Concept explainers
Interpretation:
The energies of rotation for ammonia,
Concept introduction:
Atoms of a molecule rotate in space about its moment of inertia. The rotational quantum number is represented by the symbol
Answer to Problem 14.98E
The energies of rotation for ammonia,
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
For the rotational quantum number
For the rotational quantum number
For the rotational quantum number
For the rotational quantum number
For the rotational quantum number
For the rotational quantum number
For the rotational quantum number
For the rotational quantum number
For the rotational quantum number
The energy level diagram for all the rotational levels is shown below.
Explanation of Solution
The formula to energy of rotation (
Where,
•
•
The formula for
The formula for
Where,
•
•
The value of
Substitute the value of
The value of
Substitute the value of
The value of
The degeneracy is calculated by the formula given below.
For the rotational quantum number
The value of
The value of
Substitute the value of
Therefore, the degeneracy is
Substitute the value of
Similarly the value of
1 | -1 | 3753.185 |
1 | 0 | 1264.06 |
1 | 1 | 3753.185 |
For the rotational quantum number
Substitute the value of
Therefore, the degeneracy is
Similarly the value of
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
Substitute the value of
Therefore, the degeneracy is
Similarly the value of
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
Substitute the value of
Therefore, the degeneracy is
Similarly the value of
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
Substitute the value of
Therefore, the degeneracy is
Similarly the value of
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
Substitute the value of
Therefore, the degeneracy is
Similarly the value of
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
Substitute the value of
Therefore, the degeneracy is
Similarly the value of
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
Substitute the value of
Therefore, the degeneracy is
Similarly the value of
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
Substitute the value of
Therefore, the degeneracy is
Similarly the value of
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
Substitute the value of
Therefore, the degeneracy is
Similarly the value of
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,
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Chapter 14 Solutions
Physical Chemistry