Question

What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?

Short answer

The smallest wavelength line in the Balmer series is 364.63 nm and this does not lie in the visible part of spectrum.

Step-by-step solution

What is balmer series?

The Balmer series refers to a set of spectral emission lines produced by electron transitions from higher energy levels to the energy level with main quantum number 2 in the hydrogen atom.

Determine the formula

Consider the formula for the Bohr's theory of hydrogen atom.

\(\frac{1}{\lambda } = R\left( {\frac{1}{{n_f^2}} - \frac{1}{{n_i^2}}} \right)\)

Here, wavelength of the emitted electromagnetic radiation is \(\lambda \), and the Rydberg constant is \(R = 1.097 \times {10^7}\;{{\rm{m}}^{ - 1}}\).

Determine the smallest wavelength line in the Balmer series. And also, if it is in the visible part of the spectrum.

Consider the given data:

\(\begin{array}{}{n_i} = \infty \\{n_f} = 2\\R = 1.097 \times {10^7}\;{{\rm{m}}^{ - 1}}\end{array}\)

Substitute the values and solve for the wavelength as follows:

\(\begin{array}{c}\frac{1}{\lambda } = 1.097 \times {10^7}\;\;{{\rm{m}}^{ - 1}}\left( {\frac{1}{{{2^2}}} - \frac{1}{{\;{\infty ^2}}}} \right)\\\frac{1}{\lambda } = 1.097 \times {10^7}\;\;{{\rm{m}}^{ - 1}}\left( {\frac{1}{4} - 0} \right)\\\frac{1}{\lambda } = 2742500\;{{\rm{m}}^{ - 1}}\\\lambda = 3.6463 \times {10^{ - 7}}\;{\rm{m}}\\\lambda = 364.63\;\;{\rm{nm}}\end{array}\)

The obtained wavelength is in the ultraviolet (UV) spectrum.

Hence, the smallest wavelength line in the Balmer series is 364.63 nm and this does not lie in the visible part of spectrum.