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 11, Problem 11.58E

In exercise 11.57 regarding C 60 , what are the numerical values of the total angular momenta of the electron for each state having quantum number l ? What are the z components of the angular momentum for each state?

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Interpretation Introduction

Interpretation:

The total angular momenta of the electron of the given state that has quantum number l and the z-components of the angular momentum for the state are to be calculated.

Concept introduction:

The total angular momentum for the 3-Dimensional system is given by,

L=l(l+1)

The total angular momentum does not depend on the mass of the particle, radius of the rotation and also the magnetic quantum number.

The z-component of the three dimensional angular momentum that has components in x, y and z direction is quantized. Its value depends on the magnetic quantum number and it is given by,

L^z=ml

Answer to Problem 11.58E

The total angular momenta of the electron of the given state that has quantum number l=5, l=6, l=7 and l=8 is 5.78×1034kgm2s1, 6.84×1034kgm2s1, 7.89×1034kgm2s1 and 8.95×1034kgm2s1 respectively.

The z-components of the angular momentum for the state l=5 is 5, 43, 2, 1, 0, +1, +2, +3, +4 and +5.

The z-components of the angular momentum for the state l=6 is 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5 and +6.

The z-components of the angular momentum for the state l=7 is 7, 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5, +6 and +7.

The z-components of the angular momentum for the state l=7 is 8, 7, 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5, +6, +7 and +8.

Explanation of Solution

The moment of inertia for the electron in the spherical C60 molecule is 1.12×1049kgm2

The total angular momentum for the 3-D rotational wavefunction is calculated using the formula,

L=l(l+1)

Where,

L is the total angular momentum.

l is the quantum number.

has the value h2π, h is the Planck’s constant.

For l=5, substitute the values of quantum number, moment of inertia and Planck’s constant in the given formula.

L=5(5+1)(6.626×1034kgm2s12π)L=30(6.626×1034kgm2s12π)L=5.78×1034kgm2s1

Thus, the total angular momentum of the electron in spherical C60 molecule for l=5 is 5.78×1034kgm2s1.

For l=6, substitute the values of quantum number, moment of inertia and Planck’s constant in the given formula.

L=6(6+1)(6.626×1034kgm2s12π)L=42(6.626×1034kgm2s12π)L=6.84×1034kgm2s1

Thus, the total angular momentum of the electron in spherical C60 molecule for l=6 is 6.84×1034kgm2s1.

For l=7, substitute the values of quantum number, moment of inertia and Planck’s constant in the given formula.

L=7(7+1)(6.626×1034kgm2s12π)L=56(6.626×1034kgm2s12π)L=7.89×1034kgm2s1

Thus, the total angular momentum of the electron in spherical C60 molecule for l=7 is 7.89×1034kgm2s1.

For l=8, substitute the values of quantum number, moment of inertia and Planck’s constant in the given formula.

L=8(8+1)(6.626×1034kgm2s12π)L=72(6.626×1034kgm2s12π)L=8.95×1034kgm2s1

Thus, the total angular momentum of the electron in spherical C60 molecule for l=8 is 8.95×1034kgm2s1.

The relation between the Z-component of angular momentum and magnetic quantum number for 3-dimensional rotation is

L^z=ml.

Where,

ml is the magnetic quantum number.

has the value h2π, h is the Planck’s constant.

The possible vales of ml for quantum number l=5 is 5, 43, 2, 1, 0, +1, +2, +3, +4 and +5.

The Z-component of angular momentum to by substituting the values of magnetic quantum number in the given formula is 5, 43, 2, 1, 0, +1, +2, +3, +4 and +5 respectively.

The possible vales of ml for quantum number l=6 is 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5 and +6.

The Z-component of angular momentum to by substituting the values of magnetic quantum number in the given formula is 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5 and +6 respectively.

The possible vales of ml for quantum number l=7 is 7, 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5+6 and +7.

The Z-component of angular momentum to by substituting the values of magnetic quantum number in the given formula is 7, 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5, +6 and +7 respectively.

The possible vales of ml for quantum number l=8 is 8, 7, 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5+6, +7 and +8.

The Z-component of angular momentum to by substituting the values of magnetic quantum number in the given formula is 8, 7, 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5, +6, +7 and +8 respectively.

Conclusion

The total angular momenta of the electron of the given state that has quantum number l=5, l=6, l=7 and l=8 is 5.78×1034kgm2s1, 6.84×1034kgm2s1, 7.89×1034kgm2s1 and 8.95×1034kgm2s1 respectively.

The z-components of the angular momentum for the state l=5 is 5, 43, 2, 1, 0, +1, +2, +3, +4 and +5.

The z-components of the angular momentum for the state l=6 is 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5 and +6.

The z-components of the angular momentum for the state l=7 is 7, 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5, +6 and +7.

The z-components of the angular momentum for the state l=7 is 8, 7, 6, 5, 43, 2, 1, 0, +1, +2, +3, +4, +5, +6, +7 and +8.

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Chapter 11 Solutions

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