Suppose that, in an alternate universe, the possible values of...

Question

Suppose that, in an alternate universe, the possible values of ml were the integer values including 0 ranging from −l−1 to l+1 (instead of simply −l to +l). how many orbitals would exist in each of the following sublevels?

Accepted Answer

Welcome back everyone. Assuming that instead of negative L to positive L, the possible values of magnetic quantum number M sub L are the integer values from negative L minus two to positive L plus two including zero, determine the number of orbitals that exists in the following sublevel S P and D. Let's begin by defining our variables for the quantum scale. Beginning with L in which we should recall represents our angular momentum quantum number where each of our integer values correspond to a subs shell. We'll use a different color for subs shell. So for example, recall that our L values range from zero all the way to three where our L value of zero corresponds to our S subs shell, our L value of one corresponds to our P subs shell. An L value of two corresponds to the D subs shell and an L value of three corresponds to our F subs shell. With this in mind. Let's also make note of what the prompt mentions, which is our variable for magnetic quantum number In which we are told specifically for this example that we have integers which range from negative L -2. Sorry this is a minus sign To the range of positive L-plus two. So if we are able to come up with the values or the integers that make up this range for M sub, our magnetic quantum numbers, we will then be able to count each of these integers to determine our final answer for the number of orbitals existing in each sublevel. Let's begin with our first sub sublevel or subs shell which is our S subs shell. So for the S subs shell, as we noted above, that will correspond to a angular momentum quantum number L equal to zero. So in this case, we would have our M sub L variable range from zero minus two, 20 Plus two. So in this case, taking the results, we would have M sub equal to negative two to the range of positive two. And that would correspond to the following range of negative two, comma negative one, comma zero comma positive one, comma positive two. And so counting each of these integers, we would count 12345. And this would lead us to a total of five orbitals for our S subs shell. This is going to be our first answer. Let's move on to our next subs shell which is now considering our P S shell. So based on, I'm sorry, this is a colon here. So based on our notes above our P subs shell corresponds to an angular momentum quantum number or L value equal to one. Meaning we would have magnetic quantum numbers or M sub values equal to negative one minus two to positive one. Or we can just write one plus two. This range would correspond to negative 32 positive three which will correspond to the specific range as negative three comma negative two comma negative one comma zero, comma one comma and we can say plus one comma positive two comma positive three counting each of these integers, we would count +1234567. This will correspond to seven orbitals making up our P subs shell for this example as our second answer so far. So next, we want to consider our D subs shell. So we'll do that next. So for our D subs shell recall from our notes above that this corresponds to an angular momentum quantum number or L value equal to two in which we would have M sub vari or values ranging from negative to minus two to positive to I'm sorry, we have two positive 2 plus two. So taking the results, we would have negative two minus two which would give us a range of negative four up to positive two plus two. So plus four and this would correspond to negative four comma negative three, comma negative two, comma negative one, comma zero comma positive one, comma positive two comma positive three and comma positive four counting each of these integers, you would count 123456789. So this corresponds to nine orbitals as our third answer for our DS. And each of our final answers highlighted in yellow will all correspond to choice C which states that for the S subs shell, we have five orbitals. For the P subs shell, we have seven orbitals. And for the D subs shell, we have nine orbitals as our final answer. I hope this made sense and let us know if you have any questions.

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Suppose that, in an alternate universe, the possible values of ml were the integer values including 0 ranging from −l−1 to l+1 (instead of simply −l to +l). how many orbitals would exist in each of the following sublevels?

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