How Much Energy is Released When a Uranium Nucleus Captures an Electron?

In summary, during a violent nuclear event, a uranium nucleus is stripped of all 92 electrons and then captures a single free electron from its surroundings. Given the ionization energy of hydrogen, the maximum energy of the photon emitted during this process is approximately 115,000 eV. This calculation takes into account the difference in mass and charge between uranium and hydrogen atoms.
  • #1
docnet
Gold Member
747
409
Homework Statement
What is the ionization energy for an electron of a 1-electron uranium atom?
Relevant Equations
##\frac{1}{\lambda}=R_H(\frac{1}{n_f^2}-\frac{1}{n_i^2})##
A neutral uranium atom has 92 electrons and 92 protons. in a violent nuuclear event a uranium nucleus is stripped of all 92 electrons. The resulting bare nucleus captures a single free electron from the surroundings. Given that the ionization energy for hydrogen is ##13.6eV##, derive the approximate value for the maximum energy of the photon that can be given off as the nucleus captures its first electron.

Solution:

The energy of the photon is inversely proportional to ##\lambda##

and the wavelength ##\lambda## is related to the mass of the electron and the energy levels of the atom by the equation

##\frac{1}{\lambda}=R_H(\frac{1}{n_f^2}-\frac{1}{n_i^2})##

where ##R_H=\frac{2\pi ^2 me^4}{h^3 c}##

Our mass of the uranium is be approximately 92 times larger than that of the hydrogen atom. This means the ionization energy for the hydrogen, or the energy of the photon given off by the captured electron, is given by

##92\times13.6eV=1251.2eV##
 
  • Like
Likes Delta2
Physics news on Phys.org
  • #3
haruspex said:
Venturing well outside my knowledge here, but at https://en.wikipedia.org/wiki/Energ...y_level:_atom/ion_with_nucleus_+_one_electron I see a formula with Z2, not Z. Is that a worry?

Yes you are right. ##Z^2## definitely changes our solution.

using the formula from wiki for a hydrogen-like atom with atomic number Z,

##\frac{1}{\lambda} =\frac{m_{electron}e^4}{8ε_0^2 h^3 c}Z^2(\frac{1}{n^2_f}-\frac{1}{n^2_f})##

##E= \frac{hc}{\lambda} = \frac{m_{electron}e^4}{8ε_0^2 h^2}(92)^2(1-\frac{1}{∞})\approx 115,000eV##

which is confirmed here
https://www.omnicalculator.com/physics/hydrogen-like-atom (thank you).
 
  • Like
Likes sysprog
  • #4
docnet said:
Our mass of the uranium is be approximately 92 times larger than that of the hydrogen atom.
That statement is wrong for mass as U has lots of neutrons as well as protons. Compare the U and H atomic weights of ~238 and ~1.

It is correct for charge when you remove approximately as the U nucleus has 92 protons whereas the H nucleus has only 1 proton.

Mass does not come into your calculation as gravity is so weak in comparison with electric forces. Your calculation depends on charge.
 
Last edited:
  • Like
Likes docnet

Related to How Much Energy is Released When a Uranium Nucleus Captures an Electron?

1. What is the Bohr theory problem?

The Bohr theory problem refers to the limitations and inconsistencies of Niels Bohr's atomic model, proposed in 1913. It was unable to explain certain experimental observations, such as the emission spectrum of multi-electron atoms.

2. What are the main features of the Bohr theory?

The main features of the Bohr theory include the concept of quantized energy levels in an atom, the idea that electrons orbit the nucleus in fixed circular orbits, and the postulate that electrons can only transition between energy levels by emitting or absorbing a specific amount of energy.

3. How did the Bohr theory contribute to our understanding of atomic structure?

The Bohr theory was the first successful attempt at explaining the structure of atoms, and it laid the foundation for further developments in quantum mechanics. It also introduced the concept of quantized energy levels, which is still a fundamental principle in modern atomic theory.

4. What are the limitations of the Bohr theory?

The Bohr theory was unable to explain the fine structure of spectral lines, the behavior of multi-electron atoms, and the wave-like properties of particles. It also did not account for the concept of electron spin, which is essential in understanding the behavior of atoms.

5. How was the Bohr theory eventually replaced?

The Bohr theory was eventually replaced by the more comprehensive and accurate quantum mechanical model of the atom, developed by Erwin Schrödinger and Werner Heisenberg in the 1920s. This model takes into account the wave-like properties of particles and accurately describes the behavior of multi-electron atoms.

Similar threads

Replies
13
Views
2K
  • Introductory Physics Homework Help
Replies
9
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
4
Views
3K
Replies
9
Views
2K
  • Introductory Physics Homework Help
Replies
22
Views
7K
  • Introductory Physics Homework Help
Replies
4
Views
1K
Replies
6
Views
3K
  • High Energy, Nuclear, Particle Physics
Replies
15
Views
3K
  • Sticky
  • Atomic and Condensed Matter
Replies
2
Views
7K
Back
Top