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

Expert verified

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

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.

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.

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.

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.

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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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

Step by step solution

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

Find probability along the x-axis

Find probability along the y-axis

Find probability along the z-axis

Choose the correct alternative

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