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(A) The energy difference between two states is given by $\Delta E = \frac{hc}{\lambda}$.For the transition from state $A$ to state $C$:$E_A - E_C = \

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$3000 \ \mathring A$

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(A) The energy difference between two states is given by $\Delta E = \frac{hc}{\lambda}$.For the transition from state $A$ to state $C$:$E_A - E_C = \frac{hc}{2000 \ \mathring A} \quad \dots (i)$For the transition from state $B$ to state $C$:$E_B - E_C = \frac{hc}{6000 \ \mathring A} \quad \dots (ii)$For the transition from state $A$ to state $B$,the energy difference is:$E_A - E_B = (E_A - E_C) - (E_B - E_C)$Substituting the values from equations $(i)$ and $(ii)$:$\frac{hc}{\lambda_{AB}} = \frac{hc}{2000 \ \mathring A} - \frac{hc}{6000 \ \mathring A}$$\frac{1}{\lambda_{AB}} = \frac{3 - 1}{6000 \ \mathring A} = \frac{2}{6000 \ \mathring A} = \frac{1}{3000 \ \mathring A}$Therefore,$\lambda_{AB} = 3000 \ \mathring A$.

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