Question

Name the transition in${\text{C}}^{5+}$that will lead to the absorption of green light.

Short answer

The probable transition in ${\text{C}}^{5+}$which will lead to the absorption of green light is $n_i=7$to $n_f=8$.

Where, ${\text{n}}_{\text{i}}\text{=}$lowest (initial) energy state and ${\text{n}}_{\text{f}}\text{=}$highest (final) energy state.

Step-by-step solution

Equation for the Frequency of Light Absorbance

The equation for the frequency of the light absorbed during the transition is:

$v=3.29\times {10}^5{s}^{-1}\times Z^2\times \left(\frac{1}{{\mathrm{n}}_{\mathrm{i}}^2}-\frac{1}{{\mathrm{n}}_{\mathrm{f}}^2}\right)$

Where, Z is an atomic number,$n_i$is initial energy state and$n_f$is the final energy state.

Calculate the frequency

If the${\text{C}}^{5+}$absorbs green light, whose wavelength is$550\text{nm}$, the transition will be from one to another state. To find the number of each state, first, calculate the frequency.

Before the calculation, convert m to nm. We know, $1\text{m}={10}^9\text{nm}$.

$550\text{nm}\times \frac{1\text{m}}{{10}^9\text{nm}}=5.5\times {10}^{-7}\text{m}$

Substitute the data for velocity.

The value of velocity is calculated in the following way:

$$\begin{gathered} \nu =\frac{c}{\lambda } \\ \nu =\frac{3\times {10}^8{\text{ms}}^{-1}}{5.5\times {10}^{-7}\text{m}} \\ \nu =5.45\times {10}^{14}{\text{s}}^{-1} \end{gathered}$$

Find out the ni

Equalize the expression when$Z=6$.

$$\begin{gathered} \nu =3.29\times {10}^{15}{\text{s}}^{-1}\times Z^2\times \left(\frac{1}{n_i^2}-\frac{1}{n_f^2}\right) \\ 5.45\times {10}^{14}{\text{s}}^{-1}=3.29\times {10}^{15}{\text{s}}^{-1}\times 6^2\times \left(\frac{1}{n_i^2}-\frac{1}{n_f^2}\right) \\ \left(\frac{1}{n_i^2}-\frac{1}{n_f^2}\right)=4.60\times {10}^{-3}{\text{s}}^{-1} \\ {n}_i=\sqrt{\frac{1}{4.60\times {10}^{-3}{\text{s}}^{-1}+\frac{1}{n_f^2}}} \end{gathered}$$

Finding out the nf

If we substitute numbers in$n_f$, we see the only transition that occurs is one, that is, from state 7 to state 8.

So, $n_i=7$and$n_f=8$