The energy levels of a hypothetical atom are shown in the given figure. Let $\lambda_X$ , $\lambda_Y$ , and $\lambda_Z$ be the wavelengths of the photons emitted during the transitions $X$ , $Y$ , and $Z$ respectively. Which of the following mathematical relations is correct?

Question

The energy levels of a hypothetical atom are shown in the given figure. Let

\lambda_X

,

\lambda_Y

, and

\lambda_Z

be the wavelengths of the photons emitted during the transitions

X

,

Y

, and

Z

respectively. Which of the following mathematical relations is correct?

Quantrex Academy · Accepted Answer

Correct answer: A. λ_X = 2λ_Y

The energy of a photon emitted during a transition is given by

\Delta E = \dfrac{hc}{\lambda}

.

For transition X (from

-E

to

-3E

):

\Delta E_X = -E - (-3E) = 2E

\Rightarrow \dfrac{hc}{\lambda_X} = 2E \Rightarrow \lambda_X = \dfrac{hc}{2E}

For transition Y (from

-3E

to

-7E

):

\Delta E_Y = -3E - (-7E) = 4E

\Rightarrow \dfrac{hc}{\lambda_Y} = 4E \Rightarrow \lambda_Y = \dfrac{hc}{4E}

For transition Z (from

-E

to

-7E

):

\Delta E_Z = -E - (-7E) = 6E

\Rightarrow \dfrac{hc}{\lambda_Z} = 6E \Rightarrow \lambda_Z = \dfrac{hc}{6E}

Comparing

\lambda_X

and

\lambda_Y

:

\dfrac{\lambda_X}{\lambda_Y} = \dfrac{\dfrac{hc}{2E}}{\dfrac{hc}{4E}} = \dfrac{4E}{2E} = 2

\Rightarrow \lambda_X = 2\lambda_Y

Answer:

\lambda_X = 2\lambda_Y

The energy levels of a hypothetical atom are shown in the given...