Quantum Electron Energy Level for Hydrogenic atom

In summary, the maximum wavelength of light that would completely ionize a doubly ionized Lithium (Li++) atom in its ground state (n=1) is 10.1nm. This is because the energy of a Li++ atom is -13.606eV*(3)/n^2 = 40.818eV, due to its 3 times the mass compared to a Hydrogen atom. However, the difference in mass is not the only factor, as the energy formula takes into account the mass of the electron and the charge of the nucleus and electron. Taking this into consideration, the correct answer is 10.1nm, not 30.38nm as initially calculated.
  • #1
JewishDude
2
0
1. Consider doubly ionized Lithium (Li++), which has one electron orbiting a +3 charge nucleus. Assuming the electron is in the ground state (n=1), what is the maximum wavelength of light, λ, that would completely ionize the Li++? (free the electron from the nucleus),
all variables and given/known data

2. Energy of a Hydrogen atom is [itex]\frac{-13.606eV}{n^{2}}[/itex] where 'n' is the energy level. The general formula is:
E = [itex]\frac{-1}{4\pi r^{2}}[/itex][itex]\frac{me^{4}}{2\bar{h}n^{2}}[/itex]
for atoms with one electron.

For light:
E = [itex]\frac{1240eV-nm}{λ}[/itex]

3. I assumed that Li++ would be like a hydrogen atom with 3 times the mass. Since E is proportional to m, the energy of Li should be -13.606eV*(3)/n^2 = 40.818eV. (n=1) I plugged this into the equation λ = 1240eV-nm/E, and got λ = 30.38nm.
The correct answer is 10.1nm... so I am off by a factor of three. So does this mean that the doubly ionized lithium actually has 9 times the mass?

Thank you!
 
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  • #2
Mass is not the only difference between H and Li.
 
  • #3
In the energy formula, does m stand for the mass of the atom or the mass of the electron?
 
  • #4
Hi guys,
I figured it out. As it turns out m is mass of the electron, and e^4 is actually the charge of the nucleus squared times the charge of the electron squared. Which happens to be e^4 in an H atom. In this case it was e^2*(3e)^2 That's where the factor of 9 comes from! Thanks
 
  • #5


I would like to clarify a few things about the content and provide a response to the given question.

Firstly, the energy levels of a hydrogen atom and a hydrogenic atom (such as Li++) are different. The energy levels of a hydrogenic atom are given by the equation E = -13.606eV/n^2Z^2, where Z is the atomic number and n is the energy level. In the case of Li++, Z=3, so the energy level would be -13.606eV/1^2(3)^2 = -1.5118eV.

Secondly, the formula E = -1/4πr^2(me^4/2ħ^2)n^2 is for the energy of a single electron in a hydrogenic atom, not for the entire atom. So, it cannot be used to calculate the energy of Li++.

Thirdly, the energy of a photon of light is given by the equation E = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength of light. The equation E = 1240eV-nm/λ is not applicable in this case.

Now, coming to the given question, the maximum wavelength of light that would completely ionize Li++ (remove the electron from the nucleus) can be calculated using the energy difference between the ground state (n=1) and the ionization energy of Li++. The ionization energy of Li++ is given by the difference in energy between the ground state (n=1) and the first excited state (n=2), which is -1.5118eV - (-0.37545eV) = -1.13635eV.

Using the equation E = hc/λ, we can calculate the maximum wavelength of light as λ = hc/E = (6.626x10^-34 Js)(3x10^8 m/s)/1.13635eV = 5.507x10^-7 m = 550.7 nm.

In conclusion, the maximum wavelength of light that would completely ionize Li++ is 550.7 nm. The discrepancy in the given answer of 10.1 nm is due to incorrect use of equations and not considering the difference in energy levels between a hydrogen atom and a hydrogenic atom.
 

Related to Quantum Electron Energy Level for Hydrogenic atom

1. What is a hydrogenic atom?

A hydrogenic atom is an atom that has only one electron and behaves similarly to a hydrogen atom. It is a simplified model used in quantum mechanics to study the energy levels of more complex atoms.

2. How is the energy level of a hydrogenic atom calculated?

The energy level of a hydrogenic atom is calculated using the formula En = -13.6 eV/n2, where n is the principal quantum number. This formula is derived from the Schrödinger equation for a one-electron system.

3. What is the significance of the principal quantum number in hydrogenic atoms?

The principal quantum number, denoted by n, represents the energy level of the electron in a hydrogenic atom. It determines the size and energy of the orbital in which the electron is located.

4. How does the energy level of a hydrogenic atom change with increasing principal quantum number?

The energy level of a hydrogenic atom decreases as the principal quantum number increases. This means that electrons in higher energy levels are further from the nucleus and have more energy.

5. What is the relationship between energy levels and spectral lines in hydrogenic atoms?

The energy levels of a hydrogenic atom determine the energy of the photons emitted or absorbed by the electron during transitions between levels. These energy differences correspond to specific wavelengths of light, which can be observed as spectral lines in the atom's emission or absorption spectrum.

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