Sketch the \(3 d\) orbitals. How do the \(4 d\) orbitals differ from the \(3 d\) orbitals?

Understanding the shapes of orbitals is a fundamental concept in quantum chemistry, as it provides insights into where electrons are likely to be located around a nucleus. When sketching the 3d orbitals, it is crucial to realize that these are not two-dimensional, but rather three-dimensional shapes extending into space. The five types of 3d orbitals are: dxy, dyz, dxz, dx2-y2, and dz2.

Visualize the dxy, dyz, and dxz orbitals as four leaf clovers lying in between the x, y, and z axes. The dx2-y2 orbital has its lobes directed along the axes, resulting in a cross shape. The dz2 stands out with its two lobes extending along the z-axis and a distinctive torus, or donut-shaped lobe, circling the middle. When comparing these to the 4d orbitals, remember that the overall shapes are the same, but scaled up, illustrating a larger spatial distribution due to increased principal quantum number.

The energy level of an orbital is a crucial factor in understanding electron configuration and chemical properties. In general, as you move from 3d to 4d orbitals, there is an increase in energy. This is dictated by the principal quantum number, which is higher for the 4d orbitals (n=4) relative to the 3d orbitals (n=3).

The larger principal quantum number translates to an electron being further from the nucleus and, therefore, less tightly bound – this contributes to the 4d orbitals having a higher energy. Notably, electrons in an atom will fill the lower energy 3d orbitals before occupying the 4d orbitals, a principle that's important when determining the electron configuration of elements.

Electron configuration outlines how electrons are distributed in an atom's orbitals, influencing chemical reactivity and bonding. Following the Aufbau principle, electrons fill up orbitals in order of increasing energy, starting with the lowest. Hund's rule and the Pauli Exclusion Principle also govern the arrangement of electrons in orbitals.

Considering 3d and 4d orbitals within electron configuration, the 3d orbitals generally fill after the 4s orbital and before the 4p, according to the configuration of transition metals. The 4d orbitals begin to fill in the fifth period of the periodic table, as seen with elements like Yttrium (Y) and Zirconium (Zr). The additional node in 4d orbitals mentioned in the exercise is one aspect that distinguishes them from their 3d counterparts and influences their energy and occupancy in the electron configuration.