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Chapter 6: Problem 99

For orbitals that are symmetric but not spherical, the contour representations (as in Figures 6.23 and 6.24 ) 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
The nodal planes for the orbitals are as follows: (a) For the \(p_{z}\) orbital, the nodal plane is the \(x y\) plane. (b) For the \(d_{xy}\) orbital, there are two nodal planes: the \(x\)-axis and the \(y\)-axis. (c) For the \(d_{x^2-y^2}\) orbital, there are two nodal planes: the plane where \(x=y\) and the plane where \(x=-y\).

Step by step solution

01

Determine the nodal plane of the \(p_{z}\) orbital

To find the nodal plane of the \(p_{z}\), we are asked to consider where the \(z\) value is equal to zero. This equation \((z=0)\) is satisfied by all the points on the \(x y\) plane. Therefore, the \(x y\) plane is the nodal plane for the \(p_{z}\) orbital.
02

Find the nodal planes of the \(d_{xy}\) orbital

To find the nodal planes of the \(d_{xy}\) orbital, we need to consider where the product of \(x\) and \(y\) values equals zero. This means either \(x=0\) or \(y=0\). So, there are two nodal planes here: the \(x\)-axis (where \(y=0\)) and the \(y\)-axis (where \(x=0\)).
03

Identify the nodal planes of the \(d_{x^2-y^2}\) orbital

To find the nodal planes of the \(d_{x^2-y^2}\) orbital, we need to consider where the difference of the squares of \(x\) and \(y\) values is equal to zero. This means either \(x^2=y^2 \Rightarrow x=y\) or \(x^2=y^2 \Rightarrow x=-y\). So, there are two nodal planes here: the plane where \(x=y\) (a diagonal plane from the positive \(x\)-axis to the positive \(y\)-axis) and the plane where \(x=-y\) (a diagonal plane from the positive \(x\)-axis to the negative \(y\)-axis).

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