Question

What about the winding of the frauliens $3\to 2.4\to 2$, and $5\to 2$in the hydroponics ${\mathrm{O}}^{-7}$In what spectral range do these lie?

Short answer

${\lambda }_{32}=10.26\mathrm{nm};{\lambda }_{42}=7.6\mathrm{nm};{\lambda }_{52}=6.79\mathrm{nm}$

Step-by-step solution

Step: 1 Given information

The given hydronics are${\mathrm{O}}^{-7}$

Calculation

$Wecanbeginbyexpres\sin genergylevelsasfollows:\$\$\backslash begin\left\{aligned\right\}E\_\left\{n\right\}\&=-\backslash frac\{13.6Z^\{2\left\}\right(eV\left)\right\}\{n^\{2\left\}\right\}E\_\left\{3\right\}-E\_\left\{2\right\}=-13.6Z^\left\{2\right\}(eV)\backslash cdot\backslash left(\backslash frac\{1\left\}\right\{3^\left\{2\right\}\}-\backslash frac\{1\left\}\right\{2^\left\{2\right\}\}\backslash right)\backslash \backslash E\_\left\{3\right\}-E\_\left\{2\right\}=\backslash frac\{hc\}\{\backslash lambda\_\{32\left\}\right\}\backslash \backslash \backslash lambda\_\left\{32\right\}=\backslash frac\{hc\}\left\{E\_\right\{3\}-E\_\{2\left\}\right\}\backslash \backslash \backslash lambda\_\left\{32\right\}=\backslash frac\{-13.6\backslash cdot8^\{2\}\backslash cdot1.6\backslash cdot10^\{-19\}\backslash cdot\backslash left(\backslash frac\left\{1\right\}\{3^\{2\left\}\right\}-\backslash frac\left\{1\right\}\{2^\{2\left\}\right\}\backslash right\left)\right\}\{10.26\backslash mathrm\{~nm\left\}\right\}\backslash \backslash \backslash lambda\_\left\{32\right\}\&=10^\{-34\}\backslash cdot3\backslash cdot10^\left\{8\right\}\backslash end\left\{aligned\right\}Wecanbeginbyexpres\sin genergylevelsasfollows:\$\$\backslash begin\left\{aligned\right\}E\_\left\{n\right\}\&=-\backslash frac\{13.6Z^\{2\left\}\right(eV\left)\right\}\{n^\{2\left\}\right\}E\_\left\{3\right\}-E\_\left\{2\right\}=-13.6Z^\left\{2\right\}(eV)\backslash cdot\backslash left(\backslash frac\{1\left\}\right\{3^\left\{2\right\}\}-\backslash frac\{1\left\}\right\{2^\left\{2\right\}\}\backslash right)\backslash \backslash E\_\left\{3\right\}-E\_\left\{2\right\}=\backslash frac\{hc\}\{\backslash lambda\_\{32\left\}\right\}\backslash \backslash \backslash lambda\_\left\{32\right\}=\backslash frac\{hc\}\left\{E\_\right\{3\}-E\_\{2\left\}\right\}\backslash \backslash \backslash lambda\_\left\{32\right\}=\backslash frac\{-13.6\backslash cdot8^\{2\}\backslash cdot1.6\backslash cdot10^\{-19\}\backslash cdot\backslash left(\backslash frac\left\{1\right\}\{3^\{2\left\}\right\}-\backslash frac\left\{1\right\}\{2^\{2\left\}\right\}\backslash right\left)\right\}\{10.26\backslash mathrm\{~nm\left\}\right\}\backslash \backslash \backslash lambda\_\left\{32\right\}\&=10^\{-34\}\backslash cdot3\backslash cdot10^\left\{8\right\}\backslash end\left\{aligned\right\}$uncaught exception: Invalid chunk

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$E_n=-\frac{13.6Z^2\left(eV\right)}{n^2}$$E_3-E_2=-13.6Z^2\left(eV\right)·\left(\frac{1}{3^2}-\frac{1}{2^2}\right)$

$E_3-E_2=\frac{hc}{{\lambda }_{32}}$

$$\begin{gathered} {\lambda }_{32}=\frac{hc}{E_3-E_2} \\ {\lambda }_{32}=\frac{-13.6·8^2·1.6·{10}^{-19}·\left(\frac{1}{3^2}-\frac{1}{2^2}\right)}{10.26\mathrm{nm}} \\ {\lambda }_{32}={10}^{-34}·3·{10}^8 \end{gathered}$$

In the same approach, we can determine out $4\to 2$and $5\to 2$transitions:

$$\begin{gathered} {\lambda }_{42}=\frac{hc}{E_4-E_2} \\ {\lambda }_{42}=\frac{6.63·{10}^{-34}·3·{10}^8}{-13.6·8^2·1.6·{10}^{-19}·\left(\frac{1}{4^2}-\frac{1}{2^2}\right)} \end{gathered}$$

$$\begin{gathered} {\lambda }_{42}=7.6\mathrm{nm} \\ {\lambda }_{52}=\frac{hc}{E_5-E_2} \\ {\lambda }_{52}=\frac{6.63·{10}^{-34}·3·{10}^8}{-13.6·8^2·1.6·{10}^{-19}·\left(\frac{1}{5^2}-\frac{1}{2^2}\right)} \\ {\lambda }_{52}=6.79\mathrm{nm} \end{gathered}$$