Chapter 8: Problem 44

Find the magnitude of the orbital magnetic dipole moment of the electron in in the \(4 d\) state. (Express your answer in terms of \(\mu_{\mathrm{B}}\).)

Short Answer

Expert verified

The magnitude of the orbital magnetic dipole moment of the electron in the 4d state is \( \sqrt{6}\mu_{\mathrm{B}} \).

Step by step solution

Identify the quantum numbers for the given state

We are given the electron is in a 4d state. We can find the principal quantum number (n) and the orbital quantum number (l) using the notation: n = The number in the notation (4 in this case) l = The number assigned to the letter in the notation (s=0, p=1, d=2, f=3) For the 4d state: n = 4 and l = 2.

Formula for the magnitude of the orbital magnetic dipole moment

The formula for the magnitude of the orbital magnetic dipole moment (mL) of an electron is given by: mL = -e * ħ * sqrt(l(l + 1)) / (2 * me) Where: e = elementary charge (1.602 x 10^(-19) C) ħ = reduced Planck's constant (1.05 x 10^(-34) Js) l = orbital quantum number me = electron mass (9.11 x 10^(-31) kg)

Find the magnitude of the orbital magnetic dipole moment in terms of the Bohr magneton

The Bohr magneton (µB) is a constant of proportionality that relates the magnetic dipole moment to the quantum numbers. It is given by the formula: µB = eħ / (2me) Let's divide both sides of the formula mL by µB to express mL in terms of µB: mL/µB = (-e * ħ * sqrt(l(l + 1)) / (2 * me)) / (eħ / (2me)) mL/µB = sqrt(l(l + 1)) Now we can easily plug in the value of l to find the magnitude of the orbital magnetic dipole moment: mL/µB = sqrt(2(2 + 1))

Calculate the result

Plug the value of l into the equation: mL/µB = sqrt(2 * 3) mL/µB = sqrt(6) So, the magnitude of the orbital magnetic dipole moment of the electron in the 4d state is \( \sqrt{6}\mu_{\mathrm{B}} \).

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Find the magnitude of the orbital magnetic dipole moment of the electron in in the \(4 d\) state. (Express your answer in terms of \(\mu_{\mathrm{B}}\).)

Short Answer

Step by step solution

Identify the quantum numbers for the given state

Formula for the magnitude of the orbital magnetic dipole moment

Find the magnitude of the orbital magnetic dipole moment in terms of the Bohr magneton

Calculate the result

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