Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
Question
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Chapter 14, Problem 14.32E
Interpretation Introduction

(a)

Interpretation:

The most populated rotational level for a sample of LiH at 298K is to be determined.

Concept introduction:

An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant. This is known as the Franck-Condon principle.

Expert Solution
Check Mark

Answer to Problem 14.32E

The most populated rotational level for a sample of LiH at 298K is 4.

Explanation of Solution

The most populated rotational level is calculated by the formula as shown below.

Jmax=(kT2B)12…(1)

Where,

k is the Boltzmann’s constant.

T is the temperature.

B is the rotational constant.

The rotational constant is calculated by the formula as shown below.

B=h28π2μr2…(2)

Where,

r is the bond length of LiH.

μ is the reduced mass.

h is the Planck’s constant (6.626×1034Js).

The reduced mass is calculated by the formula as shown below.

μ=mLimHmLi+mH…(3)

Where,

mLi is the mass of lithium.

mH is the mass of hydrogen.

Substitute the value of mass of lithium and hydrogen in equation (3).

μ=6.941amu×1amu6.941amu+1amu=0.874amu

Convert 0.874amu to kg.

0.874amu=0.8746.022×1026kg=1.45×1027kg

Substitute the value of reduced mass, bond length, Planck’s constant in equation (2).

B=(6.626×1034Js)28(3.14)2×1.45×1027kg×(1.60×1010)2=43.9×1068292.8×1047=1.5×1022J

Substitute the value of rotational constant, Boltzmann’s constant and Jmax equation (1).

Jmax=(1.38×1023JK1×298K2×1.5×1022J)12=411.24×1023J3×1022J=13.708=3.70

Therefore, the most populated rotational level for a sample of LiH at 298K is 4.

Conclusion

the most populated rotational level for a sample of LiH at 298K is 4.

Interpretation Introduction

(b)

Interpretation:

The most populated rotational level for a sample of LiH at 1000K is to be determined.

Concept introduction:

An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant. This is known as the Franck-Condon principle.

Expert Solution
Check Mark

Answer to Problem 14.32E

The most populated rotational level for a sample of LiH at 1000K is 7.

Explanation of Solution

The most populated rotational level is calculated by the formula as shown below.

Jmax=(kT2B)12…(1)

Where,

k is the Boltzmann’s constant.

T is the temperature.

B is the rotational constant.

The rotational constant is calculated by the formula as shown below.

B=h28π2μr2…(2)

Where,

r is the bond length of LiH.

μ is the reduced mass.

h is the Planck’s constant (6.626×1034Js).

The reduced mass is calculated by the formula as shown below.

μ=mLimHmLi+mH…(3)

Where,

mLi is the mass of lithium.

mH is the mass of hydrogen.

Substitute the value of mass of lithium and hydrogen in equation (3).

μ=6.941amu×1amu6.941amu+1amu=0.874amu

Convert 0.874amu to kg.

0.874amu=0.8746.022×1026kg=1.45×1027kg

Substitute the value of reduced mass, bond length, Planck’s constant in equation (2).

B=(6.626×1034Js)28(3.14)2×1.45×1027kg×(1.60×1010)2=43.9×1068292.8×1047=1.5×1022J

Substitute the value of rotational constant, Boltzmann’s constant and Jmax equation (1).

Jmax=(1.38×1023JK1×1000K2×1.5×1022J)12=1.38×1020J3×1022J=46=6.78

Therefore, the most populated rotational level for a sample of LiH at 1000K is 7.

Conclusion

The most populated rotational level for a sample of LiH at 1000K is 7.

Interpretation Introduction

(c)

Interpretation:

The most populated rotational level for a sample of LiH at 5000K is to be determined.

Concept introduction:

An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant. This is known as the Franck-Condon principle.

Expert Solution
Check Mark

Answer to Problem 14.32E

The most populated rotational level for a sample of LiH at 5000K is 15.

Explanation of Solution

The most populated rotational level is calculated by the formula as shown below.

Jmax=(kT2B)12…(1)

Where,

k is the Boltzmann’s constant.

T is the temperature.

B is the rotational constant.

The rotational constant is calculated by the formula as shown below.

B=h28π2μr2…(2)

Where,

r is the bond length of LiH.

μ is the reduced mass.

h is the Planck’s constant (6.626×1034Js).

The reduced mass is calculated by the formula as shown below.

μ=mLimHmLi+mH…(3)

Where,

mLi is the mass of lithium.

mH is the mass of hydrogen.

Substitute the value of mass of lithium and hydrogen in equation (3).

μ=6.941amu×1amu6.941amu+1amu=0.874amu

Convert 0.874amu to kg.

0.874amu=0.8746.022×1026kg=1.45×1027kg

Substitute the value of reduced mass, bond length, Planck’s constant in equation (2).

B=(6.626×1034Js)28(3.14)2×1.45×1027kg×(1.60×1010)2=43.9×1068292.8×1047=1.5×1022J

Substitute the value of rotational constant, Boltzmann’s constant and Jmax equation (1).

Jmax=(1.38×1023JK1×5000K2×1.5×1022J)12=6.9×1020J3×1022J=230=15.16

Therefore, the most populated rotational level for a sample of LiH at 5000K is 15.

Conclusion

The most populated rotational level for a sample of LiH at 5000K is 15.

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Chapter 14 Solutions

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