Estimating Atom Size using the Uncertainty Principle

Click For Summary
SUMMARY

The discussion focuses on estimating the size of an atom using the Uncertainty Principle, specifically through the relationship between ionization energy and atomic size. The typical energy required to ionize an atom is approximately 5 eV, leading to a calculated atomic size of about 14 nm using the formula Δx ≥ h/(4πΔp). The conversation highlights the confusion regarding the distinction between atomic and nuclear properties, emphasizing that the question pertains to an atom, not a nucleus, and clarifies that most atoms are larger than hydrogen. The participants stress the importance of understanding the relationship between ionization energy and the mass-energy equivalence in this context.

PREREQUISITES
  • Understanding of the Uncertainty Principle (ΔxΔp ≥ h/4π)
  • Familiarity with ionization energy concepts (5 eV for typical atoms)
  • Basic knowledge of mass-energy equivalence (E=mc²)
  • Ability to manipulate momentum equations (E = p²/2m)
NEXT STEPS
  • Research the implications of the Uncertainty Principle on atomic structure.
  • Study the relationship between ionization energy and atomic size in various elements.
  • Explore the concept of mass-energy equivalence in quantum mechanics.
  • Learn about the differences between atomic and nuclear properties in physics.
USEFUL FOR

Students of physics, particularly those studying quantum mechanics, atomic theory, and anyone interested in the fundamental principles governing atomic structure and behavior.

Quantum_man
Messages
8
Reaction score
0
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.
 
Physics news on Phys.org
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.
 
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..
 
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.
 
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?
 
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.
 
Yes there is - but I cannot help you if you won't answer questions.
 

Similar threads

Converting Energy to Mass: Understanding the Relationship Between MeV and kg
  • · Replies 8 ·
Replies
8
Views
5K
Uncertainty in Wavelength Calculation from Wave Crests on a Pier
  • · Replies 6 ·
Replies
6
Views
2K
Estimate the uncertainty in electron's position.
  • · Replies 4 ·
Replies
4
Views
2K
Heisenburg Uncertainty Principle - Seems like an easy question?
  • · Replies 3 ·
Replies
3
Views
2K
Uncertainty Principle and single-slit diffraction
  • · Replies 8 ·
Replies
8
Views
6K
Uncertainty principle for electron in nucleus
  • · Replies 2 ·
Replies
2
Views
3K
"Uncertainity Product for Particle in 1-D Box
  • · Replies 5 ·
Replies
5
Views
2K
Use Uncertainty principle to find gound state energy?
  • · Replies 1 ·
Replies
1
Views
5K
Heisenberg's Uncertainty Principle
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Is this a valid derivation of the Uncertainty Principle?
  • · Replies 3 ·
Replies
3
Views
2K