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(B) In the hydrogen emission spectrum,the wavelength $\lambda$ is given by the Rydberg formula: $\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{

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$\frac{n^2(n+1)^2}{R(2 n+1)}$

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(B) In the hydrogen emission spectrum,the wavelength $\lambda$ is given by the Rydberg formula: $\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$ For the maximum wavelength of a series,the energy difference between the levels must be minimum. This occurs for the transition from $n_2 = n+1$ to $n_1 = n$. Substituting these values into the formula: $\frac{1}{\lambda_{\max}} = R \left[ \frac{1}{n^2} - \frac{1}{(n+1)^2} \right]$ $= R \left[ \frac{(n+1)^2 - n^2}{n^2(n+1)^2} \right]$ $= R \left[ \frac{n^2 + 2n + 1 - n^2}{n^2(n+1)^2} \right]$ $= \frac{R(2n+1)}{n^2(n+1)^2}$ Therefore,the maximum wavelength is: $\lambda_{\max} = \frac{n^2(n+1)^2}{R(2n+1)}$

In the hydrogen emission spectrum,for any series,the principal quantum number of the higher energy level is $n+1$ and the lower energy level is $n$. The corresponding maximum wavelength $\lambda$ is ($R=$ Rydberg's constant).

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