Energy levels A, B, C of a certain atom corresponds to increasing value of energy $E_A< E_B< E_C$. If $λ_1,λ_2,λ_3$ are the wavelengths of radiation corresponding to the transitions C → B, B → A and C → A respectively then:

${λ_3}=\frac{λ_1λ_2}{λ_1+λ_2}$

The correct answer is Option (4) → ${λ_3}=\frac{λ_1λ_2}{λ_1+λ_2}$

The energy (E) is,

$E=\frac{hc}{λ}$

for the transition,

$C→B$, the energy different $E_{C→B}=E_C-E_B$

$B→A$, the energy different $E_{B→A}=E_B-E_A$

$C→A$, the energy different $E_{C→A}=E_C-E_A$

$λ_1=\frac{hc}{E_{C→B}},λ_2=\frac{hc}{E_{B→A}},λ_3=\frac{hc}{E_{C→A}}$

Now,

$E_{C→A}=E_{C→B}+E_{B→A}$

$⇒\frac{1}{λ_3}=\frac{1}{λ_1}+\frac{1}{λ_2}$

$⇒λ_3=\frac{λ_1λ_2}{λ_1+λ_2}$