An atom in a 3 d state emits a photon of wavelength 475.082 nm when it decays to a 2 p state. (a

An atom in a 3 d state emits a photon of wavelength 475.082...

An atom in a $3 d$ state emits a photon of wavelength $475.082 \mathrm{nm}$ when it decays to a $2 p$ state. (a) What is the energy (in electron volts) of the photon emitted in this transition? (b) Use the selection rules described in Section 41.4 to find the allowed transitions if the atom is now in an external magnetic field of $3.200 \mathrm{~T}$. Ignore the effects of the electron's spin. (c) For the case in part (b), if the energy of the $3 d$ state was originally $-8.50000 \mathrm{eV}$ with no magnetic field present, what will be the energies of the states into which it splits in the magnetic field? (d) What are the allowed wavelengths of the light emitted during transition in part (b)?

An atom in a $3 d$ state emits a photon of wavelength $475.082 \mathrm{nm}$ when it decays to a $2 p$ state. (a) What is the energy (in electron volts) of the photon emitted in this transition?
(b) Use the selection rules described in Section 41.4 to find the allowed transitions if the atom is now in an external magnetic field of $3.200 \mathrm{~T}$. Ignore the effects of the electron's spin. (c) For the case in part (b), if the energy of the $3 d$ state was originally $-8.50000 \mathrm{eV}$ with no magnetic field present, what will be the energies of the states into which it splits in the magnetic field? (d) What are the allowed wavelengths of the light emitted during transition in part (b)?

Key Concepts

Magnetic Quantum Number Adjustment

In the presence of an external magnetic field, energy levels associated with different magnetic quantum numbers shift differently. The shifts are typically given by the product of the Bohr magneton, the magnetic field strength, and the value of the magnetic quantum number, leading to multiple sub-levels for a given orbital state. This adjustment is a key factor in explaining the distribution of emitted photon wavelengths in a magnetic field.

Zeeman Effect

The Zeeman effect describes the splitting of atomic energy levels when an external magnetic field is applied. This splitting results from the interaction between the magnetic field and the magnetic dipole moment of the electrons, and it converts a single spectral line into multiple components. The magnitude of the energy split is proportional to the magnetic field strength and the magnetic quantum number.

Selection Rules

Selection rules are criteria that determine which transitions between quantum states are allowed or forbidden based on conservation laws and symmetries. For electric dipole transitions, these rules typically involve changes in quantum numbers, such as a change in the orbital angular momentum quantum number by plus or minus one. They ensure that only transitions compatible with angular momentum and parity conservation occur, simplifying the interpretation of atomic spectra.

Atomic Electron Transitions

This concept pertains to the jump of an electron between quantized energy states within an atom. When an electron transitions from a higher energy state to a lower one, a photon is emitted, and its energy corresponds to the difference between the two states. This process is central to atomic spectroscopy and underpins the quantized nature of atomic energy levels.

Photon Energy Calculation

This concept involves determining the energy of a photon from its wavelength using the relation E = hc/?, where h is Planck’s constant and c is the speed of light. Calculating photon energy is fundamental to understanding the quantized energy differences between atomic states, and it helps relate measurable spectral properties (wavelengths) to the atomic energy levels involved.

Transcript

00:01 The presence of an external magnetic field shifts the energy levels up or down depending upon the value of m.

00:10 So for part a, e equals hc divided by wavelength, which would be 4 .136 times 10 to the negative 15 evs times 3 .00 times 10 to the 8 meters per second.

00:29 And all of that divided by 475 .082 nanometers.

00:36 And that gives you 2 .612 evs.

00:44 For part b, for allowed transitions, we have nine allowed transitions from the 3d state in the presence of a magnetic field.

00:55 So l equals to ml equals to, ml equals to, goes to l equals 1, ml equals 1.

01:09 L equals 2, ml equals 1, goes to l equals 1, ml equals 1, ml equals 0, l equals 2, ml equals 1, goes to l equals 1, ml equals 1, l equals 2, ml equals 0, goes to l equals 1, ml equals 1, 0, l equals 2, ml equals 0 goes to l equals 1, ml equals 1, l equals 2, ml equals 0, l equals 2, ml equals 0, ml equals negative 1, l equals 2, ml equals negative 1, goes to l equals 1, ml equals 0, ml equals 0, l equals 2, ml equals negative 1, goes to l equals 1, ml equals 0, and l equals 0, l equals 1, ml equals negative 1, and l equals 2, ml equals negative 2 goes to l equals 1, ml equals negative 1, for part c, we have this equation, and so that is 5 .788 times 10 to negative 5 .5 evs divided by t times 3 .50 t's, and that leaves you with 0 .000203 evs.

02:43 So the energies of the new stats would be negative 8 .5 000 evs plus 0 and negative 8 .555000 evvs plus or minus 0 .000203 evs.

03:12 So this gives you negative 5 .002.

03:18 Sorry...

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