In the following transitions, which one has higher frequency?

Question

In the following transitions, which one has higher frequency? (A)  3→ 1  (B)  4→ 2  (C)  4→ 3  (D)  3→ 2

Tardigrade Admin · Accepted Answer

From Bohr's postulate, energy of electron in

n t h

orbit is given by

E = - \frac{M Z ^{2} e ^{4}}{8 ε _{0}^{2} h ^{2}} (\frac{1}{n ^{2}})

When electron jumps from some higher energy state

n_{2}

to a lower energy state

n_{1}

, the energy difference between these states is

E_{2} - E_{1} \propto (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})

From Bohr's third postulate, the frequency

v

of the electromagnetic wave is

v = \frac{E _{2} - E _{1}}{h} \propto (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})

First case

n_{1} = 1, n_{2} = 3

∴ v_{1} \propto (1 - \frac{1}{9}) \propto \frac{8}{9}

Second case

n_{1} = 2, n_{2} = 4

∴ v_{2} \propto (\frac{1}{4} - \frac{1}{16}) \propto \frac{3}{16}

Third case

n_{1} = 3, n_{2} = 4

∴ v_{3} \propto (\frac{1}{9} - \frac{1}{16}) \propto \frac{7}{144}

Fourth case

n_{1} = 2, n_{2} = 3

∴ v_{4} \propto (\frac{1}{4} - \frac{1}{9}) \propto \frac{5}{36}

v_{1} > v_{2} > v_{4} > v_{3}

Hence, transition

3 \to 1

has higher frequency.

Q. In the following transitions, which one has higher frequency?