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Concept explainers
Interpretation:
The energies of rotation for ammonia, NH3, as the rotational quantum number J ranges from 1 to 10 are to be determined. An energy level diagram for all the rotational levels is to be constructed. The degeneracies of the levels are to be determined.
Concept introduction:
Atoms of a molecule rotate in space about its moment of inertia. The rotational quantum number is represented by the symbol J. The z component of the rotational motion of the molecule is represented by MJ.
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Answer to Problem 14.98E
The energies of rotation for ammonia, NH3, as the rotational quantum number J ranges from 1 to 10 are determined below.
J | K | Erot/m−1 |
1 | -1 | 3753.185 |
1 | 0 | 1264.06 |
1 | 1 | 3753.185 |
2 | -2 | 13748.68 |
2 | -1 | 6281.305 |
2 | 0 | 3792.18 |
2 | 1 | 6281.305 |
2 | 2 | 13748.68 |
3 | -3 | 29986.49 |
3 | -2 | 17540.86 |
3 | -1 | 10073.49 |
3 | 0 | 7584.36 |
3 | 1 | 10073.49 |
3 | 2 | 17540.86 |
3 | 3 | 29986.49 |
4 | -4 | 52466.6 |
4 | -3 | 35042.73 |
4 | -2 | 22597.1 |
4 | -1 | 15129.73 |
4 | 0 | 12640.6 |
4 | 1 | 15129.73 |
4 | 2 | 22597.1 |
4 | 3 | 52466.6 |
4 | 4 | 52466.6 |
5 | -5 | 81189.03 |
5 | -4 | 58786.9 |
5 | -3 | 41363.03 |
5 | -2 | 28917.4 |
5 | -1 | 21450.03 |
5 | 0 | 18960.9 |
5 | 1 | 21450.03 |
5 | 2 | 28917.4 |
5 | 3 | 41363.03 |
5 | 4 | 58786.9 |
5 | 5 | 81189.03 |
6 | -6 | 116153.8 |
6 | -5 | 88773.39 |
6 | -4 | 66371.26 |
6 | -3 | 48947.39 |
6 | -2 | 36501.76 |
6 | -1 | 29034.39 |
6 | 0 | 26545.26 |
6 | 1 | 29034.39 |
6 | 2 | 36501.76 |
6 | 3 | 48947.39 |
6 | 4 | 66371.26 |
6 | 5 | 88773.39 |
6 | 6 | 116153.8 |
7 | -7 | 157360.8 |
7 | -6 | 125002.2 |
7 | -5 | 97621.81 |
7 | -4 | 75219.68 |
7 | -3 | 57795.81 |
7 | -2 | 45350.18 |
7 | -1 | 37882.81 |
7 | 0 | 35393.68 |
7 | 1 | 37882.81 |
7 | 2 | 45350.18 |
7 | 3 | 57795.81 |
7 | 4 | 75219.68 |
7 | 5 | 97621.81 |
7 | 6 | 125002.2 |
7 | 7 | 157360.8 |
8 | -8 | 204810.2 |
8 | -7 | 167473.3 |
8 | -6 | 135114.7 |
8 | -5 | 107734.3 |
8 | -4 | 85332.16 |
8 | -3 | 67908.29 |
8 | -2 | 55462.66 |
8 | -1 | 47995.29 |
8 | 0 | 45506.16 |
8 | 1 | 47995.29 |
8 | 2 | 55462.66 |
8 | 3 | 67908.29 |
8 | 4 | 85332.16 |
8 | 5 | 107734.3 |
8 | 6 | 135114.7 |
8 | 7 | 167473.3 |
8 | 8 | 204810.2 |
9 | -9 | 258501.8 |
9 | -8 | 216186.7 |
9 | -7 | 178849.8 |
9 | -6 | 146491.2 |
9 | -5 | 119110.8 |
9 | -4 | 96708.7 |
9 | -3 | 79284.83 |
9 | -2 | 66839.2 |
9 | -1 | 59371.83 |
9 | 0 | 56882.7 |
9 | 1 | 59371.83 |
9 | 2 | 66839.2 |
9 | 3 | 79284.83 |
9 | 4 | 96708.7 |
9 | 5 | 119110.8 |
9 | 6 | 146491.2 |
9 | 7 | 178849.8 |
9 | 8 | 216186.7 |
9 | 9 | 258501.8 |
10 | -10 | 318435.8 |
10 | -9 | 271142.4 |
10 | -8 | 228827.3 |
10 | -7 | 191490.4 |
10 | -6 | 159131.8 |
10 | -5 | 131751.4 |
10 | -4 | 109349.3 |
10 | -3 | 91925.43 |
10 | -2 | 79479.8 |
10 | -1 | 72012.43 |
10 | 0 | 69523.3 |
10 | 1 | 72012.43 |
10 | 2 | 79479.8 |
10 | 3 | 91925.43 |
10 | 4 | 109349.3 |
10 | 5 | 131751.4 |
10 | 6 | 159131.8 |
10 | 7 | 191490.4 |
10 | 8 | 228827.3 |
10 | 9 | 271142.4 |
10 | 10 | 318435.8 |
For the rotational quantum number J=1, the degeneracy is 3.
For the rotational quantum number J=2, the degeneracy is 5.
For the rotational quantum number J=3, the degeneracy is 7.
For the rotational quantum number J=4, the degeneracy is 9.
For the rotational quantum number J=5, the degeneracy is 11.
For the rotational quantum number J=6, the degeneracy is 13.
For the rotational quantum number J=7, the degeneracy is 15.
For the rotational quantum number J=8, the degeneracy is 17.
For the rotational quantum number J=9, the degeneracy is 19.
For the rotational quantum number J=10, the degeneracy is 21.
The energy level diagram for all the rotational levels is shown below.
Explanation of Solution
The formula to energy of rotation (Erot) is given by the formula below.
Erot=BJ(J+1)+(C−B)K2 …(1)
Where,
• J is the rotational quantum number.
• K is the quantum number bounded by J.
The formula for B is given below.
B=h8π2Ibc …(2)
The formula for C is given below.
C=h8π2Icc …(3)
Where,
• h is the Planck’s constant. (6.6×10−34 J s).
• c is the speed of light. (3×108 m s−1).
The value of Ib is 4.413×10−47 kg m2.
Substitute the value of Ib, h and c in equation (2).
B=6.6×10−34 J s8×(3.14)2×4.413×10−47 kg m2×3×108 m s−1=632.03 J s kg−1 m−2 m−1 s1×kg m2 s-21 J=632.03 m−1
The value of Ic is 2.806×10−47 kg m2.
Substitute the value of Ic, h and c in equation (3).
C=6.6×10−34 J s8×(3.14)2×2.806×10−47 kg m2×3×108 m s−1=3121.155 J s kg−1 m−2 m−1 s1×kg m2 s-21 J=3121.155 m−1
The value of K is shown by the equation below.
K=−J to +J …(4)
The degeneracy is calculated by the formula given below.
Degeneracy=2J+1 …(5)
For the rotational quantum number J=1, the value of K is calculated below.
K=−1 to +1=−1,0,1
The value of K is −1.
The value of J is 1.
Substitute the value of J in equation (5).
Degeneracy=2×1+1=3
Therefore, the degeneracy is 3.
Substitute the value of J, K, B and C in equation (1).
Erot=(632.03 m−1×1(1+1))+(3121.155 m−1−632.03 m−1)(−1)2=1264.06 m−1+2849.125 m−1=3753.185 m−1
Similarly the value of Erot for J=1 and corresponding values of K is given below.
J | K | Erot/m−1 |
1 | -1 | 3753.185 |
1 | 0 | 1264.06 |
1 | 1 | 3753.185 |
For the rotational quantum number J=2, the value of K is calculated below.
K=−2 to +2=−2,−1,0,1,2
Substitute the value of J in equation (5).
Degeneracy=2×2+1=5
Therefore, the degeneracy is 5.
Similarly the value of Erot for J=2 and corresponding values of K is given below.
J | K | Erot/m−1 |
2 | -2 | 13748.68 |
2 | -1 | 6281.305 |
2 | 0 | 3792.18 |
2 | 1 | 6281.305 |
2 | 2 | 13748.68 |
For the rotational quantum number J=3, the value of K is calculated below.
K=−3 to +3=−3,−2,−1,0,1,2,3
Substitute the value of J in equation (5).
Degeneracy=2×3+1=7
Therefore, the degeneracy is 7.
Similarly the value of Erot for J=3 and corresponding values of K is given below.
J | K | Erot/m−1 |
3 | -3 | 29986.49 |
3 | -2 | 17540.86 |
3 | -1 | 10073.49 |
3 | 0 | 7584.36 |
3 | 1 | 10073.49 |
3 | 2 | 17540.86 |
3 | 3 | 29986.49 |
For the rotational quantum number J=4, the value of K is calculated below.
K=−4 to +4=−4,−3,−2,−1,0,1,2,3,4
Substitute the value of J in equation (5).
Degeneracy=2×4+1=9
Therefore, the degeneracy is 9.
Similarly the value of Erot for J=4 and corresponding values of K is given below.
J | K | Erot/m−1 |
4 | -4 | 52466.6 |
4 | -3 | 35042.73 |
4 | -2 | 22597.1 |
4 | -1 | 15129.73 |
4 | 0 | 12640.6 |
4 | 1 | 15129.73 |
4 | 2 | 22597.1 |
4 | 3 | 52466.6 |
4 | 4 | 52466.6 |
For the rotational quantum number J=5, the value of K is calculated below.
K=−5 to +5=−5,−4,−3,−2,−1,0,1,2,3,4,5
Substitute the value of J in equation (5).
Degeneracy=2×5+1=11
Therefore, the degeneracy is 11.
Similarly the value of Erot for J=5 and corresponding values of K is given below.
J | K | Erot/m−1 |
5 | -5 | 81189.03 |
5 | -4 | 58786.9 |
5 | -3 | 41363.03 |
5 | -2 | 28917.4 |
5 | -1 | 21450.03 |
5 | 0 | 18960.9 |
5 | 1 | 21450.03 |
5 | 2 | 28917.4 |
5 | 3 | 41363.03 |
5 | 4 | 58786.9 |
5 | 5 | 81189.03 |
For the rotational quantum number J=6, the value of K is calculated below.
K=−6 to +6=−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6
Substitute the value of J in equation (5).
Degeneracy=2×6+1=13
Therefore, the degeneracy is 13.
Similarly the value of Erot for J=6 and corresponding values of K is given below.
J | K | Erot/m−1 |
6 | -6 | 116153.8 |
6 | -5 | 88773.39 |
6 | -4 | 66371.26 |
6 | -3 | 48947.39 |
6 | -2 | 36501.76 |
6 | -1 | 29034.39 |
6 | 0 | 26545.26 |
6 | 1 | 29034.39 |
6 | 2 | 36501.76 |
6 | 3 | 48947.39 |
6 | 4 | 66371.26 |
6 | 5 | 88773.39 |
6 | 6 | 116153.8 |
For the rotational quantum number J=7, the value of K is calculated below.
K=−7 to +7=−7,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,7
Substitute the value of J in equation (5).
Degeneracy=2×7+1=15
Therefore, the degeneracy is 15.
Similarly the value of Erot for J=7 and corresponding values of K is given below.
J | K | Erot/m−1 |
7 | -7 | 157360.8 |
7 | -6 | 125002.2 |
7 | -5 | 97621.81 |
7 | -4 | 75219.68 |
7 | -3 | 57795.81 |
7 | -2 | 45350.18 |
7 | -1 | 37882.81 |
7 | 0 | 35393.68 |
7 | 1 | 37882.81 |
7 | 2 | 45350.18 |
7 | 3 | 57795.81 |
7 | 4 | 75219.68 |
7 | 5 | 97621.81 |
7 | 6 | 125002.2 |
7 | 7 | 157360.8 |
For the rotational quantum number J=8, the value of K is calculated below.
K=−8 to +8=−8,−7,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,7,8
Substitute the value of J in equation (5).
Degeneracy=2×8+1=17
Therefore, the degeneracy is 17.
Similarly the value of Erot for J=8 and corresponding values of K is given below.
J | K | Erot/m−1 |
8 | -8 | 204810.2 |
8 | -7 | 167473.3 |
8 | -6 | 135114.7 |
8 | -5 | 107734.3 |
8 | -4 | 85332.16 |
8 | -3 | 67908.29 |
8 | -2 | 55462.66 |
8 | -1 | 47995.29 |
8 | 0 | 45506.16 |
8 | 1 | 47995.29 |
8 | 2 | 55462.66 |
8 | 3 | 67908.29 |
8 | 4 | 85332.16 |
8 | 5 | 107734.3 |
8 | 6 | 135114.7 |
8 | 7 | 167473.3 |
8 | 8 | 204810.2 |
For the rotational quantum number J=9, the value of K is calculated below.
K=−9 to +9=−9,−8,−7,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,7,8,9
Substitute the value of J in equation (5).
Degeneracy=2×9+1=19
Therefore, the degeneracy is 19.
Similarly the value of Erot for J=9 and corresponding values of K is given below.
J | K | Erot |
9 | -9 | 258501.8 |
9 | -8 | 216186.7 |
9 | -7 | 178849.8 |
9 | -6 | 146491.2 |
9 | -5 | 119110.8 |
9 | -4 | 96708.7 |
9 | -3 | 79284.83 |
9 | -2 | 66839.2 |
9 | -1 | 59371.83 |
9 | 0 | 56882.7 |
9 | 1 | 59371.83 |
9 | 2 | 66839.2 |
9 | 3 | 79284.83 |
9 | 4 | 96708.7 |
9 | 5 | 119110.8 |
9 | 6 | 146491.2 |
9 | 7 | 178849.8 |
9 | 8 | 216186.7 |
9 | 9 | 258501.8 |
For the rotational quantum number J=10, the value of K is calculated below.
K=−10 to +10=−10,−9,−8,−7,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,7,8,9,10
Substitute the value of J in equation (5).
Degeneracy=2×10+1=21
Therefore, the degeneracy is 21.
Similarly the value of Erot for J=10 and corresponding values of K is given below.
J | K | Erot/m−1 |
10 | -10 | 318435.8 |
10 | -9 | 271142.4 |
10 | -8 | 228827.3 |
10 | -7 | 191490.4 |
10 | -6 | 159131.8 |
10 | -5 | 131751.4 |
10 | -4 | 109349.3 |
10 | -3 | 91925.43 |
10 | -2 | 79479.8 |
10 | -1 | 72012.43 |
10 | 0 | 69523.3 |
10 | 1 | 72012.43 |
10 | 2 | 79479.8 |
10 | 3 | 91925.43 |
10 | 4 | 109349.3 |
10 | 5 | 131751.4 |
10 | 6 | 159131.8 |
10 | 7 | 191490.4 |
10 | 8 | 228827.3 |
10 | 9 | 271142.4 |
10 | 10 | 318435.8 |
The energy level diagram for all the rotational levels is shown below.
Figure 1
The energies of rotation for ammonia, NH3, as the rotational quantum number J ranges from 1 to 10 have been rightfully stated. An energy level diagram for all the rotational levels has been rightfully constructed. The degenracies of the levels are rightfully stated.
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