Question

He atom can be excited to `1s^1 2p^1` by `lambda=58.44 nm`. If lowest excited state for He lies `4857 cm^(-1)` below the above. Calculate the energy for lower excitation state?

Answer 1

`E^"*"(""^1 S) ~~ "166259 cm"^(-1)`

for the `1s^1 2s^1` configuration.

`ul(uarr color(white)(darr))`
`2s`

`ul(uarr color(white)(darr))`
`1s`

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We are told that `"He"` can be excited from `1s^2` to `1s^1 2p^1` by a `"58.44 nm"` excitation source.

`ul(color(white)(uarr darr))" "ul(color(white)(uarr darr))" "ul(color(white)(uarr darr))" "" "" "ul(color(red)(uarr) color(white)(darr))" "ul(color(white)(uarr darr))" "ul(color(white)(uarr darr))`
`underbrace(" "" "" "" "" "" "" "" ")" "" "" "underbrace(" "" "" "" "" "" "" "" ")`
`" "" "" "2p" "" "" "" "" "" "" "" "" "" "" "2p`

`ul(color(white)(uarr darr))" "" "" "" "" "" "" "" "" "ul(color(white)(uarr darr))`
`2s" "" "" "" "" "=>" "" "" "" "2s`

`ul(uarr color(red)(darr))` `" "" "" "" "" "" "" "" "" "` `ul(uarr color(white)(darr))`
`1s" "" "" "" "" "" "" "" "" "" "" "1s`

By inspection of the above energy level diagram, indeed it can. That is a diagonal excitation as seen above (legal by the selection rules), going from the ground-state `""^1 S` to the `""^1 P` state, i.e. from `E(""^1 S) = "0.000 cm"^(-1)` to `E(""^1 P) = "171134.897 cm"^(-1)`:

The state `"4857.457 cm"^(-1)` below is the `1s^1 2s^1` configuration, or the excited state `""^1 S`, with energy `E^"*"(""^1 S) ~~ "166277.440 cm"^(-1)`.

`ul(uarr color(white)(darr))`
`2s`

`ul(uarr color(white)(darr))`
`1s`

That is the state circled in red in the `""^1 S` column in the energy level diagram above.

So, we know what we should get. Let's get it the normal way now.

`E(""^1 P) = 1/(58.44 cancel"nm") xx (10^9 cancel"nm")/(cancel"1 m") xx cancel"1 m"/"100 cm"`

`= "171115.674 cm"^(-1)`

And we know that the lower-excited state that the electron relaxes to has energy

`E(""^1 P) - "4857 cm"^(-1)`

So, the energy of the lower excited state with your level of precision is:

`= ul(color(blue)("166259 cm"^(-1) ~~ E^"*"(""^1 S)))`,

keeping zero decimal places.