Chapter 4, Problem 4.66P

Interpretation Introduction

Interpretation:

Frequency of the first five lines in Lyman series has to be calculated.

Explanation of Solution

Frequency of First line:

Energy of photon emitted on relaxation from $n=2$ state to $n=1$ state in first line of Lyman series and is calculated as shown below.

    $\begin{array}{c}{E}_{photon}=\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{n_f{}^2}-\frac{1}{n_i{}^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1^2}-\frac{1}{2^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1}-\frac{1}{4}\right)\\ =1.6349\times {10}^{-18}\text{J}\end{array}$

Frequency of photon is calculated as shown below.

    $\begin{array}{c}E=hv\\ v=\frac{E}{h}\\ =\frac{1.6349\times {10}^{-18}\text{J}}{6.626\times {10}^{-34}\text{J}\cdot \text{s}}\\ =\text{0}\text{.246740114699668}\times {10}^{16}{\text{s}}^{\text{-1}}\\ =2.467\times {10}^{15}{\text{s}}^{-1}\end{array}$

Therefore, the frequency of photon is calculated as $2.467\times {10}^{15}{\text{s}}^{-1}$.

Wavelength of photon is calculated as shown below;

    $\begin{array}{c}\lambda =\frac{c}{v}\\ =\frac{2.9979\times {10}^8{\text{ms}}^{-1}}{2.467\times {10}^{15}{\text{s}}^{-1}}\\ =1.215\times {10}^{-7}\text{m}\\ =121.5\times {10}^{-7}\text{cm}\end{array}$

Therefore, wavelength of photon is $121.5\times {10}^{-7}\text{cm}$.

Frequency of Second line:

Energy of photon emitted on relaxation from $n=3$ state to $n=1$ state in second line of Lyman series and is calculated as shown below.

    $\begin{array}{c}{E}_{photon}=\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{n_f{}^2}-\frac{1}{n_i{}^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1^2}-\frac{1}{3^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1}-\frac{1}{9}\right)\\ =1.9377\times {10}^{-18}\text{J}\end{array}$

Frequency of photon is calculated as shown below.

    $\begin{array}{c}E=hv\\ v=\frac{E}{h}\\ =\frac{1.9377\times {10}^{-18}\text{J}}{6.626\times {10}^{-34}\text{J}\cdot \text{s}}\\ =\text{0}\text{.2924388771506188}\times {10}^{16}{\text{s}}^{\text{-1}}\\ =2.924\times {10}^{15}{\text{s}}^{-1}\end{array}$

Therefore, the frequency of photon is calculated as $2.924\times {10}^{15}{\text{s}}^{-1}$.

Wavelength of photon is calculated as shown below;

    $\begin{array}{c}\lambda =\frac{c}{v}\\ =\frac{2.9979\times {10}^8{\text{ms}}^{-1}}{2.924\times {10}^{15}{\text{s}}^{-1}}\\ =1.025\times {10}^{-7}\text{m}\\ =102.5\times {10}^{-7}\text{cm}\end{array}$

Therefore, wavelength of photon is $102.5\times {10}^{-7}\text{cm}$.

Frequency of Third line:

Energy of photon emitted on relaxation from $n=4$ state to $n=1$ state in third line of Lyman series and is calculated as shown below.

    $\begin{array}{c}{E}_{photon}=\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{n_f{}^2}-\frac{1}{n_i{}^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1^2}-\frac{1}{4^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1}-\frac{1}{16}\right)\\ =2.0436\times {10}^{-18}\text{J}\end{array}$

Frequency of photon is calculated as shown below.

    $\begin{array}{c}E=hv\\ v=\frac{E}{h}\\ =\frac{2.0436\times {10}^{-18}\text{J}}{6.626\times {10}^{-34}\text{J}\cdot \text{s}}\\ =\text{0}\text{.3084213703591911}\times {10}^{16}{\text{s}}^{\text{-1}}\\ =3.08\times {10}^{15}{\text{s}}^{-1}\end{array}$

Therefore, the frequency of photon is calculated as $3.08\times {10}^{15}{\text{s}}^{-1}$.

Wavelength of photon is calculated as shown below;

    $\begin{array}{c}\lambda =\frac{c}{v}\\ =\frac{2.9979\times {10}^8{\text{ms}}^{-1}}{3.08\times {10}^{15}{\text{s}}^{-1}}\\ =0.9733\times {10}^{-7}\text{m}\\ =97.33\times {10}^{-7}\text{cm}\end{array}$

Therefore, wavelength of photon is $97.33\times {10}^{-7}\text{cm}$.

Frequency of Fourth line:

Energy of photon emitted on relaxation from $n=5$ state to $n=1$ state in fourth line of Lyman series and is calculated as shown below.

    $\begin{array}{c}{E}_{photon}=\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{n_f{}^2}-\frac{1}{n_i{}^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1^2}-\frac{1}{5^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1}-\frac{1}{25}\right)\\ =2.0927\times {10}^{-18}\text{J}\end{array}$

Frequency of photon is calculated as shown below.

    $\begin{array}{c}E=hv\\ v=\frac{E}{h}\\ =\frac{2.0927\times {10}^{-18}\text{J}}{6.626\times {10}^{-34}\text{J}\cdot \text{s}}\\ =\text{0}\text{.3158315725928162}\times {10}^{16}{\text{s}}^{\text{-1}}\\ =3.16\times {10}^{15}{\text{s}}^{-1}\end{array}$

Therefore, the frequency of photon is calculated as $3.16\times {10}^{15}{\text{s}}^{-1}$.

Wavelength of photon is calculated as shown below;

    $\begin{array}{c}\lambda =\frac{c}{v}\\ =\frac{2.9979\times {10}^8{\text{ms}}^{-1}}{3.16\times {10}^{15}{\text{s}}^{-1}}\\ =0.9487\times {10}^{-7}\text{m}\\ =94.87\times {10}^{-7}\text{cm}\end{array}$

Therefore, wavelength of photon is $94.87\times {10}^{-7}\text{cm}$.

Frequency of Fifth line:

Energy of photon emitted on relaxation from $n=6$ state to $n=1$ state in fifth line of Lyman series and is calculated as shown below.

    $\begin{array}{c}{E}_{photon}=\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{n_f{}^2}-\frac{1}{n_i{}^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1^2}-\frac{1}{6^2}\right)\\ =\left(2.1799\times {10}^{-18}\text{J}\right)\left(\frac{1}{1}-\frac{1}{36}\right)\\ =2.1193\times {10}^{-18}\text{J}\end{array}$

Frequency of photon is calculated as shown below.

    $\begin{array}{c}E=hv\\ v=\frac{E}{h}\\ =\frac{2.1193\times {10}^{-18}\text{J}}{6.626\times {10}^{-34}\text{J}\cdot \text{s}}\\ =\text{0}\text{.3198460609719288}\times {10}^{16}{\text{s}}^{\text{-1}}\\ =3.19\times {10}^{15}{\text{s}}^{-1}\end{array}$

Therefore, the frequency of photon is calculated as $3.19\times {10}^{15}{\text{s}}^{-1}$.

Wavelength of photon is calculated as shown below;

    $\begin{array}{c}\lambda =\frac{c}{v}\\ =\frac{2.9979\times {10}^8{\text{ms}}^{-1}}{3.19\times {10}^{15}{\text{s}}^{-1}}\\ =0.9397\times {10}^{-7}\text{m}\\ =93.97\times {10}^{-7}\text{cm}\end{array}$

Therefore, wavelength of photon is $93.97\times {10}^{-7}\text{cm}$.

Summary of the wavelength of first five lines in Lyman series is given below.

$n_i$	2	3	4	5	6
$\text{Wavelength/}\times {\text{10}}^{\text{-7}}\text{ cm}$	$121.5$	$102.5$	$97.33$	$94.87$	$93.97$

From the above table, in order to plot graph, the wavenumber and $1/n_i{}^2$ is calculated and summarized as shown below;

$1/n_i{}^2$	$0.25$	$0.11$	$0.06$	$0.04$	$0.03$
${\text{Wavenumber (cm}}^{\text{-1}}\text{)}$	$82304$	$97560$	$102743$	$105407$	$106416$

Graph is plotted considering wavenumber in $y$ axis and $1/n_i{}^2$ in $x$ axis as shown;

From the graph that is plotted as shown, the slope of the graph is $109339{\text{cm}}^{-1}$ and this value is approximately equal to the Rydberg constant.