What is the number of possible combinations of quantum states of an electron in the 𝑛 equals five shell?
We’ll call this number of possible States 𝑁. We’re to consider all the possible combinations of quantum numbers in the 𝑛 equals five shell. And to begin, let’s recall the quantum numbers in terms of which those possible combinations are named.
There are four numbers for us to consider. The first, 𝑛, called the principal quantum number, in general can be any positive integer. The second, 𝑙, which is the orbital angular momentum of an electron, can be any integer from zero up to 𝑛 minus one. The third quantum number, 𝑚 sub 𝑙, is called the magnetic quantum number; that value can range from negative 𝑙 to positive 𝑙 inclusive, always taking on an integer value. And finally, there’s 𝑚 sub 𝑠, 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 𝑛 equals five shell might have. To figure that out, we can create a table. In this table, we’re going to write out all the possible quantum number combinations for the 𝑛 equals five shell.
We’ll start with 𝑙. And we see, according to the rule for 𝑙, that it can range from zero up to 𝑛 minus one or four. So an electron in the 𝑛 equals five shell might have 𝑙 equals zero or one or two or three or four.
Next, we consider 𝑚 sub 𝑙, the magnetic quantum number. When 𝑙 is equal to zero, 𝑚 sub 𝑙 can simply have the value of zero. When 𝑙 is equal to one, 𝑚 sub 𝑙 can have a value of negative one. It can have a value of zero and a value of positive one. When 𝑙 is two, 𝑚 sub 𝑙 can be negative two, negative one, zero, one, and two, and likewise for 𝑙 equals three and four. For 𝑙 equals three, 𝑚 sub 𝑙 can be negative three, negative two, negative one, zero, one, two, or three. And for 𝑙 equals four, 𝑚 sub 𝑙 ranges from negative four to four, including the integers in between.
Moving on to 𝑚 sub 𝑠, the spin quantum number, this can take on a value of plus or minus one-half for each value of 𝑚 sub 𝑙. For 𝑚 sub 𝑙 equals zero, there are two possible values of 𝑚 sub 𝑠. And for each of the three possible values of 𝑚 sub 𝑙 for 𝑙 equals one, 𝑚 sub 𝑠 can again be plus or minus one-half, same for 𝑙 equals two, where each value of 𝑚 sub 𝑙 can have two values of 𝑚 sub 𝑠, and so on for 𝑙 equals three and 𝑙 equals four.
We can find the total number of possible electron orientations by adding up the possible orientations for each value of 𝑙. For 𝑙 equals zero, we have one possible 𝑚 sub 𝑙 value multiplied by two possible 𝑚 sub 𝑠 values for each 𝑚 sub 𝑙 value. That gives us a total number of possible states of two for 𝑙 equals zero.
For 𝑙 equals one, we have three possible 𝑚 sub 𝑙 values and, for each of those values, two possible 𝑚 sub 𝑠 values, giving us a total of six orientations allowable at the 𝑙 equals one level. For 𝑙 equals two, we now have five possible 𝑚 sub 𝑙 values, and each one has two possible 𝑚 sub 𝑠 values, giving us a total number of 10 possible states. For 𝑙 equals three, we now have seven possible 𝑚 sub 𝑙 values, each one multiplied by two possible 𝑚 sub 𝑠 values, giving us 14 total possible states. And for 𝑙 equals four, we have nine possible 𝑚 sub 𝑙 values, each one multiplied by two possible 𝑚 sub 𝑠 values, to give 18 possible electron configurations for 𝑙 equals four.
To find 𝑁, we simply sum up these five values. And when we do, we find 𝑁 is 50. That’s the total number of possible electron configurations in the 𝑛 equals five shell.