionisation energy be greater than 13.6 eV?

Thanks for anyone that can help.

 

[Simon Bridge]

My only problem is that this nucleus is greater than the hydrogen atom. So shouldn't its ionisation energy be greater than 13.6 eV?

Does the question talk about the ionization energy of a nucleus?
Does it talk about hydrogen specifically?

What is the order-of-magnitude diameter of a "typical" atom?

Can you explain your rationale for each step?
i.e. why would the ionization energy be related to mass energy?

Note: you can simplify your calculations by using non-SI units - nm for length, eV for energy, etc. by choosing the units of the constants appropriately.

 

[Quantum_man]

It was on my physics test yesterday. Doesn't specify any of the things you mentioned. The exact wording of the question is what I've written. Thats all...I was a bit stumped to say the least..

 

[Simon Bridge]

Well - my questions concerned everything after "3. Attempt at a solution".
They were supposed to guide you to the answers you seek.

i.e. the first question was:
Does the question talk about the ionization energy of a nucleus?
This was because you wrote: My only problem is that this nucleus is greater than the hydrogen atom.

The question says: The typical energy needed to ionise an atom is around 5 eV.
So it is not a nucleus, it's an atom. Atoms are bigger than nuclei.
It is not hydrogen, it's "an atom" ... most atoms are bigger than hydrogen.
Now try the others.

 

[Quantum_man]

My rationale for trying to determine the mass of the atom is this:

To my knowledge all the equations I'd use to find the uncertainty in the position or momentum of the electron require the knowledge of at least two variables, whether it be mass, energy, wavelength or velocity. Since I'm given only the ionization energy, I tried to find a way to determine at least one of the other variables.

Is there a way to answer the question with the information given?

 

[Quantum_man]

Also when queried about the wording of the question, my lecturer gave the following response:

"The size of the atom is the uncertainty in the position of the electron, because it could be anywhere within the atom."

I find it hard to understand how the size of the atom is directly related to the uncertainty in the position of the electron. If this makes sense could you please help me understand the principle.

Thanks in advance.

 

[Simon Bridge]

Yes there is - but I cannot help you if you won't answer questions.