Group Activity:
Average Atomic Mass

Introduction

As you know, atoms of an element come in different varieties called isotopes. For example, the common isotopes of nitrogen include¹⁵₇N and¹⁴₇N. Isotopes are atoms of the same element that have different mass numbers due to the fact that they have different numbers of neutrons. The isotopes of an element are not all equally common. Of all the nitrogen atoms on Earth 99.63% of them are¹⁴₇N. Only 0.37% of all nitrogen atoms are¹⁵₇N. That is why the average atomic mass of N as given in the Periodic Table is closer to 14 amu than to 15 amu.

Percent Abundances

The percentages given above for the isotopes of nitrogen are called percent abundances. The percent abundance of an isotope tells you the fraction of all atoms of an element that are a particular isotope of that element. Think of it this way: if you have 10,000 nitrogen atoms then 9,963 of them are¹⁴₇N and 37 of them are¹⁵₇N. Percentages are just a way to write fractions so that they all have the same denominator: 100. So the percentage 99.63% means 99.63/100 and 0.37% means 0.37/100. Percentages need to be written as decimals in order to use them in calculations. Therefore 99.63% (99.63/100) is 0.9963 and 0.37% (0.37/100) is 0.0037. Just move the decimal point two places to the left.

Weighted Averages

Average atomic masses are computed using a method called weighted averages. Weighted averages are used when the importance of the numbers to be averaged are different. For example, at Scarborough High School a student’s semester average is computed based on three grades. The First Quarter grade (Q₁), the Second Quarter grade (Q₂), and the Semester Exam grade (R₁). Both quarter grades are weighted at 40% (W_Q1 = W_Q2 = 0.40) of the semester average and the exam is weighted at 20% (W_R1 = 0.20).

Say a Chemistry student named Lyle has a Q₁ grade of 85, a Q₂ grade of 80 and a R₁ grade of 87. What is Lyle’s semester 1 grade? Set up the calculation this way:

              Q1·WQ1   +  Q2·WQ2    +  R1·WR1    =  S1
             (85)·0.40 + (80)·0.40 + (87)·0.20 = 83.4

Now, what if Lyle scored poorly during the first quarter (he earned a 75) but improved during the second (he earned an 89) and he needs to know the minimum exam grade to earn an 85 for the semester? Just use a little algebra to solve the problem:

              Q1·WQ1    +  Q2·WQ2 +  R1·WR1 = S1
             (75)·0.40 + (89)·0.40  + x·0.20 = 85
             0.20x = 85 - (75)·0.40 - (89)·0.40
             0.20x = 19.4
                x = 97

In short, Lyle will have to work very hard to get the grade he needs on the semester exam.

Average Atomic Masses

Average atomic masses are calculated in just the same way. Each isotope’s exact mass (determined using a mass spectrometer) is multiplied by the decimal equivalent of its percent abundance and all the results are added together.

   38.963707 · 0.932581 
   39.963999 · 0.000117 
 + 40.961826 · 0.067302
 -------------------------------
   39.098301 amu

What if you know the percent abundance of the isotopes of an element but not all of the exact masses? Use algebra to find the missing exact mass. Take a look at the potassium data and imagine that the exact mass of ⁴¹₁₉K is unknown. Use the average mass of the element, the known exact masses, and the percent abundances to find the missing exact mass data.

38.963707 · 0.932581 + 39.963999 · 0.000117 + x · 0.067302 = 39.098301
x · 0.067302 = 39.09830144 - 36.34148863
x · 0.067302 = 2.756812809
x = 2.756812809/0.067302 = 40.961826

Since we made up this problem we can check the answer against the exact mass given in the table: it matches perfectly. The exact mass of⁴¹₁₉K is 40.961826 amu.

One thing that is important to getting the calculations of this kind exactly correct is to use your calculator to hold all the digits for you while you work. Set up your math so that you can type the numbers and operations into your calculator in one long calculation without having to stop, write down an answer and re-enter it. This prevents rounding errors which will cause your answers to be wrong.

Problems

Show your work for all the following calculations. The masses are atomic masses expressed in atomic mass units (amu). Write the symbol for each isotope. Pay attention to significant figures: do not report any more numbers after the decimal point than are reported in the data you are given.

1. Ryan’s grades for the first semester are Q₁: 76, Q₂: 84, R₁: 86. What is his semester grade?
2. Marcus would like to earn a 93 for the semester. If his Q₁ grade is 90 and his Q₂ grade is 93 what grade does he need on the semester exam?
3. Leslie’s first quarter grade was 82. She wants to know what grades she needs for Q₂ and R₁ to earn a 90 for the semester. Assume that she will earn the same grade for Q₂ and R₁.
4. Find the average atomic mass of carbon (C ).
   

   Symbol	Mass number	Exact mass	Percent abundance
   	12	12.000000	98.93
   	13	13.003355	1.07

5. Find the average atomic mass of chlorine (Cl).
   

   Symbol	Mass number	Exact mass	Percent abundance
   	35	34.968853	75.78
   	37	36.965903	24.22

6. Find the average atomic mass of nitrogen (N).

   Symbol	Mass number	Exact mass	Percent abundance
   	14	14.003074	99.63
   	15	15.000108	0.37

7. Find the average atomic mass of neon (Ne).
   

   Symbol	Mass number	Exact mass	Percent abundance
   	20	19.992440	90.48
   	21	20.993847	0.27
   	22	21.991386	9.25

8. Find the average atomic mass of magnesium (Mg).

   Symbol	Mass number	Exact mass	Percent abundance
   	24	23.985042	78.99
   	25	24.985837	10.00
   	26	25.982593	11.01

9. Find the average atomic mass of chromium (Cr).

   Symbol	Mass number	Exact mass	Percent abundance
   	50	49.946049	4.345
   	52	51.940511	83.789
   	53	52.940653	9.501
   	54	53.938884	2.365

10. Find the average atomic mass of molybdenum (Mo).

    Symbol	Mass number	Exact mass	Percent abundance
    	92	91.906808	14.84
    	94	93.905085	9.25
    	95	94.905840	15.92
    	96	95.904678	16.68
    	97	96.906020	9.55
    	98	97.905406	24.13
    	100	99.907477	9.63

11. There is an easy way to check whether your answers are correct when you calculate the average atomic mass for the naturally occurring isotopes of an element. Explain this easy way to check your answers.
12. An element “X” has five major isotopes, which are listed below along with their abundances. Calculate the average atomic mass. What is the element?

    Symbol	Exact mass	Percent abundance
    46X	45.952630	8.25
    47X	46.951764	7.44
    48X	47.947947	73.72
    49X	48.947871	5.41
    50X	49.944792	5.18

13. An element consists of 1.4% of an isotope with mass 203.973 amu, 24.1% of an isotope with mass 205.9744 amu, 22.1% of an isotope with mass 206.9759 amu, and 52.4% of an isotope with mass 207.9766 amu. Calculate the average atomic mass and identify the element.
14. The element rhenium (Re) has two naturally occurring isotopes, ¹⁸⁵Re and ¹⁸⁷Re, with an average atomic mass of 186.207 amu. Rhenium is 62.60% ¹⁸⁷Re, and the atomic mass of ¹⁸⁷Re is 186.956 amu. Calculate the mass of ¹⁸⁵Re.
15. The isotopes of silicon are²⁸ ₁₄Si,²⁹ ₁₄Si, and³⁰ ₁₄Si. Find the percent abudance and exact mass of silicon-28 using the following information:

    Symbol	Mass number	Exact mass	Percent abundance
    28 14Si	28
    29 14Si	29	28.9765	4.69
    30 14Si	30	29.9737	3.10