The size of an orbital is given by — Atomic Structure Chemistry Question

💡 Solution & Explanation

### Step 1 - Introduction to Quantum Numbers In quantum mechanics, the properties of atomic orbitals are described by four quantum numbers: 1. Principal quantum number ($n$) 2. Azimuthal (orbital angular momentum) quantum number ($l$) 3. Magnetic quantum number ($m$) 4. Spin quantum number ($s$) Each quantum number governs a distinct physical characteristic of the orbital. --- ### Step 2 - Role of the Principal Quantum Number ($n$) The **principal quantum number ($n$)** specifies the main energy shell and determines the **size** of the orbital: $$n = 1, 2, 3, 4, \dots$$ The average distance of the electron from the nucleus scales with $n^2$: $$\langle r \rangle \propto n^2$$ As $n$ increases, the orbital extends further into space: * $n=1$: smallest orbital (1s) * $n=2$: larger orbital (2s, 2p) * $n=3$: even larger orbital (3s, 3p, 3d) A $3s$ orbital is larger than a $2s$ orbital, which is larger than a $1s$ orbital — solely due to the higher principal quantum number. --- ### Step 3 - Role of the Other Quantum Numbers 1. **Azimuthal Quantum Number ($l$):** Determines the **3D shape** of the orbital ($l=0$: spherical $s$; $l=1$: dumbbell-shaped $p$). Does not determine size. 2. **Magnetic Quantum Number ($m$):** Determines the **spatial orientation** of the orbital ($p_x$, $p_y$, or $p_z$). Does not determine size. 3. **Spin Quantum Number ($s$):** Describes the **intrinsic spin state** of the electron ($\pm 1/2$). Completely unrelated to orbital size. --- ### Step 4 - Systematic Analysis of the Options * **Option (A) principal quantum number:** Correct. The principal quantum number determines the size (and average distance from the nucleus) of the orbital. * **Option (B) azimuthal quantum number:** Incorrect. The azimuthal quantum number determines the *shape* of the orbital, not its size. * **Option (C) magnetic quantum number:** Incorrect. The magnetic quantum number determines the *spatial orientation* of the orbital, not its size. * **Option (D) spin quantum number:** Incorrect. The spin quantum number describes the *spin state* of the electron and has no relationship to orbital size. $$\text{Correct Option: } \boxed{\text{A}}$$