Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n 1. Which of the following statement(s) is(are) true? [2016]

Question

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n >> 1. Which of the following statement(s) is(are) true? [2016]

Smart Achievers · Accepted Answer

(1, 2, 4)

We know radius, $r_{n} = r_{0} \frac{n^{2}}{Z},$

Energy $E_{n} = - \frac{13.6 Z^{2}}{n^{2}},$ and angular momentum, $L_{n} = \frac{n h}{2 π}$

Relative change in the radii of two consecutive orbitals,

$\frac{Δ r}{r} = \frac{r_{n} - r_{n - 1}}{r_{n}} = 1 - \frac{r_{n - 1}}{r_{n}} = 1 - \frac{{(n - 1)}^{2}}{n^{2}}$

$= \frac{2 n - 1}{n^{2}} \approx \frac{2}{n}$ $(∵ n ≫ 1),$ which is independent of $Z$ .

Relative change in the energy of two consecutive orbitals,

$\frac{Δ E}{E} = \frac{E_{n} - E_{n - 1}}{E_{n}} = 1 - \frac{E_{n - 1}}{E_{n}} = 1 - \frac{n^{2}}{{(n - 1)}^{2}}$ $= \frac{- 2 n + 1}{{(n - 1)}^{2}} \approx - \frac{2}{n}$

$\frac{Δ L}{L} = \frac{L_{n} - L_{n - 1}}{L_{n}} = 1 - \frac{L_{n - 1}}{L_{n}} = 1 - \frac{(n - 1)}{n} = \frac{1}{n}$

Solution

Q.
Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n >> 1. Which of the following statement(s) is(are) true [2016]

1 Relative change in the radii of two consecutive orbitals does not depend on Z

2 Relative change in the radii of two consecutive orbitals varies as 1/n

3 Relative change in the energy of two consecutive orbitals varies as $1 / n^{3}$

4 Relative change in the angular momentum of two consecutive orbitals varies as 1/n

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