Number of nodal surface in 5s orbital is — Atomic Structure Chemistry Question

💡 Solution & Explanation

### Step 1 - Define Nodal Surfaces (Radial Nodes) In quantum mechanics, a nodal surface (also known as a radial node or spherical node) is a spherical shell surrounding the nucleus where the probability of finding an electron is exactly zero. The number of radial nodes ($N_r$) in any atomic orbital depends on its principal quantum number ($n$) and azimuthal quantum number ($l$). --- ### Step 2 - Determine the Quantum Numbers for the 5s Orbital For a $5s$ orbital, we can identify the relevant quantum numbers directly from its designation: * The principal quantum number is $n = 5$. * Since it is an $s$-orbital, the azimuthal quantum number is $l = 0$. --- ### Step 3 - Calculate the Number of Nodal Surfaces The formula to calculate the number of radial nodes (nodal surfaces) is given by: $$N_r = n - l - 1$$ Now, we substitute the quantum numbers into this formula: $$N_r = 5 - 0 - 1$$ $$N_r = \boxed{4}$$ --- ### Step 4 - Systematic Evaluation of the Options * **Option (A) 5:** This is incorrect because $5$ represents the principal quantum number ($n = 5$) of the orbital, not the number of nodal surfaces. * **Option (B) 4:** This is correct because our formula yields exactly $4$ nodal surfaces for the $5s$ orbital. * **Option (C) 3:** This is incorrect. A value of $3$ nodal surfaces would correspond to a $4s$ orbital ($4 - 0 - 1 = 3$) or a $5p$ orbital ($5 - 1 - 1 = 3$). * **Option (D) 0:** This is incorrect because $0$ represents the number of angular nodes (given by $l = 0$ for any $s$-orbital), not the number of radial nodal surfaces.