Video Transcript
The quantum numbers for three
electrons in an atom of silicon are shown. What is the correct order from
lowest to highest energy of these electrons? (1) 𝑛 equals two, 𝑙 equals one,
𝑚 subscript 𝑙 equals positive one, 𝑚 subscript 𝑠 equals negative one-half. (2) 𝑛 equals three, 𝑙 equals one,
𝑚 subscript 𝑙 equals negative one, 𝑚 subscript 𝑠 equals negative one-half. (3) 𝑛 equals two, 𝑙 equals zero,
𝑚 subscript 𝑙 equals zero, 𝑚 subscript 𝑠 equals positive one-half. (A) (3), (2), (1); (B) (1), (2),
(3); (C) (3), (1), (2); (D) (2), (1), (3); or (E) (2), (3), (1).
We have been given a set of quantum
numbers for three electrons in an atom of silicon. We need to use these quantum
numbers to rank the electrons in order from lowest to highest energy. Quantum numbers are values that can
be used to completely describe an electron in an atom. The first quantum number is the
principal quantum number given the symbol 𝑛. This quantum number indicates which
electron shell or energy level the electron can be found in. The principal quantum number can be
any positive integer. As 𝑛 increases, the electron will
be at a higher energy and less tightly bound to the nucleus.
If we look at the principal quantum
number of the three electrons, we can see that electron number (2) has the highest
principal quantum number and will therefore have the highest energy. As we need to rank the electrons
from lowest to highest energy, we can already eliminate answer choices (A), (B),
(D), and (E) as none of these answer choices list electron number (2) as the
electron with the highest energy. But to make sure that the electrons
are listed in the correct order in answer choice (C), let’s take a look at the rest
of the quantum numbers.
The next quantum number is the
subsidiary quantum number, denoted with a lowercase 𝑙. This quantum number indicates the
type of subshell that the electron is in. The subsidiary quantum number can
be any positive integer from zero to 𝑛 minus one with each value representing a
different subshell. The four most relevant types of
subshell are s, p, d, and f. For the same principal quantum
number value, an s-type subshell will have the lowest energy and an f-type subshell
will have the highest energy of the four, so four electrons which have the same
principal quantum number. The energy of the electron will
increase with increasing subsidiary quantum number.
As we already know that electron
number (2) has the highest energy, we need to compare the energies of electrons
number (1) and (3) which have the same principal quantum number. We can see that electron number (3)
has a subsidiary quantum number of zero, which corresponds to an s-type subshell,
the lowest energy subshell for this principal quantum number. So we have confirmed that electron
number (3) should have the lowest energy while electron number (2) has the highest
energy.
The remaining two quantum numbers,
the magnetic quantum number, which represents the orientation of the individual
atomic orbitals in a subshell, and the spin quantum number, which represents the
spin of an electron in an orbital, only affect the energy of an electron when the
electron is in a magnetic field. So we don’t need to look at these
quantum numbers to determine the correct answer. Therefore, using the quantum
numbers for three electrons in an atom of silicon, the correct order from lowest to
highest energy of these electrons is answer choice (C): (3), (1), (2).
