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(C) The frequency of a spectral line in the hydrogen atom is given by $v = R c Z^2 (1/n_f^2 - 1/n_i^2)$.For the Balmer series limit,$n_f = 2$ and $n_i

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$v_1 - v_2 = v_3$

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(C) The frequency of a spectral line in the hydrogen atom is given by $v = R c Z^2 (1/n_f^2 - 1/n_i^2)$.For the Balmer series limit,$n_f = 2$ and $n_i = \infty$,so $v_1 = R c (1/2^2 - 1/\infty^2) = R c / 4$.For the Paschen series limit,$n_f = 3$ and $n_i = \infty$,so $v_3 = R c (1/3^2 - 1/\infty^2) = R c / 9$.For the first line of the Balmer series,$n_f = 2$ and $n_i = 3$,so $v_2 = R c (1/2^2 - 1/3^2) = R c (1/4 - 1/9) = R c (5/36)$.Comparing the values: $v_1 - v_3 = R c (1/4 - 1/9) = R c (5/36) = v_2$.Thus,the correct relation is $v_1 - v_3 = v_2$.

The frequencies for the series limit of the Balmer and Paschen series are $v_1$ and $v_3$ respectively. If the frequency of the first line of the Balmer series is $v_2$,then the relation between $v_1, v_2,$ and $v_3$ is:

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