Question

Please explain the following example given in the snapshot below?

Answer 1

The answers are (B) `r = 2a_0` and (D) 0.423 Å.

Explanation:

Ex. 1

The radial node occurs when

`Psi_text(2s) = 1/(4sqrt(2π)a_0^"3/2")[2-r/a_0]e^("-"r/(2a_0)) = 0`

The zeroes of this function occur at

`2 - r/a_0 = 0` and `e^("-"r/(2a_0)) = 0`

`e^(-r/(2a_0)) → 0` only as `r → ∞`.

This should be no surprise, because we know that a wave function becomes vanishingly small as distance from the nucleus increases.

If `2 - r/a_0 = 0`

then `r/a_0 = 2`

and `r = 2a_0`.

Ex. 2

The radial node occurs when

`R(r) = 1/(9sqrt6)(Z/a_0)^(3/4)(4 - σ)σe^("-"σ/2)`

The zeroes of this function occur at

`4 - σ = 0, σ = 0`, and `e^("-"σ/2) = 0`.

`σ = 0` corresponds to zero probability at the nucleus (no surprise here!).

`e^("-"σ/2) = 0` corresponds to zero probability at an infinite distance from the nucleus (again, no surprise!).

If `4 - σ = 0`

then `σ = 4`

and `(Zr)/a_0 = 4`

∴ `r = (4a_0)/Z = (color(red)(color(black)(4)) × "0.529 Å")/color(red)(color(black)(5)) = "0.423 Å"`