Question

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

The nodal plane of the \(p_{z}\) orbital is the \(xy\) plane. The two nodal planes for the \(d_{xy}\) orbital are the \(yz\) plane and the \(xz\) plane. The nodal planes for the \(d_{x^2-y^2}\) orbital are the planes \(y=x\) and \(y=-x\).

Step-by-step solution

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

For the \(p_{z}\) orbital, the electron density is zero when \(z=0\), which is the \(xy\) plane. So, the nodal plane of the \(p_{z}\) orbital is the \(xy\) plane, as this is where the electron density is zero.

(b) Determine the two nodal planes of the \(d_{x y}\) orbital

The \(d_{xy}\) orbital is antisymmetric with respect to both the \(x\) and \(y\) axes. The two nodal planes for the \(d_{xy}\) orbital are: 1. The plane when \(x=0\), which is the \(yz\) plane. 2. The plane when \(y=0\), which is the \(xz\) plane.

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

The \(d_{x^2-y^2}\) orbital has its lobes along the \(x\) and \(y\) axes and has two nodal planes. These planes are determined by points that are equidistant from the \(x\) and \(y\) axes. The nodal planes for the \(d_{x^2-y^2}\) orbital are: 1. At a 45-degree angle to the \(x\) and \(y\) axes in the positive quadrant (the plane of \(y=x\)) 2. At a 45-degree angle to the \(x\) and \(y\) axes in the negative quadrant (the plane of \(y=-x\))

Key concepts

Nodal Planes

Understanding nodal planes is crucial in the study of quantum chemistry and molecular orbitals. Essentially, they are invisible 'zones' within an atom's orbital where the probability of finding an electron is zero. This happens due to the wave nature of electrons; where wave functions corresponding to these electrons cancel each other out, resulting in no electron density at these points.

For example, the problem mentions the p_z orbital. This orbital's nodal plane lies along the xy-plane, meaning at any point where the z-coordinate is zero, electron density is also zero. Similarly, d orbitals have more complex shapes and therefore, more nodal planes. Recognizing where these planes are helps us understand the electron distribution within an orbital and anticipate how atoms will bond with each other.

Electron Density

Electron density is a term that describes the probability of an electron being present at a specific location around an atom's nucleus. In chemistry, visualizing electron density helps in predicting the behavior of atoms in different chemical reactions and bonding scenarios. The lower the electron density in a given region, the less likely we are to find an electron there.

The exercise provided illustrates how electron density is not uniformly distributed across different orbitals. In atomic orbitals like p and d orbitals, the electron density maps out where electrons are most likely to be found, helping us predict their reactivity and interactions with other atoms.

P Orbitals

P orbitals are one step more complex than the spherical s orbitals. Each p orbital is dumbbell-shaped, consisting of two lobes that are symmetrical about a specific axis. There are three p orbitals — p_x, p_y, and p_z — each aligned along one of the three Cartesian axes. This means that the p_x orbital is symmetrical about the x-axis, and similarly for the other two.

Between the lobes of a p orbital lies a nodal plane where the electron density drops to zero, meaning there's a zero probability of finding an electron there. The nodal planes are vital for understanding the chemical bonding and angular distributions of electrons associated with these orbitals.

D Orbitals

D orbitals introduce a greater level of complexity compared to s and p orbitals. These orbitals are primarily involved in the chemistry of transition metals and come in five varieties, each with distinct shapes and orientations. The d_xy orbital, for instance, has lobes lying in the xy-plane and has nodal planes along the yz and xz planes.

The d_{x^2-y^2} orbital, on the other hand, stretches along the x and y-axis, displaying nodal planes that cut through the axes at a 45-degree angle. These nodal planes and lobe orientations dictate how d orbitals engage in bonding and their impact on the molecular geometry of compounds formed by transition metals.