Question #2e787

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Question #2e787

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Answer 1 · 2017-10-06T00:05:56

$n_{i} = 8$ $n_i = 8$

Explanation:

Your tool of choice here will be the Rydberg equation, which looks like this

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

Here

$λ$ $lamda$ is the wavelength of the photon

$R$ $R$ 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 transition

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

Now, you know that the electron starts on an initial energy level $n_{i}$ $n_i$ and ends up on the third energy level, so

$n_{f} = 3$ $n_f = 3$

Moreover, you know that this transition is accompanied by the emission of a photon of wavelength

$λ = 9.54 \cdot 10^{- 7}$ $lamda = 9.54 * 10^(-7)$ $m$ $"m"$

Rearrange the Rydberg equation to solve for $n_{i}$ $n_i$

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

This is equivalent to

$n_{i}^{2} \cdot n_{f}^{2} = λ \cdot R \cdot n_{i}^{2} - λ \cdot R \cdot n_{f}^{2}$ $n_i^2 * n_f^2 = lamda * R * n_i^2 - lamda * R * n_f^2$

$λ \cdot R \cdot n_{i}^{2} - n_{i}^{2} \cdot n_{f}^{2} = λ \cdot R \cdot n_{f}^{2}$ $lamda * R * n_i^2 - n_i^2 * n_f^2 = lamda * R * n_f^2$

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

Finally, you should end up with

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

Plug in your values to find

$n_i = sqrt( (9.54 * color(blue)(cancel(color(black)(10^(-7))))color(red)(cancel(color(black)("m"))) * 1.097 * color(blue)(cancel(color(black)(10^7)))color(red)(cancel(color(black)("m"^(-1)))) * 3^2)/(9.54 * color(blue)(cancel(color(black)(10^(-7))))color(red)(cancel(color(black)("m"))) * 1.097 * color(blue)(cancel(color(black)(10^7)))color(red)(cancel(color(black)("m"^(-1)))) - 3^2))$

$n_ i = 8.017 ~~ color(darkgreen)(ul(color(black)(8)))$

Therefore, you can say that this electron underwent a $n_i = 8 -> n_f = 3$ transition.

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Question #2e787

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