General Chemistry I

Question

In Bohr's model of the hydrogen atom, which is no longer the standard for understanding the atom, the electron revolves around the nucleus in one of an array of available orbits. Each orbit is assigned a quantum number, n, starting with n = 1 for the orbit closest to the nucleus. The energy of an electron is inversely proportional to n squared, n². Calculate the energy of an electron in the n =3 orbit.

Answer 1

The Bohr result for hydrogen is

En = -2.18 x 10^-18 J / n^2

That constant, 2.18 x 10^-18 J, is the Rydberg energy - a universal value you can take from any textbook, so you do not need the problem to supply it.

For n = 3

E3 = -2.18 x 10^-18 J / 3^2 = -2.18 x 10^-18 / 9

E3 = -2.42 x 10^-19 J

The same answer in the two other units your course may want:

-1.51 eV (from -13.6 eV / 9)
-146 kJ/mol (multiply the joules per electron by Avogadro's number: -2.42 x 10^-19 x 6.022 x 10^23 = -1.46 x 10^5 J/mol)

Do not lose the minus sign

This is the part the problem's wording glosses over, and it is worth being careful about.

Strictly, the energy is not "inversely proportional to n^2" - it is the negative of something inversely proportional to n^2. The zero of energy is defined as the electron completely removed from the atom and at rest, infinitely far away. Any bound electron is lower than that, so every En is negative.

The consequence catches people out: as n increases the magnitude gets smaller but the energy gets larger, because it climbs toward zero. E1 = -2.18 x 10^-18 J is the lowest state, and E3 = -2.42 x 10^-19 J sits above it. If you wrote E3 as positive, you would be describing an electron that has already escaped the atom.

A check, and what the number is good for

Because the levels are defined this way, the energy needed to ionize an electron sitting in n = 3 is simply +2.42 x 10^-19 J - the amount required to lift it to zero.

And differences between these levels give you the hydrogen spectrum directly. For example the n = 3 to n = 2 transition releases

E3 - E2 = (-2.42 x 10^-19) - (-5.45 x 10^-19) = 3.03 x 10^-19 J

which by E = hc/lambda corresponds to 656 nm - the red line you see in a hydrogen discharge tube. That agreement between a one-line formula and something you can look at was exactly why the Bohr model was taken seriously, even though it turned out to work only for one-electron systems.