Energy of Electrons in Atom

In summary, the ground state of lithium has two electrons in the 1s level and one in the 2s level. In an excited state, the outermost electron is raised to the 3p level. By assuming that the electron is completely outside of the other electrons, the potential energy function felt by the outermost electron can be estimated using the equation U(r) = -Zeff(r)\frac{ke^{2}}{r}, where Zeff \approx 1 for the 3p electron. Using this approximation, the energy of the outer electron can be calculated using the formula \approx -\frac{ke^{2}}{n^{2}a_{B}}, where r is substituted for rmp. This
  • #1
hitmeoff
261
1

Homework Statement


The ground state of lithium (Z=3) has two electrons in the 1s level and one in the 2s. Consider an excited state in which the outermost electron has been raised to the 3p level. Since the 3p wave functions are not very penetrating, you can estimate the energy of this electron by assuming it is completely outside both the other electrons.

a) In this approximation, what is the potential energy function felt by the outermost electron?
b) IN the same approximation write the formula for the energy of the outer electron if its principle quantum number is n.
c) Estimate the energy of the 3p electron in this way and compare to the observed value of -1.556 eV
d) repeat for the case that the outer electron is in the 3d level, whose observed energy is -1.513 eV
e) Explain why the agreement is better for the 3d level than the 3p. Why is the observed energy for 3p lower than for the 3d?

Homework Equations


U(r) = -Zeff(r)[tex]\frac{ke^{2}}{r}[/tex]
Zeff [tex]\approx[/tex] Z [r inside all other electrons]
Zeff [tex]\approx[/tex] 1 [r outside all other electrons]
Most probable radius: rmp [tex]\approx[/tex] [tex]\frac{n^{2}a_{B}}{Z_{eff}}[/tex]

The Attempt at a Solution


For a, I take the first equation U(r) = -Zeff(r)[tex]\frac{ke^{2}}{r}[/tex] and since the 3p electron is so far outside of the other 2 electrons Zeff [tex]\approx[/tex] 1

For b, I substituted r for rmp, so U(r) becomes [tex]\approx[/tex] -[tex]\frac{ke^{2}}{n^{2}a_{B}}[/tex] which got me -3.02 eV

But I am completely stumped on the rest. HOw do I actually calculate the difference in energies from 3p and 3d? I know 3p should have a lower energy than 3d because it feels more of the nucleus but how do I actually work this out given my equations (or any other you may know)?
 
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  • #2
any ideas?
 
  • #3
Id still like ot be able to solve this if anyone is able to help me out?
 

1. What is the energy of an electron in an atom?

The energy of an electron in an atom is determined by its position within the atom's energy levels, also known as orbitals. The closer an electron is to the nucleus, the lower its energy will be. As it moves further away, its energy increases. The specific energy level an electron occupies depends on its quantum number, which is determined by the atom's atomic number.

2. How is the energy of an electron calculated?

The energy of an electron can be calculated using the equation E = -13.6/n^2, where E is the energy, and n is the principal quantum number. This equation is known as the Rydberg formula and is based on the Bohr model of the atom.

3. What is the significance of energy levels in an atom?

The energy levels in an atom determine the behavior and properties of electrons. Without these distinct energy levels, electrons would not be able to occupy specific orbits around the nucleus and would be more likely to collide with it. The energy levels also play a crucial role in the absorption and emission of light by atoms.

4. How do electrons transition between energy levels?

Electrons can transition between energy levels by absorbing or emitting energy in the form of photons. When an electron absorbs a photon, it gains energy and moves to a higher energy level. Similarly, when an electron emits a photon, it loses energy and moves to a lower energy level.

5. Can electrons have any energy level in an atom?

No, electrons can only have specific energy levels in an atom. This is due to the restrictions of the quantum nature of electrons, which can only occupy certain orbits around the nucleus. These energy levels are discrete and cannot be arbitrarily chosen, but rather follow a specific pattern based on the atom's atomic number.

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