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Chapter 6: Problem 92

For orbitals that are symmetric but not spherical, the contour representations (as in Figures 6.22 and 6.23 ) suggest where nodal planes exist (that is, where the electron density is zero). For example, the \(p_{x}\) orbital has a node wherever \(x=0\). This equation is satisfied by all points on the \(y z\) plane, so this plane is called a nodal plane of the \(p_{x}\) orbital. (a) Determine the nodal plane of the \(p_{z}\) orbital. (b) What are the two nodal planes of the \(d_{x y}\) orbital? (c) What are the two nodal planes of the \(d_{x^{2}-y^{2}}\) orbital?

Short Answer

Expert verified
\(p_z\) orbital nodal plane: x-y plane \(d_{xy}\) orbital nodal planes: planes containing x-axis and y-axis \(d_{x^2-y^2}\) orbital nodal planes: planes containing lines y = x and y = -x

Step by step solution

01

(a) Nodal plane of \(p_z\) orbital

The \(p_{z}\) orbital has a dumbbell shape, with two lobes along the z-axis. The point where the electron probability density is zero is the x-y plane. All points in the x-y plane satisfy the condition for zero electron density, thus the nodal plane of the \(p_z\) orbital is the x-y plane.
02

(b) Nodal planes of \(d_{xy}\) orbital

The \(d_{xy}\) orbital has four lobes, and they lie in the x-y plane between the x and y axes. The electron probability density is zero along the x and y axes. Thus, the two nodal planes of the \(d_{xy}\) orbital are the planes perpendicular to the x-y plane, containing the x-axis and y-axis respectively.
03

(c) Nodal planes of \(d_{x^2-y^2}\) orbital

The \(d_{x^2-y^2}\) orbital has four lobes, with two along the x-axis and two along the y-axis. The electron probability density is zero along the lines y = x and y = -x. Thus, the two nodal planes of the \(d_{x^2-y^2}\) orbital are the planes perpendicular to the x-y plane, containing the lines y = x and y = -x respectively.

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