Chapter 9, Problem 9.61E

Interpretation Introduction

Interpretation:

A table of first $50$ lines of the first six series of the hydrogen atom spectrum is to be constructed. The series limit in each case is to be predicted.

Concept introduction:

Rydberg equation is used to represent the wavenumber or wavelength of the lines present in the atomic spectrum of an element. The Rydberg equation for the hydrogen atom is represented as,

$\overline{\nu }=R_{\text{H}}\left(\frac{1}{n_{\text{f}}^2}-\frac{1}{n_{\text{i}}^2}\right)$

Where,

• $R_{\text{H}}$ represents Rydberg constant with a value for hydrogen $1.09737\times {10}^7{\text{m}}^{-1}$.

• $n_{\text{i}}$ represents the initial energy level.

• $n_{\text{f}}$ represents the final energy level.

Answer to Problem 9.61E

The table for the first $50$ lines of the first six series of the hydrogen atom spectrum is shown below.

S.no.	$n_1$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=1$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=2$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=3$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=4$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=5$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=6$
1.	$2$	$82302.98$
2.	$3$	$97544.28$	$15241.29$
3.	$4$	$102878.7$	$20575.75$	$5334.45$
4.	$5$	$105347.8$	$23044.84$	$7803.542$	$2469.089$
5.	$6$	$106689.1$	$24386.07$	$9144.776$	$3810.323$	$1341.23$
6.	$7$	$107497.8$	$25194.79$	$9953.498$	$4619.045$	$2149.96$	$808.722$
7.	$8$	$108022.7$	$26079.55$	$10478.39$	$5143.936$	$2674.85$	$1333.61$
8.	$9$	$108382.5$	$26336.95$	$10838.25$	$5503.8$	$3034.71$	$1693.477$
9.	$10$	$108639.9$	$26527.41$	$11095.66$	$5761.209$	$3292.119$	$1950.886$
10.	$11$	$108830.4$	$26672.26$	$11286.11$	$5951.662$	$3482.572$	$2141.339$
11.	$12$	$108975.2$	$26784.99$	$11430.97$	$6096.517$	$3627.428$	-

Table 1

The table for the series limit for first six series of the hydrogen atom spectrum is shown below.

S.no.	$n_2$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_1=\infty$
1.	$1$	$109737.32$
2.	$2$	$27434.33$
3.	$3$	$12193.04$
4.	$4$	$6858.58$
5.	$5$	$4389.49$
6.	$6$	$3048.26$

Table 2

Explanation of Solution

The wavenumber in the hydrogen atom spectrum is calculated by the formula,

$\overline{\upsilon }=R_{\text{H}}\left(\frac{1}{n_2^2}-\frac{1}{n_1^2}\right)$

Where,

• $R_{\text{H}}$ represents Rydberg constant with a value for hydrogen $109737.31{\text{cm}}^{-1}$.

• $n_1$ and $n_2$ represents the energy levels.

The first six series in the hydrogen atom spectrum have the value of $n_2$ as $1$, $2$, $3$, $4$, $5$ and $6$. The table for the first $50$ lines of the first six series of the hydrogen atom spectrum is shown below.

S.no.	$n_1$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=1$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=2$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=3$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=4$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=5$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=6$
1.	$2$	$82302.98$
2.	$3$	$97544.28$	$15241.29$
3.	$4$	$102878.7$	$20575.75$	$5334.45$
4.	$5$	$105347.8$	$23044.84$	$7803.542$	$2469.089$
5.	$6$	$106689.1$	$24386.07$	$9144.776$	$3810.323$	$1341.23$
6.	$7$	$107497.8$	$25194.79$	$9953.498$	$4619.045$	$2149.96$	$808.722$
7.	$8$	$108022.7$	$26079.55$	$10478.39$	$5143.936$	$2674.85$	$1333.61$
8.	$9$	$108382.5$	$26336.95$	$10838.25$	$5503.8$	$3034.71$	$1693.477$
9.	$10$	$108639.9$	$26527.41$	$11095.66$	$5761.209$	$3292.119$	$1950.886$
10.	$11$	$108830.4$	$26672.26$	$11286.11$	$5951.662$	$3482.572$	$2141.339$
11.	$12$	$108975.2$	$26784.99$	$11430.97$	$6096.517$	$3627.428$	-

Table 1

The series limit is when $n_1=\infty$ in each case.

The table for the series limit for first six series of the hydrogen atom spectrum is shown below.

S.no.	$n_2$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_1=\infty$
1.	$1$	$109737.32$
2.	$2$	$27434.33$
3.	$3$	$12193.04$
4.	$4$	$6858.58$
5.	$5$	$4389.49$
6.	$6$	$3048.26$

Table 2

Conclusion

The table for the first $50$ lines of the first six series of the hydrogen atom spectrum is shown below.

S.no.	$n_1$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=1$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=2$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=3$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=4$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=5$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_2=6$
1.	$2$	$82302.98$
2.	$3$	$97544.28$	$15241.29$
3.	$4$	$102878.7$	$20575.75$	$5334.45$
4.	$5$	$105347.8$	$23044.84$	$7803.542$	$2469.089$
5.	$6$	$106689.1$	$24386.07$	$9144.776$	$3810.323$	$1341.23$
6.	$7$	$107497.8$	$25194.79$	$9953.498$	$4619.045$	$2149.96$	$808.722$
7.	$8$	$108022.7$	$26079.55$	$10478.39$	$5143.936$	$2674.85$	$1333.61$
8.	$9$	$108382.5$	$26336.95$	$10838.25$	$5503.8$	$3034.71$	$1693.477$
9.	$10$	$108639.9$	$26527.41$	$11095.66$	$5761.209$	$3292.119$	$1950.886$
10.	$11$	$108830.4$	$26672.26$	$11286.11$	$5951.662$	$3482.572$	$2141.339$
11.	$12$	$108975.2$	$26784.99$	$11430.97$	$6096.517$	$3627.428$	-

Table 1

The table for the series limit for first six series of the hydrogen atom spectrum is shown below.

S.no.	$n_2$	Wavenumber $\left({\text{cm}}^{-1}\right)$$n_1=\infty$
1.	$1$	$109737.32$
2.	$2$	$27434.33$
3.	$3$	$12193.04$
4.	$4$	$6858.58$
5.	$5$	$4389.49$
6.	$6$	$3048.26$

Table 2