Question

Integrated Concepts

A galaxy moving away from the earth has a speed of 0.0100c. What wavelength do we observe for an n_i = 7 to n_f = 2 transition for hydrogen in that galaxy?

Short answer

The observed wavelength is 401 nm.

Step-by-step solution

Determine the formulas:

Consider the formula for the charge to mass ratio in the electron and the proton as follows:

\[\begin{array}{c}\frac{{{q_e}}}{{{m_e}}} = - 1.76 \times {10^{11}}\;\frac{{\rm{C}}}{{{\rm{kg}}}}\\\frac{{{q_p}}}{{{m_p}}} = 9.57 \times {10^7}\;\frac{{\rm{C}}}{{{\rm{kg}}}}\end{array}\]

Here,\[{m_e} = 9.11 \times {10^{ - 31}}\;{\rm{kg}}\]is the mass of the electron and\[{m_p} = 1.67 \times {10^{ - 27}}\;{\rm{kg}}\]is the mass of the proton.

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

\[\frac{{\bf{1}}}{{\bf{\lambda }}}{\bf{ = R}}\left( {\frac{{\bf{1}}}{{{\bf{n}}_{\bf{f}}^{\bf{2}}}}{\bf{ - }}\frac{{\bf{1}}}{{{\bf{n}}_{\bf{i}}^{\bf{2}}}}} \right)\]

Here, wavelength of the emitted electromagnetic radiation is λ, and the Rydberg constant is R = 1.097 x 10⁷ m^-1.

Find the observed wavelength

Substitute the values and determine the wavelength as:

\[\begin{array}{l}\frac{1}{\lambda } = \left( {1.097 \times {{10}^7}\;{{\rm{m}}^{ - 1}}} \right)\left( {\frac{1}{{{2^2}}} - \frac{1}{{{7^2}}}} \right)\\\frac{1}{\lambda } = \left( {1.097 \times {{10}^7}\;{{\rm{m}}^{ - 1}}} \right)\left( {\frac{1}{4} - \frac{1}{{49}}} \right)\\\frac{1}{\lambda } = 2518622.449\;{{\rm{m}}^{ - 1}}\\\lambda = \frac{1}{{2518622.449\;\;{{\rm{m}}^{ - 1}}}}\\\lambda = 397 \times {10^{ - 9}}\;{{\rm{m}}^{ - 1}}\end{array}\]

The wavelength observed is calculated as

\[{\lambda _{observed{\rm{ }}}} = \lambda \sqrt {\frac{{1 + \frac{u}{c}}}{{1 - \frac{u}{c}}}} \]

By changing the values in the formula above:

\[\begin{array}{l}{\lambda _{Observed{\rm{ }}}} = 397 \times {10^{ - 9}}\;{\rm{m}}\sqrt {\frac{{1 + \frac{{0.01c}}{c}}}{{1 - \frac{{0.01}}{c}}}} \\{\lambda _{Observed{\rm{ }}}} = 397 \times {10^{ - 9}}\;{\rm{m}}\sqrt {\frac{{1 + 0.01}}{{1 - 0.01}}} \\{\lambda _{Observed{\rm{ }}}} = 401 \times {10^{ - 9}}\;{\rm{m}}\\{\lambda _{Observed{\rm{ }}}} = 401\;\;{\rm{nm}}\end{array}\]

Therefore,therequired observed wavelength is 401 nm.