What will be the ratio of the wavelength of the first line to that of the second line of paschen series of H atom? A)256:175 B)175:256 C)15:16 D)24:27

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What will be the ratio of the wavelength of the first line to that of the second line of paschen series of H atom? A)256:175 B)175:256 C)15:16 D)24:27

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Answer 1 · 2017-09-30T00:33:33

The answer is (A) $256 : 175$ $256:175$

Explanation:

Your tool of choice here will be the Rydberg equation, which tells you the wavelength, $λ$ $lamda$ , of the photon emitted by an electron that makes a $n_{i} \to n_{f}$ $n_i -> n_f$ transition in a hydrogen atom.

$\frac{1}{λ} = R \cdot (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}})$ $1/(lamda) = R * (1/n_f^2 - 1/n_i^2)$

Here

$R$ $R$ is the Rydberg constant, equal to $1.097 \cdot 10^{7}$ $1.097 * 10^(7)$ $m^{- 1}$ $"m"^(-1)$

$n_{i}$ $n_i$ is the initial energy level of the electron

$n_{f}$ $n_f$ is the final energy level of the electron

Now, the Paschen series is characterized by $n_{f} = 3$ $n_f = 3$ .

![https://thecuriousastronomer.wordpress.com/2013/08/20/emission-line-spectra/]( useruploads.socratic.org )

The first transition in the Paschen series corresponds to

$n_{i} = 4 \to n_{f} = 3$ $n_i = 4" " -> " " n_f = 3$

In this transition, the electron drops from the fourth energy level to the third energy level.

You will have

$\frac{1}{λ_{1}} = R \cdot (\frac{1}{3^{2}} - \frac{1}{4^{2}})$ $1/(lamda_1) = R * (1/3^2 - 1/4^2)$

The second transition in the Paschen series corresponds to

$n_{i} = 5 \to n_{f} = 3$ $n_i = 5 " " -> " " n_f = 3$

This time, you have

$\frac{1}{λ_{2}} = R \cdot (\frac{1}{3^{2}} - \frac{1}{5^{2}})$ $1/(lamda_2) = R * (1/3^2 - 1/5^2)$

Now, to get the ratio of the first line to that of the second line, you need to divide the second equation by the first one.

$(1/(lamda_2))/(1/(lamda_1)) = (color(red)(cancel(color(black)(R))) * (1/3^2 - 1/4^2))/(color(red)(cancel(color(black)(R))) * (1/3^2 - 1/5^2))$

This will be equivalent to

$(lamda_1)/(lamda_2) =(1/9 - 1/25)/ (1/9 - 1/16)$

$(lamda_1)/(lamda_2) = ((25 - 9)/(25 * 9))/((16 - 9)/(16 * 9)) = 16/(25 * color(red)(cancel(color(black)(9)))) * (16 * color(red)(cancel(color(black)(9))))/7 = 16^2/(25 * 7)$

Therefore, you can say that you have

$(lamda_1)/(lamda_2) = 256/175$

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What will be the ratio of the wavelength of the first line to that of the second line of paschen series of H atom? A)256:175 B)175:256 C)15:16 D)24:27

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