# Question #03297

Nov 21, 2017

${n}_{f} = 3$

#### Explanation:

Your tool of choice here will be the Rydberg equation, which allows you to calculate the wavelength of the photon emitted when an electron in a hydrogen atom makes an ${n}_{i} \to {n}_{f}$ transition.

$\frac{1}{l a m \mathrm{da}} = R \cdot \left(\frac{1}{n} _ {f}^{2} - \frac{1}{n} _ {i}^{2}\right)$

Here

• $R$ is the Rydberg constant, equal to $1.097 \cdot {10}^{7}$ ${\text{m}}^{- 1}$
• ${n}_{i}$ is the energy level from which the electron falls
• ${n}_{f}$ is the energy level to which the electron falls

In your case, you know that the electron falls from the seventh energy level, so

${n}_{i} = 7$

Your goal here is to find the value of the final energy level, ${n}_{f}$. Rearrange the Rydberg equation to solve for ${n}_{f}$.

$\frac{1}{l} a m \mathrm{da} = R \cdot \frac{{n}_{i}^{2} - {n}_{f}^{2}}{{n}_{i}^{2} \cdot {n}_{f}^{2}}$

${n}_{i}^{2} \cdot {n}_{f}^{2} = l a m \mathrm{da} \cdot R \cdot \left({n}_{i}^{2} - {n}_{f}^{2}\right)$

${n}_{i}^{2} \cdot {n}_{f}^{2} + l a m \mathrm{da} \cdot R \cdot {n}_{f}^{2} = l a m \mathrm{da} \cdot R \cdot {n}_{i}^{2}$

This is equivalent to

${n}_{f}^{2} \cdot \left({n}_{i}^{2} + l a m \mathrm{da} \cdot R\right) = l a m \mathrm{da} \cdot R \cdot {n}_{i}^{2}$

which gets you

${n}_{f} = \sqrt{\frac{{n}_{i}^{2} \cdot l a m \mathrm{da} \cdot R}{{n}_{i}^{2} + l a m \mathrm{da} \cdot R}}$

Now all you have to do is to plug in the values that you have--do not forget that the wavelength of the photon must be expressed in meters!

${n}_{f} = \sqrt{\left({7}^{2} \cdot 1005 \cdot {10}^{- 9} \textcolor{red}{\cancel{\textcolor{b l a c k}{{\text{m"))) * 1.097 * 10^7 color(red)(cancel(color(black)("m"^(-1)))))/(7^2 + 1005 * 10^(-9)color(red)(cancel(color(black)("m"))) * 10.97 * 10^7 color(red)(cancel(color(black)("m}}^{- 1}}}}\right)}$

${n}_{f} = 2.99983 \approx 3$

Therefore, you can say that a photon of wavelength $\text{1005 nM}$ is emitted when an electron in a hydrogen atom falls from ${n}_{i} = 7$ to ${n}_{f} = 3$, a transition that is part of the Paschen series. 