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Chapter 3: Problem 27

For each of the questions, four choices have been provided. Select the correct alternative. Probability of finding a \(\mathrm{d}_{\mathrm{yz}}\) electron is zero along the (a) \(x\) -axis (b) \(y\) -axis (c) \(z\) -axis (d) all of these

Short Answer

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Short Answer: The probability of finding a d₍yz₎ electron is zero along all of the x-, y-, and z-axes.

Step by step solution

01

Identify the wavefunction of the \(\mathrm{d}_{\mathrm{yz}}\) electron

To start with, let's recall the wavefunction of the \(\mathrm{d}_{\mathrm{yz}}\) orbital. The wavefunction is proportional to: $$ \psi_{\mathrm{d}_{\mathrm{yz}}} \propto Y_{2,1} \propto r^2 (\sin\theta \cos\theta)\, \sin\phi. $$ Here, \(r, \theta,\) and \(\phi\) are the spherical polar coordinates of the \(\mathrm{d}_{\mathrm{yz}}\) electron. Now we will find out the probability of finding an electron along each axis.
02

Find probability along the x-axis

The x-axis corresponds to \(\phi=0\) and \(\phi=\pi\). Substituting these values in the wavefunction, we get: $$ \psi_{\mathrm{d}_{\mathrm{yz}}} \propto r^2 (\sin\theta \cos\theta) \sin0 = 0 $$ and $$ \psi_{\mathrm{d}_{\mathrm{yz}}} \propto r^2 (\sin\theta \cos\theta) \sin\pi = 0 $$ Thus, the probability of finding a \(\mathrm{d}_{\mathrm{yz}}\) electron is zero along the x-axis.
03

Find probability along the y-axis

The y-axis corresponds to \(\theta = \frac{\pi}{2}\) and \(\phi = \frac{\pi}{2}\). Substituting these values in the wavefunction, we get: $$ \psi_{\mathrm{d}_{\mathrm{yz}}} \propto r^2 (\sin\frac{\pi}{2} \cos\frac{\pi}{2}) \sin\frac{\pi}{2} = 0 $$ Thus, the probability of finding a \(\mathrm{d}_{\mathrm{yz}}\) electron is zero along the y-axis.
04

Find probability along the z-axis

The z-axis corresponds to \(\theta = 0\) and \(\theta = \pi\). Substituting these values in the wavefunction, we get: $$ \psi_{\mathrm{d}_{\mathrm{yz}}} \propto r^2 (\sin0 \cos0) \sin\phi = 0 $$ and $$ \psi_{\mathrm{d}_{\mathrm{yz}}} \propto r^2 (\sin\pi \cos\pi) \sin\phi = 0 $$ Thus, the probability of finding a \(\mathrm{d}_{\mathrm{yz}}\) electron is zero along the z-axis.
05

Choose the correct alternative

From Steps 2-4, we find that the probability of finding a \(\mathrm{d}_{\mathrm{yz}}\) electron is zero along all of the x-, y-, and z-axes. Therefore, the correct answer is (d) all of these.

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