# which emission line in the hydrogen spectrum occurs at the highest frequency?

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n=3 —> n=1

n=4 —> n=2

n=7 —> n=5

n=10 —> n=8

Can you explain how you get the answer? Please and thank you in advance

• n=3 —> n=1

The explanation requires a number of steps…

The highest frequency means the biggest photon energy (E=hf).

All the junps in the list of choices involve dropping down 2 levels.

The biggest photon energy means then biggest gap (energy difference) between the 2 energy levels.

The gap between energy levels gets smaller as n gets bigger – see diagram in link.

So the biggest gap is between the smallest n values.

The smallest n values are n=3 —> n=1.

Look at the diagram carefully and it should be clear.

• Photon energy = energy of the difference between two energy levels. But you also know the photon energy E is

E = hf where h = planck’s constant and f = frequency.

Now the energy difference between two states is DE = -E0*(1/n1^2 – 1/n2^2) where E0 = ground state

So the photon energy is DE and the photon frequency is

f =- (E0/h)(1/n1^2 – 1/n2^2) –> so all that matters is how big the number 1/n1^2 – 1/n2^2 is. So let’s make a table

n1 n2 -(1/n1^2 – 1/n2^2)

3 1 8/9

4 2 3/16

7 5 24/1225

10 8 9/1600

Clear the transition from n=3 to n=1 has an energy closest to the ground state so that is the transition whose photon has the highest frequency

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