Question

Prove that the probability to find an electron in $2p_z$ orbital anywhere in$x-y$ plane is zero. Also, determine the nodal planes for $d_{xz}$ and $d_{x^2-y^2}$.

Short answer

The nodal planes are two planes that contain the -axis at a $45°$ from the $X$-and $Y$-axes.

Step-by-step solution

Definition of the wave function

A mathematical representation of an electron in an orbital is a wave function. It is

created by multiplying the radial and angular wave functions.

Equation for the angular wave function of the 2pz orbital

The equation for the angular wave function of the $2p_z$ orbital is as follows:

$Y_{p_z}=\sqrt{\frac{3}{4\pi }}\cos \theta$

In the $x$-y coordinate system, the angle formed by the $2p_z$ orbital will be $\left(\frac{\pi }{2}\right)$, which is data-custom-editor="chemistry" $90°$. In this case, $\theta$ has a value of 90.

Calculation of Ypz'  substitute the value in the preceding equation

Calculate ${\mathbf{Y}}_{p_{z^{\text{'}}}}$ and substitute the value in the preceding equation for the angular component of the wave function:

Recall: $\theta =\left(\frac{\pi }{2}\right)$

$$\begin{gathered} {\mathbf{Y}}_{p_z}=\sqrt{\frac{3}{4\pi }}\cos \theta \\ =\sqrt{\frac{3}{4\pi }}\cos \left(\frac{\pi }{2}\right) \\ =\sqrt{\frac{3}{4\pi }}\times 0=0 \\ \theta =\left(\frac{\pi }{2}\right),\cos \theta =0 \end{gathered}$$

Stating the reason to prove that the probability of finding an electron in an 2pz  orbital anywhere in the -y plane is zero.

Since, the total wave function of an $2p_z$ orbital is the product of a radial wave function and an angular wave function, the total wave function of an $2p_z$ orbital will also be zero in the $x$-y plane.

Thus, the probability of finding an electron in a $2p_z$ orbital anywhere in the $x$-y plane is zero.

For an atom, the radial part of wave functions is affected solely by $r$, while the angular part of wave functions is affected solely by the direction and shape of an orbital.

The region where the radial part of a wave function passes through zero is referred to as a radial node. Angular nodes or nodal planes are regions where the angular wave function passes through zero.

Specifying the nodal plane for dxzYdxz=154πsinθcosθcosϕ       =0     θ=π2,cosθ=0 

Therefore, the nodal planes are the $y-z$and $x-y$planes

Specifying the nodal plane for dx2-y2