What is the change in energy, ΔE, in kilojoules per mole of hydrogen atoms for an electron transition from n = 7 to n = 2?

The Rydberg equation expresses the wavelength, λ, of emitted light based on the initial and final energy states (n_{i} and n_{f}) of an electron in a hydrogen atom:

$\frac{\mathbf{1}}{\mathbf{\lambda}}\mathbf{=}\mathbf{}{\mathbf{R}}_{\mathbf{H}}\mathbf{\times}\mathbf{(}\frac{\mathbf{1}}{{{\mathbf{n}}_{\mathbf{f}}}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{1}}{{{\mathbf{n}}_{\mathbf{i}}}^{\mathbf{2}}}\mathbf{)}$

where R_{H} = 1.097 × 10^{7} m^{− 1} = 1.097 × 10^{−2} nm^{−1}.

You may also see this equation written as

$\frac{\mathbf{1}}{\mathbf{\lambda}}\mathbf{=}\mathbf{}\mathbf{-}{\mathbf{R}}_{\mathbf{H}}\mathbf{\times}\mathbf{(}\frac{\mathbf{1}}{{{\mathbf{n}}_{\mathbf{i}}}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{1}}{{{\mathbf{n}}_{\mathbf{f}}}^{\mathbf{2}}}\mathbf{)}$

Since

$(\frac{1}{{{n}_{f}}^{2}}-\frac{1}{{{n}_{i}}^{2}})\mathbf{=}\mathbf{-}(\frac{1}{{{n}_{i}}^{2}}-\frac{1}{{{n}_{f}}^{2}})$

the two formulas are equivalent and sometimes used interchangeably. It can help to remember that when light is emitted, E is negative. When light is absorbed, E is positive.

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