Video Transcript
Which of these sets of four quantum numbers does not exist? (A) 𝑛 equals three, 𝑙 equals two, 𝑚 subscript 𝑙 equals negative one, 𝑚 subscript
𝑠 equals positive one- half. (B) 𝑛 equals one, 𝑙 equals zero, 𝑚 subscript 𝑙 equals zero, 𝑚 subscript 𝑠
equals positive one-half. (C) 𝑛 equals four, 𝑙 equals three, 𝑚 subscript 𝑙 equals negative two, 𝑚
subscript 𝑠 equals positive one-half. (D) 𝑛 equals two, 𝑙 equals two, 𝑚 subscript 𝑙 equals zero, 𝑚 subscript 𝑠 equals
positive one-half.
To answer this question, we need to identify which of the sets of quantum numbers
does not exist. Quantum numbers are values that can be used to describe an electron in an atom. There are four quantum numbers each represented by a different symbol. The first quantum number is the principal quantum number represented by a lowercase
𝑛. This quantum number can be any positive integer. The value represents the electron shell or energy level where an electron is
found. The principal quantum number can also be used to represent the relative size of an
atomic orbital. An atomic orbital with a larger value for 𝑛 will extend further from the
nucleus.
Looking at the answer choices, we can see that all of the principal quantum numbers
are positive integers. So based on the principal quantum number alone, we cannot determine which of the sets
of quantum numbers does not exist. The next quantum number is the subsidiary quantum number represented by a lowercase
𝑙. The possible values for the subsidiary quantum number depend on the principal quantum
number. 𝑙 can have any integer value ranging from zero to 𝑛 minus one. The subsidiary quantum number indicates the type of subshell or orbital shape.
Let’s go back to the answer choices. When the principal quantum number is equal to three, the subsidiary quantum number
could be zero, one, or two. So, answer choice (A) is still a valid set of quantum numbers. When 𝑛 equals one, 𝑙 can only equal zero. So answer choice (B) is still a valid set of quantum numbers. When 𝑛 equals four, 𝑙 can be zero, one, two, or three. So answer choice (C) is still a valid set of quantum numbers. When 𝑛 equals two, 𝑙 can be zero or one. So when the principal quantum number is two, the subsidiary quantum number cannot be
equal to two. Therefore, this set of quantum numbers does not exist and must be the answer to this
question.
Even though we’ve correctly identified the answer, let’s briefly discuss the last two
quantum numbers. The third quantum number is the magnetic quantum number represented by 𝑚 subscript
𝑙. This quantum number represents the orientation of each orbital in a subshell. The magnetic quantum number depends on the subsidiary quantum number and can be any
integer from negative 𝑙 to zero to positive 𝑙. The final quantum number is the spin quantum number represented by 𝑚 subscript
𝑠. The spin quantum number does not depend on any of the other quantum numbers and can
either have a value of positive one-half or negative one-half. The spin quantum number as the name implies indicates the spin state of the
electron.
Returning to the question, we have determined that the set of quantum numbers that
does not exist is answer choice (D): 𝑛 equals two, 𝑙 equals two, 𝑚 subscript 𝑙
equals zero, 𝑚 subscript 𝑠 equals positive one-half.
