For more information, please see full course syllabus of Physical Chemistry

For more information, please see full course syllabus of Physical Chemistry

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### Spin Quantum Number: Term Symbols II

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Microstates
- We Started With 21 Possible Microstates
- ³P State
- Microstates in ³P Level
- ¹D State
- ³P State
- ²P₂ State
- ³P₁ State
- ³P₀ State
- 9 Microstates in ³P are Subdivided
- ¹S State
- Quicker Way to Find the Different Values of J for a Given Basic Term Symbol
- Ground State
- Hund's Empirical Rules for Specifying the Term Symbol for the Ground Electronic State

- Intro 0:00
- Microstates 0:25
- We Started With 21 Possible Microstates
- ³P State
- Microstates in ³P Level
- ¹D State
- ³P State
- ²P₂ State
- ³P₁ State
- ³P₀ State
- 9 Microstates in ³P are Subdivided
- ¹S State
- Quicker Way to Find the Different Values of J for a Given Basic Term Symbol
- Ground State
- Hund's Empirical Rules for Specifying the Term Symbol for the Ground Electronic State 27:29
- Hund's Empirical Rules: 1
- Hund's Empirical Rules: 2
- Hund's Empirical Rules: 3 - Part A
- Hund's Empirical Rules: 3 - Part B
- Example: 1s²2s²2p²

### Physical Chemistry Online Course

### Transcription: Spin Quantum Number: Term Symbols II

*Hello, and welcome to www.educator.com, welcome back to Physical Chemistry.*0000

*As I stated in the last lesson, this particular lesson is a continuation of the example that was started in the last lesson which was,*0003

*we are developing term symbols for the electron configuration for carbon 1S2, 2S2, 2P2.*0013

*The P2 electron configuration, we want to find the term symbols for that.*0019

*Let us go ahead and continue that example.*0024

*We started with 21 possible microstates.*0028

*Let me do this in light blue.*0033

*We started with 21 possible microstates.*0038

*And by microstates, we just mean the arrangements of electron in a particular orbital.*0047

*We started with 21 microstates and 6 of these, 6 violated the exclusion principle.*0056

*That parallel spins in the same suborbital, it is not going to happen.*0071

*6 violated the exclusion principle.*0075

*We found that 5 of them were in the D1 level, which is what we wanted.*0077

*We want to find the term symbol representing that particular electron configuration.*0088

*One of the term symbols is D1.*0092

*We have 10 configurations left, we have 10 microstates left, and we need to group those.*0095

*Let us do the same thing.*0107

*I’m going to ask you actually to look back over that particular page of microstates.*0108

*It is a good idea to copy that page, make a screenshot of it so that you can keep referring back and forth to it.*0119

*Looking back at our page of microstates of the 10 remaining, when you actually chose it, when you actually grouped a certain number of them,*0126

*like when we took the 5 and we found that D1, let us go ahead and cross them out, so you do not confuse them with any others.*0155

*Looking back at our page of microstates.*0160

*Of the 10 remaining stases, the next largest value of M sub L is equal to 1.*0162

*And for this value of M sub L, the largest M sub S is also equal to 1.*0189

*What we have is the following.*0213

*The fact that M sub L is equal to 1 it implies that L is equal to 1.*0215

*The possible values of M sub L that are going to be represented are 1, 0, -1.*0218

*M sub S equaling 1 it implies that S is equal to 1.*0233

*The possible values, we will find a better way, M sub S is also 1, 0, and -1.*0242

*L1, L equal to 1, you remember the correlation, when L is equal to 0 that is the S term.*0254

*When L is equal to 1 that is the P term.*0267

*S equal to 1 means that 2S + 1 = 2 × 1 + 1 that is equal to 3.*0276

*Together, we have the P3 term.*0288

*We found a term of the next set of microstates that fall into a given level, it is the P3 level.*0301

*The number of the M sub L × the number of the M sub S is equal to 3 × 3 = 9.*0313

*All those 10 microstates, 9 of those microstates fall into the P3 level.*0326

*We actually do not say 3P, we say triplet P.*0332

*We will talk a little bit more about what we actually call these.*0334

*For example, the 1 D it is actually called singlet D.*0337

*If you were let us say 2P, they would be doublet P.*0342

*In this case, it is 3P or it is triplet P state.*0346

*There are 9 microstates in the triplet P level.*0350

*All those 9 states, they have the same energy.*0366

*When I look back, this is what I find.*0371

*I'm finding all combinations of M sub L and M sub S.*0376

*The M sub L values are 1, 0, -1.*0381

*The M sub S values are 1, 0, -1.*0386

*I’m going to look back and in my microstate, I’m going to find the M sub L, M sub S combination 1 1, 1 0, 1- 1.*0390

*I’m going to find 0 1, 0 0, 0 -1.*0398

*I’m going to find -1 1, -1 0, -1 -1.*0401

*These 9 microstates belong to the triplet P level.*0405

*They are as follows, we have 1, 2, 3, 4, 5, 6, 7, 8, 9.*0411

*And we have 123, 123, 123.*0418

*And we have 123, 123, 123.*0424

*We have up spin up spin, here M sub L = 1 and M sub S = 1.*0431

*That is the 1 1 combination, down spin and up spin.*0440

*M sub L = 1 M sub S =0, that is the 1 -0 combination, down spin down spin.*0451

*M sub L = 1 M sub S = -1, 1 1, 1 0, 1 -1, 1 1, 1 0, 1 -1 that is a microstate, a microstate.*0461

*We have this and this, let me just go ahead and draw them all out.*0474

*That and that, that and that, we have up and up, we have down and up, we have down and down here.*0481

*M sub L = 0 M sub S = 1, M sub L = 0 M sub S = 0, M sub L = 0 and M sub = -1.*0499

*This is the 0 1, 0 0, 0 -1, combination.*0514

*And here we have, M sub L =-1 M sub S = 1, we have M sub L = -1 and M sub S = 0.*0518

*And we have M sub L =-1 and M sub S is equal to -1.*0535

*These 9 belong to the triplet P level, these 9 have the same energy.*0547

*I apologize, it is an old habit of mine to always say 3P, 1D.*0564

*I really should say triplet P, singlet D, doublet, things like that, quadruplet, quintuplet, whatever.*0565

*Just old habits or just for odd reason had not been able to shake things up.*0579

*We have got 9 microstates of the 10 remaining.*0584

*This accounts for 9 of them, that leaves 1 microstate.*0587

*On our page of microstates, there is only one left, its ML value is equal to 0 and it is MS value is equal to 0.*0595

*M sub L = 0 that implies that L is equal to 0, which means that all the possible M sub L values are just 0.*0619

*M sub S = 0 implies that S is equal to 0.*0644

*Those are the ones that we are going to use for term symbol, that implies that all the values for M sub S are also just plain old 0.*0649

*L equaling 0 means the S term.*0659

*S = 0 means 2 × 0 + 1 is equal to 1.*0665

*We have a singlet S level.*0673

*This microstate is this only one left.*0681

*For every value, it is the microstate with ML = 0 MS = 0.*0687

*One value of ML, one value of MS, 1 × 1 is 1.*0694

*This microstate has M sub L = 0 and M sub S = 0 and it is the microstate with the electrons in the Z orbital paired spin.*0698

*We have a singlet D state, we have a triplet P state, and we have a singlet S state.*0732

*15 viable microstates, 15 viable arrangements of 2 electrons in 6 spin orbital arrange themselves in 3 different energies.*0747

*5 of them are here in the singlet D, 9 of them are going to be the triplet P and 1 of them is going to be in the singlet S.*0759

*This is how we derive the term symbols.*0770

*Let us go ahead and finish these.*0772

*Let us finish the term symbols by finding J, the right subscript, the total angular momentum.*0779

*Let us go ahead and do the singlet D state first.*0798

*In this particular case, D means that L is equal to 2.*0801

*The one here, 2S + 1 = 1, 2S = 0, that means that S =0.*0808

*This primary term symbol can actually give you the value of L and S.*0815

*We have used to say configuration to find the term symbol.*0819

*But if you are just given the term symbol, you can get these numbers.*0822

*If L is equal to 2, M sub L is equal to 2, 1, 0, -1, and 2, and M sub S is equal to 0.*0826

*We are going to calculate the M sub J values.*0840

*We do not need to do the M sub J on that page where we are given the microstates, but we did it anyway, it is not a problem.*0842

*M sub J, you are going to take all the values of M sub L.*0848

*Remember, we said M sub J was M sub L + M sub S.*0852

*We are going to add 2 + 0, 1 + 0, 0 + 0, -1 + 0, -2 + 0.*0858

*When we do that, we are going to get the 5 values.*0866

*We add each value of ML with each value of MS to get 2, 1, 0, -1, -2.*0871

*This sequence 2, 1, 0, -1, -2 for the M sub J means this sequence, that the number J itself is equal to 2.*0882

*The singlet D state has a total angular momentum equal to 2.*0904

*The term symbol, the complete term symbol is singlet D2.*0909

*Notice the correlation that we had.*0915

*With the hydrogen atom we had L and we had M sub L which takes on the values from -L to + L passing through 0.*0917

*L, these values of M sub L take on all the values from +L to -L passing through 0.*0926

*S, all the values of M and S run from -S to + S.*0937

*We have J, all the values of M sub J run from -J to +J passing through 0.*0943

*It is that same correlation that goes through all of this.*0953

*Our complete term symbol is singlet D2, it has 5 microstates in that energy level.*0956

*Let us go ahead and take care of the P3 or the triplet P state.*0963

*In this particular case, the P means that the L is equal to 1, the 3 here means 2S + 1 = 3, 2S = 2, S = 1.*0978

*If L is equal to 1 that means the M sub L values are 1, 0, -1.*0991

*S is equal to 1 means that the M sub S values are equal to 1, 0, and -1.*0997

*The M sub J is equal to M sub L + M sub S, I'm going to add each one of these to each one of these.*1005

*1 + 1, 1 + 0, 1 + -1, I get 2, 1, 0.*1020

*Next one, 0 + 1, 0 + 0, 0 + -1, I get 1, 0, -1.*1027

*-1 + 1, -1 + 0, -1 + -1, I get 0, -1, and 2.*1035

*All of these matter, I look in here and I see a sequence of 2, 1, 0, -1, 2.*1045

*There is a sequence 2, 1, 0, -1, 2, for the M sub J.*1064

*That means that J is equal to 2, for one of those.*1081

*We have a triplet P2 state.*1089

*However, that just takes care of that.*1096

*There is also a sequence of 1, 0, -1.*1106

*There is also a sequence of the M sub J =1, 0, -1.*1123

*This implies that J is equal to 1.*1135

*We also have a triplet P1 state.*1141

*Notice the triplet P states actually have more sub states.*1148

*What is left over, there is a N sub J = 0 leftover, that implies that J is equal to 0.*1153

*We also have a triplet P0 state.*1168

*The 9 microstates in the triplet P level are subdivided as follows.*1185

*5 of the microstates are in triplet P2.*1208

*3 microstates in triplet P1 and 1 microstate in triplet P0.*1227

*If we want to know where this 5, 3, and 1 came from, they are actually degeneracy of each individual fully complete term.*1246

*These come from 2J + 1.*1253

*For a complete term symbol, one that actually includes the J value, the degeneracy is 2J + 1.*1259

*In this case, J is equal to 0, 2 × 0 + 1, 1 microstate falling in that term symbol energy level.*1278

*Here J is equal to 1, 2 × 1 + 1 is equal to 3, there are 3 microstates in that particular term symbol energy level.*1286

*Here J is equal to 2, 2 × 2 + 1 is equal to 5.*1295

*There are 5 microstates in the triplet P2 level.*1299

*Our last one, we have our singlet S.*1307

*In this case, L is equal to 0 and S is equal to 0, that means the M sub L is equal to 0, that means M sub S is equal to 0.*1311

*M sub J is equal to M sub L + M sub S = 0 + 0 is equal to 0, that implies that J is equal to 0.*1321

*We have singlet S0, that is the complete term symbol.*1335

*There is actually a quicker way, once you have the initial grouping, the singlet D, the triplet P, singlet S.*1345

*There is a quicker way to find the different values of J for a given basic term symbol.*1360

*The basic term symbol is just the left subscript and it is the 2S + 1 and is the L value, without the J.*1388

*There is a quicker way to find the different values of J for a given basic term symbol.*1397

*The allowed values of J are, you start by adding L + S and then you do L + S -1, L + S -2, L + S -3, and you keep going.*1408

*The lowest number is going to be the absolute value of L – S.*1426

*Let us do the J values.*1430

*For the singlet D state, we have L is equal to 2 and we have S is equal to 0.*1433

*L + S is equal to 2 + 0 it is equal to 2.*1440

*The absolute value of L - S is the absolute value of 2 -0, which is 2.*1445

*2 down to 2, only 2.*1455

*Singlet D2 state, that is how we get that one.*1459

*Let us do the triplet P state, L is equal to 1, S is equal to 1.*1469

*L + S = 1 + 1 is equal to 2.*1476

*The absolute value of L - S = absolute value of 1 -1 which is 0.*1482

*We have 2, 1, 0, that gives us the triplet P2, triplet P1, triplet P0.*1487

*Nice and quick.*1501

*Singlet S state, L = 0 S = 0 L + S 0 + 0 is equal to 0.*1503

*The absolute value of L - S is 0, that means that 0 is the only one.*1512

*We have a singlet S0.*1521

*Let us see, where are we?*1530

*We have the triplet P2, we have the triplet P1, we have the triplet P0, and we have the singlet S0.*1548

*All of those 15 viable microstates are distributed among these.*1560

*And these are the terms symbols that represent the energy of those.*1565

*It is giving us more information than just a normal electron configuration, 1S2 2S2 2P2.*1570

*This is telling us something about the spin angular momentum, the orbital angular momentum, and the total angular momentum.*1578

*All of these have different energies.*1585

*The question is which one was the ground state?*1588

*Which do we call the ground state, the state of lowest energy.*1597

*The one with the lowest energy is the ground state.*1609

*Which level has the lowest energy, that is the question we are asking.*1621

*Which level, which term level has the lowest energy?*1624

*The rules that we use to find that are Hund’s rules.*1638

*They are the empirical rules that decide which is the lowest energy state for a particular term.*1642

*Hund’s empirical rules for specifying the term symbol for the ground electronic state.*1667

*The first rule says this, the state with the largest value of S is the lowest in energy.*1706

*In other words, the most stable.*1737

*And as S decreases, stability decreases.*1745

*Energy goes up, stability decreases.*1757

*For states that have the same value of S like we just saw second ago.*1764

*The triplet P has 3 states, triplet P2, triplet P1, triplet P0.*1768

*How do we decide?*1774

*For states having the same value of S, the state with the largest value of L is lowest in energy.*1779

*In other words, the most stable.*1819

*For states having the same values of S and L, break it down a little bit further.*1829

*We have part A, if the shell is less than half filled show smallest J, smallest J is the lowest energy.*1847

*And B, for more than half filled shell show it is the largest J, it is lowest in energy.*1882

*For our 1S2, 2S2, 2P configuration, we have a singlet D2, triplet P2, a triplet P1, a triplet P0, and we have a singlet S0.*1919

*Hund’s first rule, largest S means the triplet P state.*1942

*The second rule did not tell us anything because the L value is the same for each PPP, L = 1.*1955

*The second one, L was the same for all 3 triplet P states.*1963

*Let me make this 3 a little bit clearer here.*1978

*Our third one, 2P2 it is a less than half filled shell that means less than half filled.*1982

*Therefore, we pick the smallest J between the 2, 1, 0, we choose the 0.*1998

*Less than half filled, the triplet P0 state is the ground state.*2004

*The triplet P0 state, we said that there were 5 in this one, 3 in this one, 1 in this one.*2023

*One particular configuration, that is going to be the ground state.*2028

*That takes care of term symbol but we are not exactly finished discussing it.*2036

*But for this particular example, given a configuration in particular the P2 state,*2044

*we were able to elucidate 5 term symbols, singlet D2, triplet P2, triplet P1, triplet P0, and a singlet S0.*2050

*And using Hund’s rules, we were able to actually find out that the ground state is actually the triplet P0.*2061

*These term symbols give us more information than just the basic electron configuration.*2068

*They tell us about the spin angular momentum.*2073

*They tell us about the orbital angular momentum.*2078

*And it tells us about the total angular momentum.*2081

*That is all, thank you so much for joining us here at www.educator.com.*2086

*We will see you next time for a further discussion of term symbols.*2090

*Take care.*2094

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