In the Classroom
Using Punnett Squares To Facilitate Students’ Understanding of Isotopic Distributions in Mass Spectrometry Lawrence T. Sein, Jr.† School of Science and Health, Philadelphia University, Philadelphia, PA 19144;
[email protected] Francis William Aston’s discovery of isotopes in 1912 complicated the understanding of the relationship between nuclear physics and chemistry. Dalton’s theory of the atom claimed that all atoms of a particular element were identical (1). For most elements, this is not true; there are species of the same element, with different masses, called isotopes. With the 1919 invention by Aston of the mass spectrometer, it is possible to weigh individual nuclei, thereby distinguishing element from element, and isotope from isotope. Using a mass spectrometer themselves, undergraduate students can experimentally weigh atoms as a part of their training and dispel the impression that atoms are somehow “unreal”. Mass spectrometry is of more than theoretical interest. Geologists can establish the relative dates of rocks by determining the ratio of isotopes present in a sample: potassium– argon, rubidium, and uranium–lead dating permit us a better informed understanding of the processes involved in the evolution of Earth. By making it possible to date human artifacts, radiocarbon dating (dating organic fragments—Willard F. Libby received the Nobel Prize in chemistry in 1960 for his work on this) allows us a greater understanding of the evolution of human culture (2). As the examples above show, mass spectrometry (MS) is an important topic in the university science curriculum because the isotopic information provided by MS is critical to applications in nuclear physics, geology, biology, anthropology, environmental science, and analytical chemistry. MS is useful for identifying pure compounds, being able to distinguish molecular fragments by their masses. Additional insight into the molecular structure is provided by the detection of isotopes of particular elements. While fluorine and phosphorus are examples of elements that exist in only a single stable isotope (3), important elements such as bromine, carbon, chlorine, and sulfur have distinctive abundance ratios among their isotopes, and these ratios can reveal the presence of the particular element in the compound (4). Despite the utility of mass spectrometry, interpretation of spectra by undergraduate students is frequently labored. Often, the rules presented in lecture are memorized without context; practice examples are either too complicated to be grasped, or too simple to simulate molecules of interest. Using an algorithm for the analysis of isotopic abundances would simplify instructors’ presentations, minimize students’ frustration, and increase students’ understanding. Punnett squares (5) are widely used in biology to predict the distribution of genetic traits in the offspring of sexual reproduction. So long as Mendel’s laws apply (the law of independent assortment, in particular), the Punnett square yields excellent agreement with experiment. Punnett squares are taught as early as the freshman year of high school, so
university students already have experience in their use. This adaptation of a technique borrowed from biology is particularly relevant to environmental science students. These students are frequently required to take a course in instrumental analysis, with mass spectrometry as a significant component of that course. Since these students may not have had as rigorous a training in mathematics as chemistry, physics, or engineering majors have had, using a biology technique to assist their understanding is particularly effective. Moreover, the key feature of the Punnett square is that it is a multiplication table. Students have experience with multiplication tables from their elementary education. Since the formulae of probability are conveniently written as a series of products, these multiplication tables have powerful predictive value for chemists and students of chemistry. Punnett Square Examples
Isotopic Distributions Resembling Genetic Crosses: Br The simplest nontrivial cases in mass spectrometry consist of those elements that exist in two isotopes. Bromine exists as 79Br and 81Br, and each isotope is almost equally abundant, resulting in a relative ratio of approximately oneto-one. This one-to-one ratio immediately suggests the oneto-one ratio found in the generation of sex cells by a parent heterozygous in a particular trait, or the progeny of a cross between one heterozygous parent and one homozygous parent (Table 1). Half of the parental sex cells will carry the A (dominant) allele, and the other half will carry the a (recessive) allele. A Punnett square (Table 2) can be drawn immediately for the case of a molecule (such as methyl bromide) with exactly one bromine atom. Notice that the probability of possessing one 79Br is virtually the same as possessing one 81 Br, demonstrating a one-to-one ratio.
Table 1. Punnett Square for Cross of Parent Homozygous for Trait with Parent Heterozygous for Trait Aa AA
A
a
A
0.2500
0.2500
A
0.2500
0.2500
Table 2. Normalized Isotopic Distribution of Molecules Containing Exactly One Bromine Atom Isotopes of Bromine 79
Br
0.5069
81
Br
0.4931
†
Current address: Science Department, Cabrini College, Radnor, PA 19087.
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Isotopic Distributions Resembling Genetic Crosses: Cl 35
Chlorine, on the other hand, exists as isotopes Cl and Cl, with 35Cl three times more abundant than 37Cl (6), accounting for chlorine’s peculiar molecule weight of 35.45. For a compound such as methyl chloride, which possesses exactly one chlorine atom, we are reminded of a first generation cross between parents that are heterozygous for a single trait (i.e., a cross Aa Aa). In the second generation, the offspring will be found with genotypes AA, Aa, aa, in ratio of 1:2:1 (Table 3). If A is dominant over a, then the phenotype (actual appearance of the offspring) will exist in a 3:1 ratio (AA and Aa exhibiting the dominant trait, and only aa exhibiting the recessive trait). Notice that the relative ratio of probabilities in Table 4 is three to one. This, so far, demonstrates the utility of drawing an analogy between genetic processes and nomenclature with which students may already be familiar, and chemical processes and nomenclature with which they may not. Nevertheless, there are two difficulties with this approach. First, the ability to utilize Punnett squares seems linked to only those cases for which the probabilities commonly occur in genetics. Sec-
37
Table 3. Punnett Square for Cross of Parents Each Heterozygous for Trait Aa Aa
A
a
A
0.2500
0.2500
a
0.2500
0.2500
Table 4. Normalized Isotopic Distribution of Molecules Containing Exactly One Chlorine Atom Isotopes of Chlorine 35 37
P-Values for Isotopic Distribution
Cl
0.7577
Cl
0.2423
Table 5. Normalized Isotopic Distribution of Molecules Containing Exactly Two Bromine Atoms 79
Br Br
79
Br
81
Br
81
Br
Br
0.5069
0.4931
0.5069
0.2569
0.2500
0.4931
0.2500
0.2431
Table 6. Normalized Isotopic Distribution of Molecules Containing Exactly Two Chlorine Atoms 35
Cl Cl
37
Cl
Cl
0.7577
0.2423
35
0.7577
0.5741
0.1836
37
0.2423
0.1836
0.0587
Cl Cl
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ondly, it does not seem possible to use the approach to account for probabilities that are not exactly integral (e.g., a one-to-one ratio, or a three-to-one ratio).
Compounds Whose Isotopic Distributions Do Not Resemble Genetic Crosses Both these difficulties can be overcome by returning to our conception of the Punnett square as a multiplication table (7). The law of multiplication for probabilities states that if the probability of event A is PA, and the probability of event B is PB, then the probability of both A and B occurring is PA times PB (8). If we know the exact probability of finding isotope A (call it PA) and the exact probability of isotope B (call it PB) then the probability of find exactly one isotope A and exactly one isotope B would be PA times PB. PA times PB is the entry in the multiplication table where column A intersects column B. Because isotopes of the same mass are indistinguishable, a complication arises. Therefore, the total probability of a combination of isotopes requires the addition of all the probabilities found on the multiplication table for a single event. The product of the appropriate rows in the multiplication table gives the relative weight of each distribution. In the case of a compound with exactly two bromine atoms (Table 5), such as methylene bromide, the probability of two 79 Br’s is 1 4 1/4, the probability of one 79Br and one 81 Br is (1 1) 4 1/2, and the probability of two 81Br is 1 4 = 1/4. The sum of all possible combinations in the Punnett square is four. The ratio of 1:2:1 is obtained easily, and is easily understood in the context of genetics, as a similar ratio is obtained when crossing two parents homozygous for the same trait ( Aa Aa ). In the case of a compound with exactly two chlorine atoms (Table 6), for example, methylene chloride, the probability of two 35Cl’s is 9 16 0.5625, the probability of one 35 Cl atom and one 37Cl atom is (3 3) 16 0.375, and the probability of two 37Cl atoms is 1 16 0.0625. The sum of all possible combinations in the Punnett square is 16. By using the ratios of whole numbers in the square, we immediately obtain the distinctive 9:6:1 ratio for chlorine. The product of the appropriate rows in the multiplication table gives the normalized weight of each distribution. The probability of two 35Cl’s is 0.5741, the probability of one 35Cl and one 37Cl is 0.1836 0.1836 0.3672, and the probability of two 37Cl is 0.0587. The sum of all possible combinations in the Punnett square is 1; therefore the probabilities are normalized. Note that probabilities calculated by the two methods described in the previous two methods differ slightly, due to the fact that in the previous paragraph we rounded the isotopic probability for chlorine to the nearest whole numbers (i.e., three to one). We can modify the Punnett square to contain graphical information (Figures 1 and 2). We do this by making the width of each column in the multiplication table proportional to the probability of its entry. For example, a column containing the factor 2 will be twice as wide as a column containing the factor 1, and a column containing the factor 0.5 will be half as wide. Equivalent “proportioning” is effected on the rows of the multiplication table. The areas of the factors in the table are now strictly proportional to the probabilities. It becomes very easy, without even looking at the
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numbers, to determine the relative probability of events. Incorporating chemical information in a graphical image that is easy to remember increases the likelihood that students will remember the derivation of the information—and that they will be able to reproduce it later, unassisted. The situation with carbon is more complex (9). The isotopic distribution of 13C is small, so the probability for more than one carbon atom in a small molecule being 13C can be ignored. For large molecules, the probability cannot be ignored, and poses difficulties. While the technique of Punnett squares can be applied in this circumstance also, we restrict this presentation to elements whose isotopic distributions are more nearly equal, so as to be able to apply the technique to the relatively small molecules usually studied in an introductory analytical chemistry–instrumental methods course. In this instance, experimental determinations are obtained by GC–MS using electron ionization, which restricts the compounds in question to low-boiling liquids, effectively limiting the maximum molecular mass. Applying the Punnett Square Analogy
Experiments for Undergraduates Obtaining the low-resolution mass spectra of the following series of compounds can effect a detailed application to an actual course: methyl chloride, methylene chloride, chloroform, methyl bromide, methylene bromide, and bromoform.
This provides examples of compounds with exactly one, two, or three chlorine atoms, and exactly one, two, or three bromine atoms. For greater complexity, a compound with two different halogens, such as 1,1-bromochloro-ethane can be employed. The most convenient single sulfur compound is probably dimethyl sulfoxide. For examples of compounds with exactly two sulfur atoms, a number of α, ω-alkanethiols are liquids at room temperature (10).
Determining Molecular Peaks In the low-resolution MS experiment, the spectrum for each compound will consist of a distribution of molecular fragments (peaks), located at the appropriate mass/charge (m/z) ratio along with their relative peak heights, which are proportional to the probabilities. One peak, called the monoisotopic molecular peak, denoted M, is the peak corresponding to the molecular mass of the compound for which each atom is the lowest mass isotope. The inability to determine M for many classes of compounds is a complicating factor in identification. Fortunately, there is little difficulty in determining the M peak for the suggested compounds listed above. For methyl chloride (a molecule containing exactly one chlorine atom), the M peak will be present at 50, and an additional peak (the M2 peak) at 52. The ratio of the heights M:M2 will be 3:1. Often, the value of the M peak is set to 100, and the other peaks given as a fraction of that. In this case, the M peak will have a height of 100, and M2 will be a height of 33 (Table 7). A similar analysis can be applied to a molecule that contains two chlorine atoms, such as methylene chloride. The molecular peak, consisting of the molecule with both chlorines 35Cl, is M. If there is exactly one 37Cl and one 35Cl,
Table 7. Scaled Isotopic Distribution of Molecules Containing Exactly Two Chlorine Atoms
Figure 1. Normalized Punnett square for molecule (e.g., methylene bromide) with exactly two bromine atoms. Areas of boxes are proportional to the probability of a particular distribution. Identical distributions (except for ordering of isotopes) are displayed in the same color. For the purposes of calculating peak heights, areas of the same color must be added together.
35
Cl Cl
35
Cl
37
Cl
37
Cl
Cl
0.7577
0.2423
0.7577
100.0
31.9
0.2423
131.9
10.2
Table 8. Additive Punnett Square for Molecules Containing Two Bromine Atoms 79
Br Br
Figure 2. Normalized Punnett square for molecule (e.g., methylene chloride) with exactly two chlorine atoms. Areas of boxes are proportional to the probability of a particular distribution. Identical distributions (except for ordering of isotopes) are displayed in the same color. For the purposes of calculating peak heights, areas of the same color must be added together.
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Br
81
Br
79
Br
M
M2
81
Br
M2
M4
Table 9. Additive Punnett Square for Molecules Containing Two Chlorine Atoms 35
Cl Cl
Cl
37
Cl
35
M
M2
37
M2
M4
Cl Cl
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In the Classroom
the m/z ratio will be M2, and if both are 37Cl, m/z will equal M4. The situation becomes more complicated when there are multiple isotope effects to consider. The additive Punnett square (Tables 8 and 9) correlates a specific peak to a particular isotopic combination. It is common to assign the M peak in such a way that it is the smallest of the possible values, since the most abundant isotope of an element is usually the one with the smallest mass. This way, there will be no terms in the additive Punnett square less than M. Paradoxically, this illustrates the conversion of the Punnett square (formerly a multiplication table) into an addition table. Notice that the peaks (M, M2, M4) will be located at the same m/z values, whether the compound has exactly two bromine atoms or has exactly two chlorine atoms. Though not shown, these values will also be the same in the case of a molecule with one chlorine atom and one bromine atom. Therefore, the additive Punnett square is not sufficient in itself, in many circumstances, for positive identification of a compound, without the additional information as to the relative peak heights determined from the multiplicative square. A more complex example of the additive square is shown in Table 10, with its normalized version shown in Table 11, for comparison. There, the peak locations are shown for the case of a molecule with exactly two sulfur atoms. But we can do even more. If we make the probability of each factor in the Punnett square multiplication table proportional to its probability, we retain the interpretation of areas being proportional to their probabilities. In addition, if we write the value of the molecular peak index (e.g., M4) in the appropriate box, we can combine both additive and multiplicative information into the same square (Figure 3). Now students can conveniently correlate the particular peak
Table 10. Additive Punnett Square for Molecules Containing Exactly Two Sulfur Atoms SS
32
33
S
34
S
36
S
S
32
S
M
M1
M2
M4
33
S
M1
M2
M3
M5
34
S
M2
M3
M4
M6
36
S
M4
M5
M6
M8
Figure 3. Additive Punnett square for molecule (e.g., methylene chloride) with exactly two chlorine atoms. Areas of boxes are proportional to the probability of a particular distribution. Identical distributions (except for ordering of isotopes) are displayed in the same color. For the purposes of calculating peak heights, areas of the same color must be added together.
Figure 4. Normalized Punnett cube for molecule (e.g., bromoform) with exactly three bromine atoms. Volumes of boxes are proportional to the probability of a particular distribution. Identical distributions (except for ordering of isotopes) are displayed in the same color. For the purposes of calculating peak heights, volumes of the same color must be added together.
Table 11. Normalized Isotopic Distribution for Molecules Containing Exactly Two Sulfur Atoms 32
SS
S
33
34
S
36
S
S
0.9502
0.0075
0.0421
0.0002
0.9502
0.9529
0.0071
0.0400
0.0002
S
0.0075
0.0071
0.0001
0.0003
0.0000
34
S
0.0421
0.0400
0.0003
0.0018
0.0000
36
S
0.0002
0.0002
0.0000
0.0000
0.0000
32
S
33
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Figure 5. Normalized Punnett cube for molecule (e.g., chloroform) with exactly three chlorine atoms. Volumes of boxes are proportional to the probability of particular distribution. Identical distributions (except for ordering of isotopes) are displayed in the same color. For the purposes of calculating peak heights, volumes of the same color must be added together.
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in the mass spectrograph with the isotopic combination that produced it.
Extensions to Multiple Dimensions We are not limited to a maximum of two elements when using Punnett squares: there are two methods for extending to higher dimensions. The first is to utilize the 3-D analog of the square, the cube. Each of the three mutually perpendicular edges represents the distribution of one of three elements, so we can easily visualize a compound with three elements of interest. Such molecules are bromoform (Figure 4) and chloroform (Figure 5). If we utilize our previous convention so that lengths are proportional to probabilities, each component of the cube (each a parallelepiped) has a volume proportional to its probability. Again, we must be careful to take account of the indistinguishability of isotopes of the same mass, and add together the volumes corresponding to equivalent cases. In Figures 4 and 5, the equivalent cases are noted by properly coloring the parallelepiped. There are four different cases: all three halogens the same isotope (white), two of the lighter isotope and one of the heavier (light gray), one of the lighter and two of the heavier isotopes (medium gray), and all three of the heaviest isotopes (dark gray). This technique can be applied to elements with any number of isotopes, but it is limited to a maximum of three elements. A second technique that can be applied to three or more elements exploits the fact that the multiplication of probabilities is associative. If an is the probability corresponding to the nth of N elements, the total probability of an a1 a2 a3 ... aN. We can always convert this into a twodimensional problem by grouping a1 a2 a3 ... aN1 and calling it b. Now the total probability is b aN. (There are also other ways of grouping that will be completely legitimate). We form the Punnett square by listing the probabilities corresponding to b as the rows of the square, and those corresponding to aN on the columns (Figure 6).
Algorithm for Using Punnett Squares with MS Having described several different applications of Punnett square to mass spectrometry, we can formally state an algorithm applicable to an introductory course in instrumental analysis. 1. Form a normalized Punnett square locating one element on each axis. 2. Form an additive Punnett square in which M corresponds to the monoisotopic peak. 3. Form a scaled Punnett square by dividing each entry in the normalized Punnett square by the entry corresponding to the M peak.
If there are more than three polyisotopic atoms in the compound, it is necessary to form a Punnett square for exactly two of the elements, and then combine the entries into a single row. This single row is used to form a second Punnett square with the third element. These probabilities are again combined into a single row, to form a third Punnett square with the fourth element. This process continues until all the polyisotopic elements have been accounted for.
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Figure 6. Normalized Punnett square for molecule (e.g., chloroform) with exactly three chlorine atoms. Areas of boxes are proportional to the probability of a particular distribution. Identical distributions (except for ordering of isotopes) are displayed in the same color. For the purposes of calculating peak heights, areas of the same color must be added together. Notice that three dimensions of information can be condensed into the two dimensions of the Punnett square.
Conclusion Punnett squares are familiar, easy to compute, and often graphical—this makes them helpful to students. The relative distribution of isotopic combinations is easily generated for even extraordinarily complicated molecules. Further, the isotopic distribution (the molecular-level view) can be correlated with the peak heights (the laboratory-level view). This is sufficient to make the Punnett square even more useful in teaching of chemistry than in the teaching of biology, because biological scenarios are always limited to two parents, whereas elements can have up to ten naturally occurring stable isotopes. Literature Cited 1. Aston, Francis W. Mass Spectra and Isotopes. In Nobel Lectures, Chemistry 1922–1941; Elsevier Publishing Company: Amsterdam, 1966; http://nobelprize.org/chemistry/laureates/ 1922/aston-lecture.pdf (accessed Oct 2005). 2. Schwarcz, Henry P. Acc. Chem. Res. 2002, 35 (8), 637–643. 3. For up-to-date isotopic abundances, see: Lite, D. R. CRC Handbook of Chemistry and Physics, 81st ed.; CRC Publishing: New York, 2000. 4. Richardson, S. D. Chem. Rev. 2001, 101 (2), 211–254. 5. Campbell, N. A. Biology; Benjamin/Cummings Publishing Company, Inc.: Menlo Park, CA, 1987; pp 263–264. 6. Emsley, J. The Elements, 2nd ed.; Oxford University Press, Inc.: New York, 1991; pp 37, 47, 51. 7. Fraleigh, J. B. A First Course in Abstract Algebra; AddisonWesley Publishing Company, Inc.: Reading, MA, 1982; p 13. 8. Ghahramani, S. Fundamentals of Probability; Prentice-Hall, Inc.: Upper Saddle River, NJ, 1996. 9. Silverstein, R. M. Spectrometric Identification of Organic Compounds, 6th edition; John Wiley and Sons, Inc.: New York, 1998; pp 7–8, 34. 10. Thalladi, Venkat R.; Boese, Roland; Weiss, Hans-Christoph. J. Am. Chem. Soc. 2000, 122 (6), 1186–1190.
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