Hydrogen Atom Energy Levels

\[E = \frac{-13.6\text{ eV}}{n^2}\quad[\text{Eq. }(8.4)]\]

Note: If you feel uncomfortable with working with variables, fractions, or exponents, you should consider reviewing one or more of the following tutorials before working through this tutorial:

Sections

Introduction

Johann Balmer was the first to recognize that there was a pattern to the visible-light spectral lines of hydrogen that fit an equation for wavelength, and involved different, but sequential, integers, (2, 3, 4, …), to distinguish each line in the sequence. Based in part on Balmer’s discovery, Niels Bohr developed a “semi-classical” model of the hydrogen atom to explain the individual spectral lines that were observed. He was also able to extend Balmer’s wavelength formula to include almost all hydrogen lines, including those outside of the visible range (e.g. ultraviolet and infrared). (Hydrogen can produce a very long wavelength spectral line of 21.1 cm that is purely quantum-mechanical in nature; something Bohr’s model cannot account for.)

Bohr’s model was based on imagining that a negatively charged electron literally orbits a positively charged nucleus and that each discrete orbit is uniquely identified with a principal quantum number, \(n\), that corresponds to the orbit’s radius and its potential energy. Bohr’s model is referred to as being semi-classical because it used Newton’s laws of motion and Coulomb’s equation for the force between charged particles [Equation (6.10) in your text], much like Newton’s use of his universal law of gravitation to understand orbits of astronomical objects. The “semi-classical” part was the somewhat arbitrary insertion of the principal quantum number that specified which orbits could exist by restricting orbital angular momentum to only “allowed” values. This led to Bohr’s derivation of Equation (8.4) for the potential energy of each orbit, and the derivation of an extended version of Balmer’s visible spectral lines for hydrogen. Bohr showed that the observed wavelengths, along with spectral lines outside of the visible range, were due to electrons jumping between allowed energy levels while emitting or absorbing photons with energies equal to the difference in energies of the orbits.

The fully-quantum mechanical derivation of Bohr’s energy level equation [Equation (8.4)] came about a decade later. For the lowest-possible energy level of hydrogen with the smallest allowed “orbital”, \(n = 1\) and so \(E_1 = \frac{-13.6}{1^2}\text{ eV} = -13.6\text{ eV}\). At the other extreme, the highest hydrogen energy level for which the electron is still attached to the atom, and for which the orbital radius is infinitely large, \(n = \infty\), and \(E_\infty = \frac{-13.6}{\infty^2}\text{ eV} = \frac{-13.6}{\infty}\text{ eV} = 0\text{ eV}.\) [Notes: (a) Any number (except infinity) that is divided by infinity equals zero. (b) \(\infty^2\) is equal to a bigger \(\infty\) than \(\infty\) itself! Weird and mind-boggling, I know, but cool. ☺ If you are not already planning on it, consider taking more math; it is a powerful, and very relevant, language in today’s world. It can also be fun!]

Examples

  1. What is the hydrogen energy level associated with \(n = 3\)?

    \[E_3 = \frac{-13.6\text{ eV}}{3^2} = \frac{-13.6\text{ eV}}{9} = -1.51\text{ eV}\]
  2. What is the hydrogen energy level associated with \(n = 5\)?

    \[E_5 = \frac{-13.6\text{ eV}}{5^2} = \frac{-13.6\text{ eV}}{25} = -0.544\text{ eV}\]
  3. What is the hydrogen energy level associated with \(n = 10\)? (You shouldn’t need a calculator for this one.)

    \[E_{10} = \frac{-13.6\text{ eV}}{10^2} = \frac{-13.6\text{ eV}}{100} = -0.136\text{ eV}\]
  4. An electron in a hydrogen atom “jumps” from the \(n = 4\) energy level to the \(n = 2\) energy level.
    1. What is the energy of the \(n = 4\) orbital?

      \[E_4 = \frac{-13.6\text{ eV}}{4^2} = \frac{-13.6\text{ eV}}{16} = -0.85\text{ eV}\]
    2. What is the energy of the \(n = 2\) orbital?

      \[E_2 = \frac{-13.6\text{ eV}}{2^2} = \frac{-13.6\text{ eV}}{4} = 3.4\text{ eV}\]
    3. How much energy is released as a photon due to the “jump” (or transition)?

      \[E_\text{eV} = E_4 – E_2 = -0.85\text{ eV} – (-3.4\text{ eV}) = -0.85\text{ eV} + 3.4\text{ eV} = 2.55\text{ eV}\]
    4. Solving Equation (8.3), the energy of a photon in electron volts, for wavelength and substituting the result of part (c) for the energy (\(E_\text{eV}\)), what is the wavelength of the photon in nanometers?

      \[\lambda_\text{nm} = \frac{1240}{E_\text{eV}}\text{ nm} = \frac{1240}{2.55}\text{ nm} = 486\text{ nm}\]
    5. In which portion of the electromagnetic spectrum does the photon belong?

      Visible
    6. Is the wavelength you calculated listed in Table 7.2 of your text?

      Yes

Quiz Yourself

Try the following exercises to make sure that you really do understand the material just presented. By the way, computers and many calculators write scientific notation in “computer-speak,” where \(10\) raised to an integer (either positive or negative) replaces \(\times 10^\text{exponent}\) with \({\rm E(exponent)}\) or \({\rm e(exponent)}\); in other words, \(3.14159 \times 10^9\) is represented by \(3.14159\text{E}9\). If your calculator has the \(\Large{\hat{ }}\) symbol, then you would enter the number as \(3.14159 \times 10{\Large{\hat{ }}}9\). If your calculator has a \(10^x\) button then you would enter the number as \(3.14159\times9\,\,10^x\). If you have having problems with entering numbers in scientific notation, read the manual or help screens, or ask your instructor.

  1. What is the lowest possible energy level for hydrogen?
  2. What is the energy associated with the \(n = 2\) orbital of hydrogen?
  3. What is the wavelength of a photon that is emitted when an electron “falls” from the \(n = 2\) energy level to the \(n = 1\) energy level? What portion of the electromagnetic spectrum does the wavelength corresponds to?
  4. An electron in a hydrogen atom makes a transition from the \(n = 6\) energy level to the \(n = 3\) energy level. What are the energies associated with the two levels? What is the energy of the photon that is produced during the transition? What is the electron’s wavelength? What portion of the electromagnetic spectrum does the photon’s wavelength correspond to?

(Answers are available below.)

Answers
  1. \(-13.6\text{ eV}\)
  2. \(-3.4\text{ eV}\)
  3. \(121\text{ nm}\) is in the ultraviolet portion of the EM spectrum.
  4. \(E_6 = -0.378\text{ eV}\), \(E_3 = -1.51\text{ eV}\), \(1.13\text{ eV}\), \(1100\text{ nm}\), infrared
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