Please explain the following example given in the snapshot below?

Question

1840 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-07-13T15:55:52

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

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 $(Zr)/a_0 = 4$

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