Question

Which transition emits photons of the minimum wavelength? `n = 4->1` or `n = 2->1`?

`a)` `Deltan = 4->1`
`b)` `Deltan = 1->2`
`c)` `Deltan = 3->2`
`d)` `Deltan = 2->1`

I know it's not `(b)` or `(c)`, since `(b)` is an absorption, and `(c)` and `(d)` seem to be almost the same???

Answer 1

The energy difference for each electronic transition must be quantized according to

`DeltaE = E_"photon" = hnu = (hc)/lambda`,

where:

• `DeltaE` is the difference in energy between two chosen quantum levels.
• `E_"photon"` is the energy of the photon emitted when an electron relaxes down from a higher quantum level to a lower one.
• `h = 6.626 xx 10^(-34)` `"J"cdot"s"` is Planck's constant.
• `nu` is the frequency of the emitted photon in `"s"^(-1)`.
• `lambda` is its wavelength.
• `c = 2.998 xx 10^8` `"m/s"` is the speed of light.

Since the energy difference, `DeltaE` is inversely proportional to `lambda`, i.e.

`DeltaE prop 1/lambda`,

we should be able to immediately say that if we search for the minimum wavelength, we want the maximum energy difference.

The maximum energy difference is not all that tricky here. The ending point of each transition shown is the same, and only the starting point differs.

Here I show the relative energy differences (almost to scale):

`ul(color(white)(uarr darr))`
`n = 4`

`ul(color(white)(uarr darr))`
`n = 3`
`" "`
`" "`
`" "`
`" "`
`ul(color(white)(uarr darr))`
`n = 2`
`" "`
`" "`
`" "`
`" "`
`" "`
`" "`
`" "`
`" "`
`" "`
`" "`
`" "`
`ul(color(white)(uarr darr))`
`n = 1`

So, can you now say which transition involves a larger energy difference? `n = 4->1`, or `n = 2->1`?