Chapter 28, Problem 1CQ

To determine

Does the frequency of possible spectral lines from hydrogen increase without limit.

Answer to Problem 1CQ

The frequency will not increase without limit as the principle quantum number is increasing without limit in the hydrogen atom.

Explanation of Solution

Expressing the relation between energy of the spectrum of the hydrogen atom and the principle quantum number is,

$E_n=-\frac{13.6\text{eV}}{n^2}$ (I)

• $E_n$ is the n^th energy level
• $n$ is the principle quantum number, where, $n=1,2,\mathrm{3....}$

Expressing the relation between energy of the spectrum of the hydrogen atom and the frequency is,

$E_n=h\nu$ (II)

• $h$ is the Plank’s constant
• $\nu$ is the frequency

Comparing equation (I) and (II), the relation between frequency and the principle quantum number is,

$h\nu =-\frac{13.6\text{eV}}{n^2}$

Thus, the frequency and the principle quantum number is inversely proportional. Thus, the frequency decreases as the principle quantum number is increasing without limit in the hydrogen atom.

Conclusion:

If the principle quantum number is increasing without limit in the hydrogen atom, it does not mean that the frequency will increase without limit.