Answer :
The shortest wavelength limit of the Balmer series for the hydrogen spectrum, based on the Rydberg formula, is 364.6 nm. This lies in the ultraviolet region of the electromagnetic spectrum.
Calculating the Shortest Wavelength of the Balmer Series
The question asks for the calculation of the shortest wavelength limit for the Balmer series in the hydrogen spectrum. Given that the shortest wavelength limit for the Lyman series is 913, we can use the relation of the series limits based on the Rydberg formula to find the corresponding limit for the Balmer series.
The Balmer series limit is obtained when n = 2 (the lower energy level) and the electron transitions are from a higher energy level to n = 2.Using the Rydberg formula for the shortest wavelength (series limit):R_{H} \times \left( \frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}} \right), where R_{H} is the Rydberg constant, n_{1} is the lower energy level, and n_{2} is infinity for the series limit. For the Balmer series, n_{1} = 2 and n_{2} = \infty.
The Rydberg constant in terms of wavelength is approximately 1.097 x 10^7 m-1.Thus, the series limit for the Balmer series is given by:R_{H} \times \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) = R_{H} \times \frac{1}{4}Substituting the values, we get:1.097 \times 10^7 m^{-1} \times \frac{1}{4} = 2.7425 \times 10^6 m^{-1}.Converting this wavenumber to wavelength (\lambda = 1/\nu):\lambda = \frac{1}{2.7425 \times 10^6 m^{-1}} = 364.6 nm
This value corresponds to the series limit of the Balmer series which lies in the ultraviolet region of the electromagnetic spectrum.
