The number of radial nodes of 3s, 3p and 3d electrons are, respectively, — Atomic Structure Chemistry Question

💡 Solution & Explanation

**Step 1 - Understanding Atomic Nodes and the Radial Node Formula** In quantum mechanics, a node represents a region of space where the probability of finding an electron is exactly zero (i.e., the wave function $\psi = 0$ and the probability density $|\psi|^2 = 0$). There are two types of nodes in an atomic orbital: 1. **Radial Nodes (Spherical Nodes):** Spherical surfaces centered at the nucleus where the probability of finding an electron is zero. 2. **Angular Nodes (Nodal Planes/Cones):** Planes or cones passing through the nucleus with zero electron density. The number of radial nodes ($N_r$) of an orbital is calculated using the formula: $$N_r = n - l - 1$$ Where: * $n$ is the principal quantum number. * $l$ is the azimuthal quantum number. The value of the azimuthal quantum number ($l$) depends on the type of subshell: * For $s$-orbitals: $l = 0$ * For $p$-orbitals: $l = 1$ * For $d$-orbitals: $l = 2$ --- **Step 2 - Determining Quantum Numbers and Calculating Radial Nodes** Let us identify the quantum numbers and calculate the number of radial nodes for each orbital: 1. **For $3s$ orbital:** * Principal quantum number: $n = 3$ * Azimuthal quantum number: $l = 0$ * Applying the formula: $$N_{r,\text{ }3s} = n - l - 1$$ Substituting the values: $$N_{r,\text{ }3s} = 3 - 0 - 1 = 2$$ 2. **For $3p$ orbital:** * Principal quantum number: $n = 3$ * Azimuthal quantum number: $l = 1$ * Applying the formula: $$N_{r,\text{ }3p} = n - l - 1$$ Substituting the values: $$N_{r,\text{ }3p} = 3 - 1 - 1 = 1$$ 3. **For $3d$ orbital:** * Principal quantum number: $n = 3$ * Azimuthal quantum number: $l = 2$ * Applying the formula: $$N_{r,\text{ }3d} = n - l - 1$$ Substituting the values: $$N_{r,\text{ }3d} = 3 - 2 - 1 = 0$$ Thus, the number of radial nodes for $3s$, $3p$, and $3d$ are $2$, $1$, and $0$, respectively. --- **Step 3 - Analysis of Options** * **Option (A) 0, 1, 2:** This set of numbers corresponds to the azimuthal quantum number ($l$) of the respective orbitals, which also equals the number of angular nodes, not the radial nodes. Therefore, this option is incorrect. * **Option (B) 2, 1, 0:** This set of numbers matches our calculated values of radial nodes for $3s$, $3p$, and $3d$ orbitals. Therefore, this option is correct. * **Option (C) 2, 2, 2:** This is incorrect because radial nodes vary for different subshells within the same principal shell since the value of $l$ is different. * **Option (D) 1, 3, 5:** This represents the number of degenerate orbitals in the $s$, $p$, and $d$ subshells respectively (given by $2l + 1$), rather than the count of radial nodes. Therefore, this option is incorrect. $$\text{Correct Option: } \boxed{\text{B}}$$