Chapter 8: Problem 41

Find the magnitude of the orbital magnetic dipole moment of the electron in in the $3 p$ 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 3p state is approximately equal to one Bohr magneton: $|\mu_L| \approx \mu_B$.

Step by step solution

Recall the formula for orbital magnetic dipole moment

The formula for the orbital magnetic dipole moment of an electron is given by: $$ \mu_L = -\mu_B \cdot g_L \cdot L $$ where $\mu_L$ is the orbital magnetic dipole moment, $\mu_B$ is the Bohr magneton, $g_L \approx 1$ is the Landé g-factor for orbital angular momentum (which is approximately equal to 1 for an electron in an atom), and $L$ is the orbital angular momentum quantum number.

Determine the value of the quantum number $L$

For the 3p state, the values of the principal quantum number ($n$) and the orbit quantum number ($\ell$) are given. In this case, $n = 3$, and since it is a p-state, $\ell = 1$. The magnetic quantum number can have values $m_\ell = -\ell, -\ell+1, ..., \ell-1, \ell$, so for a p-state, $m_\ell = -1, 0, 1$. However, for finding the magnitude of the orbital magnetic dipole moment, we are only concerned with the value of $\ell$ which is equal to 1.

Plug the value of $L$ into the formula

Now that we have determined the value of $\ell$, we can plug it into the formula for the orbital magnetic dipole moment: $$ \mu_L \approx -\mu_B \cdot L $$ Substitute the value of $\ell = 1$: $$ \mu_L \approx -\mu_B \cdot 1 $$ Since we are asked to find the magnitude of the orbital magnetic dipole moment, we can ignore the negative sign and get: $$ |\mu_L| \approx \mu_B $$ Thus, the magnitude of the orbital magnetic dipole moment of the electron in the 3p state is approximately equal to one Bohr magneton.

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Find the magnitude of the orbital magnetic dipole moment of the electron in in the \(3 p\) state. (Express your answer in terms of \(\mu_{\mathrm{B}}\).)

Short Answer

Step by step solution

Recall the formula for orbital magnetic dipole moment

Determine the value of the quantum number \(L\)

Plug the value of \(L\) into the formula

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