Question

The radial distribution curve of the orbital with double dumbbell shape in the 4th principal shell consists of N nodes, N is:
A)2
B)0
C)1
D)3

Answer 1

We are familiar with principal quantum number (n) and azimuthal quantum number (l). These two are used to find the nodes in any orbital. Shapes of orbitals determine the azimuthal quantum number such that double dumbbell shaped orbitals are d-orbitals having l value equals to 2. So, we can also say that number of nodes will be one less than the l value.

Formula used: $N = n - l - 1$

Complete step-by-step answer:
Atomic orbitals are 3D regions of space around the nucleus of an atom. They allow the atoms to make covalent bonds and s, p, d and f are the most commonly filled orbitals as they are available at the ground state. As stated by Pauli Exclusion Principal, an orbital can accommodate only two electrons. All the electrons have same value for n or we can say that the principal quantum number remains same in the same shell.
When the electrons share same value of n, l and m, they are said to be in the same orbital and have same energy but differ only in spin orbital quantum number. The value of l gives the number of orbitals of a type within a subshell such as for the four types of atomic orbitals that we mentioned above, the value of l will be 0, 1, 2, and 3 respectively.
Node is a region where the probability of finding an electron is minimum i.e. zero. The nodal plane passes through the nucleus on which the probability is zero. The number of nodal planes is equal to the azimuthal quantum number in an orbit. Basically, there are two types of nodes known to us: angular and radial nodes. Angular nodes are flat at fixed angles and radial nodes are spheres at a fixed radius and occurs as the principal quantum number increases.

The total number of nodes in an orbit are the sum of angular and radial nodes and the formula of calculating radial nodes can be expressed in terms of n and l quantum number as $N = n - l - 1$ . Whereas angular nodes will be equal to the l value of an orbital.
We are given in the question that the orbital is double dumbbell shape in the 4th principal shell. This means that it is d orbital as d-orbitals have double dumbbell shape, p has dumbbell shape and s are spherical. Since the principal shell is 4, the orbital can be written as 4d. Now, for this particular orbital, n is 4 and l is 2 as we discussed above. So, putting these values in the above formula, we get the radial node, N as
$N = n - l - 1$

$$N = 4 - 2 - 1 \\\ N = 1 \\\$$

Therefore, the radial distribution curve of the orbital with double dumbbell shape in the 4th principal shell consists of N nodes, N is 1.
Hence, the correct option is (C).

Note: There are five d orbitals and are represented as ${d_{{x^2}}}_{ - {y^2}},{d_{{z^2}}},{d_{xy}},{d_{yx}},{d_{xz}}$ . Their energies are equivalent but their shape differs from one another depending on m values which lie from -l to +l where $l = 2$ for d-orbitals i.e. -2, -1, 0, +1, +2. The orbitals that have maximum probability distribution in between the axis are ${d_{xy}},{d_{yx}},{d_{xz}}$ and along the axis are ${d_{{x^2}}}_{ - {y^2}},{d_{{z^2}}}$ .