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What is the total number of microstates in a five π· term of the ground state iron atom?

Okay, so in this question, weβre looking at a five π· term and weβre trying to work out the total number of possible microstates in this five π· term. So what does a five π· term actually mean?

Well, we can recall that term symbols are written using the following conventions. The superscript, which in this case is five, represents two π plus one, where π is the total spin quantum number. And the large value, which in this case is π·, represents πΏ, where πΏ is the total orbital angular momentum quantum number.

However, with πΏ, we donβt actually write the quantum number itself, but instead we write the letter that represents the orbital that weβre studying. So our value of πΏ could actually be zero or one or two or three or so on, but in the term symbol, we donβt write zero, one, two, or three. We actually write π or π or π· or πΉ or so on and so forth, where πΏ is equal to zero corresponds to π, πΏ is equal to one corresponds to π; you get the idea.

For the two π plus one part, however, itβs a lot simpler. We actually use the numerical value of π. So now that we know that thatβs how term symbols are written, we can calculate the values of π and πΏ for this term. So in our case, weβve got a five π· term, and this is equal to two π plus one in the superscript and πΏ as the main character, which means that two π plus one is equal to five and the numerical value of πΏ is the one which corresponds with π·. So in other words, πΏ is equal to two, and we can write that down here.

We can also rearrange this first equation here to find out what π is. When we do that, we find that π is also equal to two. So for our five π· term, we find that π is equal to two and πΏ is equal to two. In other words, the total spin quantum number is equal to two and the total orbital angular momentum quantum number is also equal to two. So why is this relevant?

Well, itβs because weβre trying to work out the number of microstates. And all the possible microstates are just all possible combinations of π sub π and π sub πΏ for this term. Now π sub π is the spin projection quantum number, and π sub πΏ is known as the magnetic quantum number. And we work out the possible values of π sub π and π sub πΏ using the values of π and πΏ, because we can recall that π sub π can have a possible set of values starting from negative π and going up in integer steps, so negative π plus one, negative π plus two, and so on and so forth, until we get to π minus one and then finally positive π. And these are all the possible values of π sub π.

And the same is true for π sub πΏ, except this time you start with negative πΏ and then go up in integer steps, negative πΏ plus one, so on and so forth, until we get to πΏ minus one and then finally positive πΏ. And those are all the possible values of π sub πΏ.

So we know how to calculate all of the possible values of π sub π and π sub πΏ, and these depend on the values of π and πΏ. So in a five π· state where π is equal to two and πΏ is equal to two, the possible values of π sub π are starting with negative π, which is negative two, and then go up in integer steps, so negative one and then zero and then one and then two, and we stop there because π is equal to two and we stop at positive π. Similarly, with π sub πΏ, all of its possible values are starting with negative πΏ, which is negative two, and then going up in integer steps and then stopping at positive πΏ. So those are all the possible values of π sub πΏ.

Now the values of π sub π and π sub πΏ are independent of each other, or in other words an electron can have any combination of the values of π sub π and π sub πΏ. So the total possible number of microstates is just all possible combinations of π sub π and π sub πΏ added together. In other words, all possible microstates include π sub π and π sub πΏ is equal to, letβs start with negative two, negative two. Then thereβs negative two, negative one, and so on and so forth, and we can keep counting all of the possible values of π sub π combined with all possible values of π sub πΏ.

And an intuitive way to think about this is that we could start with one value of π sub π β letβs say negative two β and then switch between all values of π sub πΏ β so thatβs five different values. So thatβs already five different possible microstates. Then we switch to the next value of π sub π, which is negative one, and once again we go through all five possible values for π sub πΏ.

So now weβve got another five possible microstates. So, so far, we have five states from when ππ is equal to negative two and another five from when ππ is equal to negative one. And weβll get another five from when ππ is equal to zero and another five from when itβs one and another five from when itβs two. In other words, the total number of microstates possible is five times five, which turns out to be 25. And that is the total possible number of microstates in a five π· term. And so weβve arrived at our final answer. The total number of microstates is 25.