If an electron in a hydrogen atom is at a distance of 1.32 nanometers from the nucleus, what energy level is it in? Use a value of 5.29 times 10 to the negative 11 meters for the Bohr radius.
A neutral hydrogen atom has one proton in its nucleus and one orbiting electron. According to the Bohr model of the atom, this electron moves around the nucleus in a circular orbit. As such, it has some orbital radius, and that radius is given by the following general equation. The orbital radius of an electron in the 𝑛th energy level of a hydrogen atom, 𝑟 sub 𝑛, is equal to what’s called the Bohr radius multiplied by 𝑛 squared. The Bohr radius is the orbital radius of an electron in a hydrogen atom if it’s in the ground energy state. That state has a principal quantum number value of one. And if we substitute one in for 𝑛 in this equation, we see that indeed the orbital radius of an electron in the first energy level in a hydrogen atom, that is, the ground state, is equal to the Bohr radius.
In our case, we’re given a value for 𝑟 sub 𝑛, 1.32 nanometers. By our equation, We know that this equals the Bohr radius 𝑎 sub zero times 𝑛 squared. Here, we want to solve for the principal quantum number 𝑛 corresponding to the energy level of our electron. If we divide both sides of this equation by the Bohr radius 𝑎 sub zero, canceling that factor on the right, and if we then take the square root of both sides so that the square root of 𝑛 squared is equal simply to 𝑛, we find that the energy level 𝑛 is equal to the square root of 𝑟 sub 𝑛 divided by 𝑎 sub zero.
As we’ve seen, our electron’s orbital radius is 1.32 nanometers. And the Bohr radius 𝑎 sub zero is 5.29 times 10 to the negative 11 meters. To move ahead with this calculation, let’s convert the units of our numerator from nanometers into meters. One nanometer, we recall, is equal to one one billionth of a meter. And therefore 1.32 nanometers equals 1.32 times 10 to the negative nine meters.
Notice that now the units in our expression cancel out entirely so that 𝑛 will be unitless. When we calculate 𝑛 to the nearest whole number, it’s equal to five. This is the energy level of our hydrogen atom electron.