The wave function (Ψ) of 2 s is given by
Ψ₂ₛ=1/(2 √(2 π))(1/a₀)^(1 / 2)(2-r/a₀) e^(-r / 2 a₀)
At r=r₀, radial node is formed. Thus, r₀ in terms of a₀

Solution

Chemistry ma’am · 4 steps

A radial node is where Ψ becomes zero, so set the only factor that can vanish equal to zero.

1. What a node means. At a radial node the wave function is zero, so the electron has zero probability of being found there. We need the value of r that makes Ψ₂ₛ = 0.
2. Check each factor. The constant 1/2√2π(1/a₀)^(1/2) is never zero. The exponential e^(-r/2a₀) is never zero for any finite r. Only the bracket (2 - r/a₀) can vanish.

   2 - r₀/a₀ = 0 ⇒ r₀ = 2a₀

3. Cross-check. A 2s orbital has n - l - 1 = 2 - 0 - 1 = 1 radial node, and we found exactly one value of r.
4. So. The radial node is at r₀ = 2a₀, which is option (c).

The idea to keep. To locate a radial node, set the polynomial part of the radial wave function to zero. The exponential and constant factors never vanish, so ignore them.

Answer from the NTA final answer key. Solution by PrepWiser.