# Video: Maximum Number of Electron States in a Shell

What is the number of possible combinations of quantum states of an electron in the π = 5 shell?

05:24

### Video Transcript

What is the number of possible combinations of quantum states of an electron in the π equals five shell?

Weβll call this number of possible States π. Weβre to consider all the possible combinations of quantum numbers in the π equals five shell. And to begin, letβs recall the quantum numbers in terms of which those possible combinations are named.

There are four numbers for us to consider. The first, π, called the principal quantum number, in general can be any positive integer. The second, π, which is the orbital angular momentum of an electron, can be any integer from zero up to π minus one. The third quantum number, π sub π, is called the magnetic quantum number; that value can range from negative π to positive π inclusive, always taking on an integer value. And finally, thereβs π sub π , the spin quantum number; any electron can have plus or minus one-half spin.

Based on these quantum number rules, we want to find the total number of possible combinations or states an electron in the π equals five shell might have. To figure that out, we can create a table. In this table, weβre going to write out all the possible quantum number combinations for the π equals five shell.

Weβll start with π. And we see, according to the rule for π, that it can range from zero up to π minus one or four. So an electron in the π equals five shell might have π equals zero or one or two or three or four.

Next, we consider π sub π, the magnetic quantum number. When π is equal to zero, π sub π can simply have the value of zero. When π is equal to one, π sub π can have a value of negative one. It can have a value of zero and a value of positive one. When π is two, π sub π can be negative two, negative one, zero, one, and two, and likewise for π equals three and four. For π equals three, π sub π can be negative three, negative two, negative one, zero, one, two, or three. And for π equals four, π sub π ranges from negative four to four, including the integers in between.

Moving on to π sub π , the spin quantum number, this can take on a value of plus or minus one-half for each value of π sub π. For π sub π equals zero, there are two possible values of π sub π . And for each of the three possible values of π sub π for π equals one, π sub π  can again be plus or minus one-half, same for π equals two, where each value of π sub π can have two values of π sub π , and so on for π equals three and π equals four.

We can find the total number of possible electron orientations by adding up the possible orientations for each value of π. For π equals zero, we have one possible π sub π value multiplied by two possible π sub π  values for each π sub π value. That gives us a total number of possible states of two for π equals zero.

For π equals one, we have three possible π sub π values and, for each of those values, two possible π sub π  values, giving us a total of six orientations allowable at the π equals one level. For π equals two, we now have five possible π sub π values, and each one has two possible π sub π  values, giving us a total number of 10 possible states. For π equals three, we now have seven possible π sub π values, each one multiplied by two possible π sub π  values, giving us 14 total possible states. And for π equals four, we have nine possible π sub π values, each one multiplied by two possible π sub π  values, to give 18 possible electron configurations for π equals four.

To find π, we simply sum up these five values. And when we do, we find π is 50. Thatβs the total number of possible electron configurations in the π equals five shell.