One series of lines of the hydrogen spectrum is caused by emission of energy acc

Question

One series of lines of the hydrogen spectrum is caused by emission of energy accompanying the fall of an electron from outer shells to the third shell. The lines can be calculated using the Balmer-Rydberg equation: 1/lambda = R_infinity [1/m^2 - 1/n^2] where m = 3, R_infinity = 1.097 times 10^-2 nm^-1, and n is an integer greater than 3. Calculate the wavelengths in nanometers and energies in kilojoules per mole of the first two lines in the series. In what region of the electromagnetic spectrum do they fall?

Explanation / Answer

This is an example of Paschen series where m value is 3. The electrons can make transition from 4th level to 3rd level or infinite level to 3rd level. The first two lines will be transition from 4th level to 3rd level and transition from 5th level to 3rd level. The wavelengths corresponding to these transitions are given as

1/ = R nm-1 [1/m2 – 1/n2]

When n=4, 1 will be:

1/1 = 1.097 X 10-2 [1/32 – 1/42] nm-1

1/1 = 1.097 X 10-2 [1/9– 1/16] nm-1

1/1 = 1.097 X 10-2 [(16-9)/ 16•9] nm-1

1/1 = 1.097 X 10-2 [7/ 16•9] nm-1

1/1 = 5.332 X 10-4 nm-1

1 = 1/ 5.332 X 10-4 nm

1 = 1876 nm

E = E0 [1/m2 – 1/n2]

E0 = 13.6 eV

E1 = 13.6 eV [1/32 – 1/42]

E1 = 13.6 [7/ 16•9] eV

E1 = 0.6611 eV

1 eV = 96.485 kJ/ mol

E1 = 0.6611 X 96.485 kJ/ mol

E1 = 63.787 kJ/ mol

When n=5, 2 will be:

1/2 = 1.097 X 10-2 [1/32 – 1/52] nm-1

1/2 = 1.097 X 10-2 [1/9– 1/25] nm-1

1/2 = 1.097 X 10-2 [(25-9)/ 25•9] nm-1

1/2 = 1.097 X 10-2 [16/ 25•9] nm-1

1/2 = 7.801 X 10-4 nm-1

2 = 1/ 7.801 X 10-4 nm

2 = 1282 nm

E2 = 13.6 eV [1/32 – 1/52]

E2 = 13.6 [16/ 25•9] eV

E2 = 0.9671 eV

1 eV = 96.485 kJ/ mol

E2 = 0.9671 X 96.485 kJ/ mol

E2 = 93.312 kJ/ mol

So the wavelengths for the first two transitions of the Paschen series are 1876 nm and 1282 nm, respectively, and are in infrared region. The energies associated with these two transitions are 63.787 kJ/ mol and 93.312 kJ/ mol, respectively.

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