This paper discusses our “Dry Lab” experiment where students are shown the procedure for calculating the nominal and precise M+ values of a compound. However, our main focus deals with the detailed step-by-step process for calculating the precise M+1 and M+2 values of a compound, as well as any of its higher order M+ values, when its molecular formula is known.
One of the topics taught in all organic chemistry courses is the identification of organic compounds. There are a number of wet chemistry methods as well as a number of instrumental methods for classifying and identifying organic compounds. Some very powerful instrumental methods involve the spectral data, and its interpretation, obtained from NMR, IR, UV, and mass spectrometry. Of the instrumental methods just mentioned, this paper will concentrate on mass spectroscopy. Mass spectral data not only provides the masses of the various fragments of molecules that are produced when molecules in the vapor phase are bombarded with a high-energy electron beam, but can furnish the mass of the molecular ion, M+. A mass spectrum can also display one or more higher order M+ peaks, for example, M+1 and M+2, which taken together with the M+ peak can be of value when identifying an unknown compound. In addition to a brief discussion regarding the calculation of a compound's M+ value, a detailed step-by-step calculation of the higher order M+ peaks, M+1 and M+2, which arise as a consequence of the various isotopes of the elements that are in the compound being studied, are the main focus of this paper. We have included our computer program to aid instructors in preparing keys for the students which shows in detail all of the steps involved in these calculations.
The information needed to make these calculations for this "Dry Lab" experiment are the molecular formula of the compound, some isotopic data of the elements in the compound, and the equations needed to calculate the value of M+, M+1, and M+2. The isotopic data needed are the nominal and precise atomic masses of each of the major isotopes of each element in the compound, the percentage of each isotope of each element in the compound, and the number of significant figures in each of these percentages. The isotopic data used in this paper is listed in Table 1.
There are two M+ quantities that are calculated below: the nominal and precise M+ values. As can be seen in Table 2, these calculations are straight forward requiring one to simply multiply the number of atoms of each element in the compound by its nominal mass found in Table 1 and then add each of these elements contribution to obtain the nominal M+ value. The precise M+ value, to six decimal places, is calculated in a like manner as shown in Table 3.
There is only one case or one group of isotopes to consider when calculating the value of the M+1 peak of bromothymol blue, the elements in the compound that have an M+1 isotope. Thus, one could begin by listing all of these elements as shown in the first column of Table 4. Then calculate the contribution that each one of these isotopes contributes toward the total value of the M+1 peak, starting in this example with hydrogen, the first isotope listed in Table 4. The quantity that hydrogen would contribute to the M+1 peak is calculated by taking the ratio of the percentage of the H-2 isotope divided by the percentage of the H-1 isotope times the number of hydrogens atoms in the compound times 100% 5. This calculation just described is shown in red in Table 4. The next line in Table 4 has the appropriate numerical data substituted into the formula with the results of the calculation given in the right hand column. This process is repeated for each of the remaining isotopes listed in column 1 of Table 4. Adding all of these contributions and rounding the answer, showing just the significant figures, gives the value of the M+1 peak as a percentage of the intensity of the molecular ion peak, M+. The reason there are so many decimal places included in these calculations was to demonstrate to the students how little some of the sets of isotopes actually contribute to these final calculated values.
There are three cases to consider when calculating the M+2 value for bromothymol blue: (1) two M+1 isotopes of the same element, (2) two M+1 isotopes of different elements, and (3) one M+2 isotope of one element. In order to calculate the M+2 value, one could start by listing all the sets of the isotopes that will produce an M+2 value in the first column of Table 5. Then begin the calculations with the first set of isotopes listed for case 1 in Table 5, two H-2 isotopes of hydrogen. If there were just two hydrogen atoms in the molecule, the quantity they would contribute to the M+2 peak would be calculated by taking the ratio of the percentage of the H-2 isotope divided by the percentage of the H-1 isotope. This factor is repeated a second time for the second H-2 isotope, thus this ratio appears twice, once for each H-2 isotope. Another way of expressing this mathematically would be to write this ratio once and then square it, and then multiply that result by 100% 5. However, bromothymol blue doesn’t have just two hydrogen atoms in its formula, it has 28, so this squared ratio just mentioned must be multiplied be the number of ways that the two H-2 isotopes can occur among the 28 hydrogen atoms in the molecule 6, 7, which mathematically speaking is the combination of 28 objects selected two at a time. The formula for calculating the number of combinations possible for this case is as follows: [(the total number of hydrogen atoms in the molecule) factorial] divided by [(the number of H-2 isotopes in the molecule) factorial times (the number of H-1 isotopes in the molecule) factorial)] 6, 7. This calculation just described is shown in red in Table 5 below.
The number of combinations that two H-2 isotopes have when the molecule contains 28 hydrogens is calculated as described above: (28!)/[(2!)(26!)] 6, 7. Another way of expressing this quantity mathematically is (28)(27)(26!)/[((2!)(26!)], which reduces to (28)(27)/2. The next line in Table 5 has the appropriate numerical data substituted into this formula with the results of the calculation being given in the right hand column. This process is then repeated for the two remaining examples of case 1 in Table 5, 2 C-13 and 2 O-17.
The next calculation discussed involves an example from case 2 where there are two different M+1 isotopes, 1 H-2 and 1 C-13, contributing to the value of M+2 for bromothymol blue. This calculation is similar to the calculation made for the M+1 value. Here the ratio of the percentage of the H-2 isotope is divided by the percentage of the H-1 isotope times the number of hydrogen atoms in the compound times the ratio of the percentage C-13 isotope divided by the percentage of the C-12 isotope times the number of carbon atoms in the compound 5, 6, 7 times 100%. This calculation just described is shown in red in Table 5. The next line in Table 5 has the appropriate values substituted into the formula with the results of the calculation given in the right hand column. This process is then repeated for the five remaining examples of case 2 in Table 5, 1 H-2 and 1 O-17, 1 H-2 and 1 S-33, 1 C-13 and 1 O-17, 1 C-13 and 1 S-33, 1 O-17 and 1 S-33.
The next calculation discussed involves an example from case 3 where there is only one atom of one isotope, 1 O-18, required to generate its contribution to the M+2 value for bromothymol blue. This calculation is similar to the calculations made for the M+1 value. Here the ratio of the percentage of the O-18 isotope is divided by the percentage of the O-16 isotope times the number of oxygen atoms in the compound times 100% 5, 6, 7. This calculation just described is shown in red in Table 5. The next line in Table 5 has the values substituted into the formula with the results of the calculation given in the right hand column. This process is then repeated for the two remaining examples of case 3 in Table 5, 1 S-34 and 1 Br-81. Adding each of these contributions and rounding the answer, showing just the significant figures, gives the value of the M+2 peak as a percentage of the intensity of the molecular ion peak, M+.
After a detailed discussion of this material with the lab students, they were told to calculate the nominal and precise M+ values as well as the M+1 and M+2 values for all the sets of isotopes that produce these values for butane, 1-bromobutane, 1,2-dibromobutane, 1,2,3-tribromobutane, 1,2,3,4-tetrabromobutane, and bromothymol blue. These papers were graded and returned the students, along with the key for each of these problems. There were three problems on the midterm lab test covering these calculations worth a total of 16 points. The students were asked to calculate the precise M+ value worth 1 point, M+1 value worth 4 points, and the M+2 value worth 11 points, for C5H9OBr. Table 6 lists the grade distribution and average grade for each of these problems in columns 2, 3, and 4 with column 5 listing the students grade distribution based on their total points earned on these three problems. The class average of 60 students was 83.9.
When calculating any of the higher order M+ values of a compound, be it M+1, M+2, M+3, M+4, etcetera, one should first list all of the sets of isotopes that will produce the M+ value being sought. If there is just one element present in a set of isotopes, one takes the ratio of this minor isotope percentage to its major isotope percentage and raises that result to the power of the number of the isotopes of that minor isotope in the compound. This quantity is then multiplied by the number of combinations possible for the minor isotope amongst the total number of atoms of that element in the compound 5. The number of combinations possible is calculated by taking the factorial of the total number of atoms of that element in the compound and divide it by the (factorial of the number of atoms of the minor isotope times the factorial of its number of atoms of the major isotope) 6, 7. Then multiply this result by 100%. The mathematical formula just described is shown below.
 | (1) |
If there are two different elements in a set of isotopes, the formula above is applied to each of the elements as shown below multiplied by 100%. Thus, when there are two elements in a set of isotopes when calculating their contribution to a higher order M+ value the formula used would be:
 | (2) |
Thus, in order to calculate the value of any of the higher order M+ values, all one needs to do is to apply formula 1 to each element that appears in a set of isotopes and then multiply that result by 100% 5. This process is then repeated for all of the sets of isotopes that will produce the higher order M+ value being calculated. The value of the desired higher order M+ value is obtained when the sum of each of these calculations is rounded showing just the significant figures.
All that the program requires of the user is to enter the formula and its name or identifier. That is all there is to it. All of the step-by-step calculations will be displayed on the screen as well as recorded in a disk file named from the formula and name or identifier entered into the program as mentioned above. All of this is explained in detail in the supporting information that will be provided along with the program, upon request. A typical run of the program is shown below for bromothymol blue.
The authors have no competing interest.
We thank the Department of Chemistry and Physics of West Texas A&M University for their financial support of this project.
| [1] | J. S. Coursey, D. J. Schwab, J. J. Tsai, and R. A. Dragoset, "Atomic Weights and Isotopic Compositions with Relative Atomic Masses", NIST Physical Measurement Laboratory, Last update: January 2015. | ||
| In article | |||
| [2] | Commission on Isotopic Abundances and Atomic Weights, International Union of Pure and Applied Chemistry, Last update, August 21, 2018. | ||
| In article | |||
| [3] | Meng Wang, G., F.G. Kondev, W.J. Huang, S. Naimi, and Xing Xu, "The AME2016Aatomic Mass Evaluation (II). Tables, Graphs and References", Chinese Physics C 41(3), 30003-6-30003-74, 2017. | ||
| In article | View Article | ||
| [4] | Michael Berglund and Michael E. Wieser, "Isotopic Compositions of the Elements 2009" (IUPAC Technical Report), Pure Appl. Chem., 83(2), 397-410, 2011. | ||
| In article | View Article | ||
| [5] | J. H. Beynon, "Mass Spectrometry and its Application to Organic Chemistry," Elsevier, Amsterdam, 1960, 296. | ||
| In article | |||
| [6] | Joseph K. Blizstein and Jessica Hwang, "Introduction to Probability", CRC Press, Taylor & Francis Group, 2015, 14. | ||
| In article | |||
| [7] | Harold J. Larson, "Introduction to Probability," Addison-Wesley, Publishing Company Inc., 1995, 33. | ||
| In article | |||
Published with license by Science and Education Publishing, Copyright © 2020 Andrew M. Kelley, Stephanie Garcia, David R. Khan, William G. Ambrose, Jason C. Yarbrough, James D. Woodyard and James C. Espinosa
This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit
https://creativecommons.org/licenses/by/4.0/
| [1] | J. S. Coursey, D. J. Schwab, J. J. Tsai, and R. A. Dragoset, "Atomic Weights and Isotopic Compositions with Relative Atomic Masses", NIST Physical Measurement Laboratory, Last update: January 2015. | ||
| In article | |||
| [2] | Commission on Isotopic Abundances and Atomic Weights, International Union of Pure and Applied Chemistry, Last update, August 21, 2018. | ||
| In article | |||
| [3] | Meng Wang, G., F.G. Kondev, W.J. Huang, S. Naimi, and Xing Xu, "The AME2016Aatomic Mass Evaluation (II). Tables, Graphs and References", Chinese Physics C 41(3), 30003-6-30003-74, 2017. | ||
| In article | View Article | ||
| [4] | Michael Berglund and Michael E. Wieser, "Isotopic Compositions of the Elements 2009" (IUPAC Technical Report), Pure Appl. Chem., 83(2), 397-410, 2011. | ||
| In article | View Article | ||
| [5] | J. H. Beynon, "Mass Spectrometry and its Application to Organic Chemistry," Elsevier, Amsterdam, 1960, 296. | ||
| In article | |||
| [6] | Joseph K. Blizstein and Jessica Hwang, "Introduction to Probability", CRC Press, Taylor & Francis Group, 2015, 14. | ||
| In article | |||
| [7] | Harold J. Larson, "Introduction to Probability," Addison-Wesley, Publishing Company Inc., 1995, 33. | ||
| In article | |||
