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(C) The frequency of a spectral line in the hydrogen atom is given by the formula: $
u = Rc \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} ight)$.For the

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$\frac{9 R c}{400}$

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(C) The frequency of a spectral line in the hydrogen atom is given by the formula: $
u = Rc \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} ight)$.For the Paschen series,the lower energy level is $n_1 = 3$.The first Paschen line corresponds to the transition from $n_2 = 4$ to $n_1 = 3$. Its frequency is $
u_1 = Rc \left( \frac{1}{3^2} - \frac{1}{4^2} ight) = Rc \left( \frac{1}{9} - \frac{1}{16} ight) = Rc \left( \frac{16 - 9}{144} ight) = \frac{7Rc}{144}$.The second Paschen line corresponds to the transition from $n_2 = 5$ to $n_1 = 3$. Its frequency is $
u_2 = Rc \left( \frac{1}{3^2} - \frac{1}{5^2} ight) = Rc \left( \frac{1}{9} - \frac{1}{25} ight) = Rc \left( \frac{25 - 9}{225} ight) = \frac{16Rc}{225}$.The difference between the frequencies is $\Delta 
u = 
u_2 - 
u_1 = Rc \left( \frac{16}{225} - \frac{7}{144} ight)$.Calculating the common denominator: $225 = 3^2 	imes 5^2$ and $144 = 3^2 	imes 4^2$. The $LCM$ is $3^2 	imes 5^2 	imes 4^2 = 9 	imes 25 	imes 16 = 3600$.$\Delta 
u = Rc \left( \frac{16 	imes 16 - 7 	imes 25}{3600} ight) = Rc \left( \frac{256 - 175}{3600} ight) = Rc \left( \frac{81}{3600} ight) = \frac{9Rc}{400}$.

The difference between the frequencies of the second and first Paschen lines of the hydrogen atom is (where $R$ is the Rydberg constant and $c$ is the speed of light in vacuum).

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