The wave function () of 2s is given by: _{2s} = {1}{2{2}}({1}{a_0})^{1/2}(2 - {r}{a_0})e^{-r/2a_0} At r = r0, radial…

🧠 The Vanishing Wave A "radial node" is a mathematical solution where the wave function ($\Psi$) equals zero. For the 2s orbital, this happens at a specific distance from the nucleus where the electron has zero chance of being found.

🗺️ The Root Calculation

1. The Function: $\Psi_{2s} \propto \left( 2 - \frac{r}{a_0} \right) e^{-r/2a_0}$
2. Setting to Zero: The constant term and the exponential term ($e^{-x}$) can never be zero for finite $r$. Thus, the polynomial term must be zero: $2 - \frac{r}{a_0} = 0$
3. Solving for $r$: $\frac{r}{a_0} = 2 \implies r = 2a_0$
4. Conclusion: At $r_0 = 2a_0$, the 2s electron hits its nodal shell.

⚡ The "n-Shell" Anchor For Hydrogen-like atoms, the node for 2s is always at $2a_0$. For 3s, there would be two roots. Just look for the term inside the parenthesis!

⚠️ Common Traps Don't be intimidated by the messy constants outside the parenthesis. The node is determined entirely by the polynomial "root" inside.

$\boxed{\text{Answer: (d)}}$