Please explain the following example given in the snapshot below?

1 Answer

Answer:

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"#