Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Chapter 15, Problem 15.12E
Interpretation Introduction

(a)

Interpretation:

The possible observable values of orbital and spin angular momenta and their z components for a single d electron are to be calculated. The values of j and mj are to be calculated.

Concept introduction:

The magnetic fields generated by the spin angular momentum and the orbital angular momentum of an electron interact with each other to generate an overall angular momentum. The interaction between the orbital angular momentum and the spin angular momentum is termed as spin-orbit coupling.

Expert Solution
Check Mark

Answer to Problem 15.12E

The possible observable values of orbital angular momenta for a single d electron is 2. The possible value of spin angular momenta for a single d electron is 12. The values of z components of orbital angular momentum, ml for a single d electron, are 2, 1, 0, 1, 2. The values of z components of spin angular momentum, ms for a single d electron, are 12,12. The values of j are 52or32. The values of mj are 52,2,32,1,12,0,12,1,32,2,52.

Explanation of Solution

The value of orbital angular momentum (l) for a d electron is 2.

The total number of z components of orbital angular momentum (ml) is given by the formula shown below.

ml=2l+1 …(1)

Substitute the value of l of a d electron in equation (1).

ml=2×2+1=5

The value of z components of orbital angular momentum is given below.

ml=ll

Therefore, the values of z components of orbital angular momentum, ml are 2, 1, 0, 1, 2.

The spin angular momentum (s) is 12 for electrons in all subshells.

The total number of z components of spin angular momentum (ms) is given by the formula below.

ms=2s+1 …(2)

Substitute the value of s in equation (2).

ms=2×12+1=2

The value of z components of spin angular momentum is given below.

ms=ss

Therefore, the values of z components of spin angular momentum, ms are 12,12.

The total angular momentum (j) can have two values as shown below.

j=l+sorlsj=2+12or212        Fordsubshell,l=2;s=±12j=52or32

The value of z components of total orbital angular momentum is given below.

mj=jj

Therefore, the values of z components of total orbital angular momentum, mj are 52,2,32,1,12,0,12,1,32,2,52.

Conclusion

The possible observable values of orbital angular momenta for a single d electron is 2. The possible value of spin angular momenta for a single d electron is 12. The values of z components of orbital angular momentum, ml for a single d electron, are 2, 1, 0, 1, 2. The values of z components of spin angular momentum, ms for a single d electron, are 12,12. The values of j are 52or32. The values of mj are 52,2,32,1,12,0,12,1,32,2,52.

Interpretation Introduction

(b)

Interpretation:

The possible observable values of orbital and spin angular momenta and their z components for a single f electron are to be calculated. The values of j and mj are to be calculated.

Concept introduction:

The magnetic fields generated by the spin angular momentum and the orbital angular momentum of an electron interact with each other to generate an overall angular momentum. The interaction between the orbital angular momentum and the spin angular momentum is termed as spin-orbit coupling.

Expert Solution
Check Mark

Answer to Problem 15.12E

The possible observable values of orbital angular momenta for a single f electron is 3. The possible value of spin angular momenta for a single f electron is 12. The values of z components of orbital angular momentum, ml for a single f electron, are 3, 2, 1, 0, 1, 2, 3. The values of z components of spin angular momentum, ms for a single f electron, are 12,12. The values of j are 72or52. The values of mj are 72,3,52,2,321,0,1,32,2,52,3,72.

Explanation of Solution

The value of orbital angular momentum (l) for an f electron is 3.

The total number of z components of orbital angular momentum (ml) is given by the formula shown below.

ml=2l+1 …(1)

Substitute the value of l of an f electron in equation (1).

ml=2×3+1=7

The value of z components of orbital angular momentum is given below.

ml=ll

Therefore, the values of z components of orbital angular momentum, ml are 3, 2, 1, 0, 1, 2, 3.

The spin angular momentum (s) is 12 for electrons in all subshells.

The total number of z components of spin angular momentum (ms) is given by the formula below.

ms=2s+1 …(2)

Substitute the value of s in equation (2).

ms=2×12+1=2

The value of z components of spin angular momentum is given below.

ms=ss

Therefore, the values of z components of spin angular momentum, ms are 12,12.

The total angular momentum (j) can have two values as shown below.

j=l+sorlsj=3+12or312        Forfsubshell,l=3;s=±12j=72or52

The value of z components of total orbital angular momentum is given below.

mj=jj

Therefore, the values of z components of total orbital angular momentum, mj are 72,3,52,2,321,0,1,32,2,52,3,72.

Conclusion

The possible observable values of orbital angular momenta for a single f electron is 3. The possible value of spin angular momenta for a single f electron is 12. The values of z components of orbital angular momentum, ml for a single f electron, are 3, 2, 1, 0, 1, 2, 3. The values of z components of spin angular momentum, ms for a single f electron, are 12,12. The values of j are 72or52. The values of mj are 72,3,52,2,321,0,1,32,2,52,3,72.

Interpretation Introduction

(c)

Interpretation:

The possible observable values of orbital and spin angular momenta and their z components for a single g electron are to be calculated. The values of j and mj are to be calculated.

Concept introduction:

The magnetic fields generated by the spin angular momentum and the orbital angular momentum of an electron interact with each other to generate an overall angular momentum. The interaction between the orbital angular momentum and the spin angular momentum is termed as spin-orbit coupling.

Expert Solution
Check Mark

Answer to Problem 15.12E

The possible observable values of orbital angular momenta for a single g electron is 4. The possible value of spin angular momenta for a single g electron is 12. The values of z components of orbital angular momentum, ml for a single g electron, are 4, 3, 2, 1, 0, 1, 2, 3, 4 . The values of z components of spin angular momentum, ms for a single g electron, are 12,12. The values of j are 92or72. The values of mj are 92,4,72,3,52,2,321,0,1,32,2,52,3,72,4,92.

Explanation of Solution

The value of orbital angular momentum (l) for a g electron is 4.

The total number of z components of orbital angular momentum (ml) is given by the formula shown below.

ml=2l+1 …(1)

Substitute the value of l of a g electron in equation (1).

ml=2×4+1=9

The value of z components of orbital angular momentum is given below.

ml=ll

Therefore, the values of z components of orbital angular momentum, ml are 4, 3, 2, 1, 0, 1, 2, 3, 4.

The spin angular momentum (s) is 12 for electrons in all subshells.

The total number of z components of spin angular momentum (ms) is given by the formula below.

ms=2s+1 …(2)

Substitute the value of s in equation (2).

ms=2×12+1=2

The value of z components of spin angular momentum is given below.

ms=ss

Therefore, the values of z components of spin angular momentum, ms are 12,12.

The total angular momentum (j) can have two values as shown below.

j=l+sorlsj=4+12or412        Forgsubshell,l=4;s=±12j=92or72

The value of z components of total orbital angular momentum is given below.

mj=jj

Therefore, the values of z components of total orbital angular momentum, mj are 92,4,72,3,52,2,321,0,1,32,2,52,3,72,4,92.

Conclusion

The possible observable values of orbital angular momenta for a single g electron is 4. The possible value of spin angular momenta for a single g electron is 12. The values of z components of orbital angular momentum, ml for a single g electron, are 4, 3, 2, 1, 0, 1, 2, 3, 4. The values of z components of spin angular momentum, ms for a single g electron, are 12,12. The values of j are 92or72. The values of mj are 92,4,72,3,52,2,321,0,1,32,2,52,3,72,4,92.

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