hydrogen\nuse the spectroscope to analyze the light emitted by the hydrogen lamp. (remember to record your…

Explanation:

Step1: Recall Rydberg equation

The Rydberg equation is $\frac{1}{\lambda}=R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$, where $\lambda$ is the wavelength, $R_H = 1.097\times10^{7}\ m^{- 1}$ is the Rydberg constant for hydrogen, $n_f$ is the final energy - level and $n_i$ is the initial energy - level.

For the violet line ($n_i = 5$, $n_f=2$):

Step2: Substitute values into the equation

$\frac{1}{\lambda}=1.097\times 10^{7}\left(\frac{1}{2^{2}}-\frac{1}{5^{2}}\right)$ $=1.097\times 10^{7}\left(\frac{1}{4}-\frac{1}{25}\right)$ $=1.097\times 10^{7}\left(\frac{25 - 4}{100}\right)$ $=1.097\times 10^{7}\times\frac{21}{100}$ $=1.097\times10^{7}\times0.21$ $ = 2.3037\times 10^{6}\ m^{-1}$

Step3: Solve for $\lambda$

$\lambda=\frac{1}{2.3037\times 10^{6}}\ m\approx434\times10^{-9}\ m = 434\ nm$

For the green line ($n_i = 4$, $n_f = 2$):

Step4: Substitute values into the equation

$\frac{1}{\lambda}=1.097\times 10^{7}\left(\frac{1}{2^{2}}-\frac{1}{4^{2}}\right)$ $=1.097\times 10^{7}\left(\frac{1}{4}-\frac{1}{16}\right)$ $=1.097\times 10^{7}\left(\frac{4 - 1}{16}\right)$ $=1.097\times 10^{7}\times\frac{3}{16}$ $= 2.056875\times 10^{6}\ m^{-1}$

Step5: Solve for $\lambda$

$\lambda=\frac{1}{2.056875\times 10^{6}}\ m\approx486\times 10^{-9}\ m=486\ nm$

For the red line ($n_i = 3$, $n_f = 2$):

Step6: Substitute values into the equation

$\frac{1}{\lambda}=1.097\times 10^{7}\left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right)$ $=1.097\times 10^{7}\left(\frac{1}{4}-\frac{1}{9}\right)$ $=1.097\times 10^{7}\left(\frac{9 - 4}{36}\right)$ $=1.097\times 10^{7}\times\frac{5}{36}$ $\approx1.5236\times 10^{6}\ m^{-1}$

Step7: Solve for $\lambda$

$\lambda=\frac{1}{1.5236\times 10^{6}}\ m\approx656\times 10^{-9}\ m = 656\ nm$

Answer:

1. $434\ nm$
2. $486\ nm$
3. $656\ nm$