The diagram shows the binding energy of each energy level of a hydrogen atom. If an electron is in the ground state, what wavelength of photon must it absorb in order for the hydrogen atom to become completely ionized? Use a value of 4.14 times 10 to the negative 15 electron volt seconds for the Planck constant. Give your answer to one decimal place.
Before we begin, let’s quickly recall that to completely ionize an atom means to remove all of its electrons. Also keep in mind that the binding energy of an energy level tells the minimum amount of energy required to remove an electron in that level from the atom completely.
Now, we’ve been told that an electron begins in the ground state or energy level one, which has a binding energy of negative 13.6 electron volts. Say that the electron absorbs a photon and is then expelled, leaving the hydrogen atom completely ionized. This means the electron must have received the required 13.6 electron volts from the photon. It’s worth mentioning that the electron would also be ejected if it absorbed a photon with a higher energy than this. But in this question, let’s just use the limiting case, in which the photon’s energy matches the binding energy, 13.6 electron volts.
We want to find that photon’s wavelength. Recall that we can relate the energy 𝐸 of a photon to its wavelength 𝜆 using the formula 𝐸 equals ℎ𝑐 divided by 𝜆, where ℎ is the Planck constant, whose value we’ve been given, and 𝑐 is the speed of light, 3.0 times 10 to the eight meters per second. Since we wanna solve for a wavelength, let’s rearrange this formula to make 𝜆 the subject. We can do this by multiplying both sides of the formula by 𝜆 over 𝐸 so that 𝐸 cancels out of the left-hand side and 𝜆 cancels out of the right-hand side. Thus, we have 𝜆 equals ℎ𝑐 divided by 𝐸. We know values for all of the terms on the right-hand side of the formula. So, substituting them in, we have that the photon’s wavelength equals the Planck constant times the speed of light divided by the photon’s energy.
And looking at the units, notice that we can cancel electron volts out of the numerator and denominator as well as seconds and per seconds in the numerator, leaving only units of meters, which is a sign that we’re on the right track because we wanna come up with a distance measurement after all, wavelength. Now, calculating, this comes out to about 9.132 times 10 to the negative eight meters. Wavelengths of this size are typically expressed in nanometers or 10 to the negative nine meters. So, to convert, we’ll move the decimal point on our value one place to the right.
Now, we have 91.32 times 10 to the negative nine meters, or 91.32 nanometers. Finally, rounding our answer to one decimal place, we found that in order to completely ionize the hydrogen atom, an electron in the ground state must absorb a photon with a wavelength of 91.3 nanometers.