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Calculate the average atomic weight when given isotopic weights and abundances

Let's start by repeating the solution for nitrogen from the Average Atomic Weight tutorial:

(14.003074) (0.9963) + (15.000108) (0.0037) = 14.007

The solution is laid out like this:

(exact weight of isotope #1) (abundance of isotope #1) + (exact weight of isotope #2) (abundance of isotope #2) = average atomic weight of the element

In the other tutorial, the average atomic weight is the unknown value calculated. In this tutorial, the unknown values calculated are the **TWO** percent abundances.

However, you might protest that we have two unknowns, but only one equation. You would be right, except that there is a trick. Ah, the ChemTeam loves tricks!!

Think about the sum of the percent abundances of the TWO isotopes. That's right, they add up to 100% (or, since we use decimal abundances in the calculation, 1.00). So, the trick is to express both abundances using only one unknown. I'll show how using the above nitrogen example.

**Problem #1:** Nitrogen is made up of two isotopes, N-14 and N-15. Given nitrogen's atomic weight of 14.007, what is the percent abundance of each isotope?

Here's the solution:

(14.003074) (x) + (15.000108) (1 - x) = 14.007

Notice that the abundance of N-14 is assigned 'x' and the N-15 is 'one minus x.' This is the "trick" refered to above. The two abundances always add up to one (or, if you prefer, 100%)

Comment One (see end of problem for another comment): sometimes the exact weights of the isotopes are not provided in the problem. However, in this Internet era, it is easy to look up the values on-line. I advise you to ask your instructor what the question wants you to do if only the mass numbers are provided in the problem. Please notice that the mass numbers (14 and 15) are quite close to the exact values. The consequence of using 14 and 15 rather than the exact values is that we will get slightly approximate answers for the two abundances.

For example, I might have this:

(14) (x) + (15) (1 - x) = 14.007

Solving gives:

14x + 15 - 15x = 14.007

After some more simple algebra, we get:

x = 15 - 14.007 = 0.993

so

1 - x = 0.007

You may want to compare them to the more exact abundances which are in the equation set up at the top of the page.

Comment Two: only the mass numbers may be given if the element is a made-up one. For example: "ChemTeam-42 and ChemTeamium-44 exist. Calculate the abundances if the atomic weight is 42.68." Since ChemTeamium doesn't exist, you can't look up exact isotopic weights. So, use 42 and 44 in the solution to the problem.

**Problem #2a:** Copper is made up of two isotopes, Cu-63 (62.9296 amu) and Cu-65 (64.9278 amu). Given copper's atomic weight of 63.546, what is the percent abundance of each isotope?

**Solution:**

1) Write the following equation:

(62.9296) (x) + (64.9278) (1 - x) = 63.546

Once again, notice that 'x' and 'one minus x' add up to one.

2) Solve for x:

x = 0.6915 (the decimal abundance for Cu-63)

Note that this calculation technique works only with two isotopes. If you have three or more, there are too many variables and not enough equations. I hope it's obvious why you wouldn't do this with an element that has only one stable isotope!

By the way, the 'trick' works because the other equation required is:

x + y = 1

We simply went right to y = 1 - x and substitued it immediately without ever writing down the second equation required.

**Problem #2b:** Chlorine is made up of two isotopes, Cl-35 (34.969 amu) and Cl-37 (36.966 amu). Given chlorine's atomic weight of 35.453 amu, what is the percent abundance of each isotope?

1) Write the following equation:

(34.969) (x) + (36.966) (1 - x) = 35.453

2) Solve for x:

x = 0.7576 (the decimal abundance for Cl-35)

**Problem #3a:** A sample of naturally occurring silicon consists Si-28 (amu = 27.9769), Si-29 (amu = 28.9765) and Si-30 (amu = 29.9738). If the atomic mass of silicon is 28.0855 and the natural abundance of Si-29 is 4.67%, what are the natural abundances of Si-28 and Si-30?

Solution:

1) Set up a system of two equations in two unknowns:

Let x = isotopic abundance of Si-28 (as a decimal)

Let y = isotopic abundance of Si-30 (as a decimal)

Therefore:

(27.9769) (x) + (28.9765) (0.0467) + (29.9738) (y) = 28.0855and

x + 0.0467 + y = 1.000

2) Rearrange the second equation to:

y = 1.000 - 0.0467 - x

then simplify it, substitute into the first equation and solve.

Note: isotopic abundances of the elements are well-known values. You can find the answers to Problem #3 here.

**Problem #3b:** Naturally occurring silicon consists of 3 isotopes, Si-28, Si-29 and Si-30, whose atomic masses are 27.9769, 28.9765 and 29.9738 respectively. The most abundant isotope is Si-28 which accounts for 92.23% of naturally occurring silicon. Given that the observed atomic mass of silicon is 28.0855 calculate the percentages of Si-29 and Si-30 in nature.

**Solution:**

1) Set up a system of two equations in two unknowns:

Let x = isotopic abundance of Si-29 (as a decimal)

Let y = isotopic abundance of Si-30 (as a decimal)

Therefore:

(27.9769) (0.9223) + (28.9765) (x) + (29.9738) (y) = 28.0855and

0.9223 + x + y = 1.000

2) Rearrange the second equation to:

y = 1.000 - 0.9223 - x = 0.0777 - x

3) Substitute into the first equation and solve:

(27.9769) (0.9223) + (28.9765) (x) + (29.9738) (0.0777 - x) = 28.0855Algebra!

x = 0.04668518 = 4.67%

Comment: when I first encountered this problem (Sept 2011), the given masses for Si-29 and Si-30 were 28.9865 and 29.9838. Notice the value in the 0.01 place and compare it to the ones used in the problem. It is my opinion that these small changes were made by the author of the problem in order to discourage someone from going to a source like Wikipedia and simply copying down the abundances given there.

Therein lies a warning to all students in this, the age of the Internet.

**Problem #4:** Determine the percent abundance for Fe-57 and Fe-58, given the following data:

Isotope Atomic Weight Percent

AbundanceFe-54 53.9396 5.845 Fe-56 55.9349 91.754 Fe-57 56.9354 ??? Fe-58 57.9333 ???

**Solution:**

1) Assign the percent abundance of Fe-57 to the variable 'x'

2) We need to get the percent abundance for Fe-58 in terms of x. Like this:

1 minus (0.05845 + 0.91754 + x)the term (0.05845 + 0.91754 + x) is the total of the other three isotopic abundances.

The '1' represents 100% but expressed as a decimal value.

I will use (0.02401 minus x) just below.

3) Set up (and solve) the following equation:

(53.9396) (0.05845) + (55.9349) (0.91754) + (56.9354) (x) + (57.9333) (0.02401 - x) = 55.845Algebra ensues.

x = 0.0213

The Fe-58 percent abundance is left to the reader. You can verify that 0.0213 is the correct answer by searching for 'isotopes of iron' on-line.

Note that the problem does not supply the average atomic weight of iron. This value is easily available, both in books and on the Internet.

**Problem #5:** Naturally occurring strontium consists of four isotopes, Sr-84, Sr-86, Sr-87 and Sr-88, whose atomic masses are 83.9134, 85.9094, 86.9089 and 87.9056 amu respectively. The most abundant isotope is Sr-88 which accounts for 82.58% of naturally occurring strontium and the least abundant isotope is Sr-84 which accounts for 0.56% of naturally occurring strontium. Given that the observed atomic mass of strontium is 87.62 amu, calculate the percentages of Sr-86 and Sr-87 in nature

**Solution:**

1) Assign the percent abundance of Sr-86 to the variable 'x'

2) We need to get the percent abundance for Sr-87 in terms of x. Like this:

1 minus (0.0056 + 0.8258 + x)the term (0.0056 + 0.8258 + x) is the total of the other three isotopic abundances.

The '1' represents 100% but expressed as a decimal value.

I will use (0.1686 minus x) just below.

3) Set up (and solve) the following equation:

(83.9134) (0.0056) + (85.9094) (x) + (86.9089) (0.1686 - x) + (87.9056) (0.8258) = 87.62Algebra ensues.

x = 0.09525

When you search for information about the isotopes of strontium, you will find that the abundance for Sr-86 is given as 0.0986. I may have made a mistake somewhere, probably in the calculation to get the value for x. I did not go back and double-check my calculations.

**Problem #6:** Oxygen is composed of three isotopes: One has a a mass of 16.999 amu. This isotope makes up 0.037% of oxygen. Of the other two, one has a mass of 15.995 amu, and the other has a mass of 17.999 amu. Calculate the abundance of the other two isotopes, using the average atomic mass of 15.9994 amu.

**Solution:**

1) Set abundances (as decimal percents):

O-16: x

O-17: 0.00037

O-18: 0.99963 - xNote that 0.99963 is (1 minus 0.00037)

2) Set up average atomic weight equation:

(15.995) (x) + (16.999) (0.00037) + (17.999) (0.99963 - x) = 15.9994Algebra ensues.

x = 0.9976

3) The percent abundances are as follows:

O-16: 99.76%

O-17: 0.037%

O-18: 0.203%

**Problem #7:** Pb has an average atomic mass of 207.19 amu. The three major isotopes of Pb are Pb-206 (205.98 amu); Pb-207 (206.98 amu); and Pb-208 (207.98 amu). If the isotopes of Pb-207 and Pb-208 are present in equal amounts, calculate the percent abundance of Pb-206, Pb-207, Pb-208.

**Solution:**

1) Set the percent abundances:

Pb-206: 1 minus 2x

Pb-207: x

Pb-208: x

2) Set up average atomic weight equation:

(205.98) (1 - 2x) + (206.98) (x) + (207.98) (x) = 207.19Algebra ensues.

x = 0.403

3) The percent abundances are as follows:

Pb-206: 19.4%

Pb-207: 40.3%

Pb-208: 40.3%

Here is another version of the above question: The atomic weight of lead is 207.19 amu. Assuming that it's made up of entirely the isotopes of Pb-206 (205.98 amu); Pb-207 (206.98 amu); and Pb-208 (207.98 amu) and that the percentages of the two lightest isotopes are equal, calculate the relative percentages of these isotopes in the natural element.

**Solution:**

1) Set up average atomic weight equation:

(205.98) (x) + (206.98) (x) + (207.98) (1 - 2x) = 207.19Algebra ensues.

x = 0.263

2) The percent abundances are as follows:

Pb-206: 26.3%

Pb-207: 26.3%

Pb-208: 47.4%

This last form of the question actually falls closer to the real percent abundances of lead than the first form of the question.

**Problem #8:** If you select one hundred carbon atoms at random, what would the total mass be?

a) 1200.00 amu

b) slightly more than 1200.00 amu

c) slightly less than 1200.00 amu

d) 1300.34 amu

e) slightly less than 1300.34 amu

**Solution:**

The relevant facts to answer this question are:

There are two naturally-occuring isotopes of carbon: C-12 (12.0 amu) and C-13 (13.0 amu).

The abundance of C-12 is very high, well above 90%.

Let us assume all 100 carbon atoms are C-12. The total weight would be:

12.0 amu x 100 = 1200 amu

Let us now replace a few, say 5, C-12 atoms with C-13. The total weight of the 100 atoms would be:

(12) (95) + (13) (5) = 1205 amu

The correct answer is (b), slightly more than 1200

**Problem #9:** The atomic weight of naturally occuring neon is 20.18 amu. Naturally occuring neon is composed of two isotopes:

Ne-20: 19.99 amu

Ne-22: 21.99 amu

Calculate the number of Ne-22 atoms in a 12.55 g sample of naturally occuring neon.

**Solution:**

1) Calculate the percent abundance of the two isotopes:

(19.99) (1-x) + (21.99) (x) = 20.18x = 0.095 (this is the Ne-22)

2) Calculate moles of Ne in 12.55 g:

12.55 g / 20.18 g/mol = 0.6219 mol

3) Calculate atoms of Ne in 0.6219 mol:

0.6219 mol times 6.022 x 10^{23}mol^{-1}= 3.745 x 10^{23}

4) Calculate number of Ne-22 atoms:

3.745 x 10^{23}times 0.095 = 3.558 x 10^{22}

Note: the problem above is simplified. There are actually three stable isotopes of neon, with the unmentioned one being Ne-21. Here is a neon problem involving all three isotopes:

The percent abundance for Ne-21 is 0.2700%. Find the percent abundance for Ne-22.

Ne-20 (mass = 19.9924 amu)

Ne-21 (mass = 20.9938 amu)

Ne-22 (mass = 21.9914 amu)

**Solution:**

Ne-20: abundance = 1 minus (x + 0.002700)

Ne-21: abundance = 0.002700

Ne-22: abundance = x(19.9924) (0.997300 - x) + (20.9938) (0.002700) + (21.9914) (x) = 20.1797

The solution for x is left to the reader.

Note: this problem does not provide the average atomic weight of neon, so it had to be looked up. Also note how 1 minus (x + 0.002700) was rearranged.

**Problem #10:** Natural rubidium has the average mass 85.4678 amu and is composed of isotopes Rb-85 (mass = 84.9117 amu) and Rb-87. The ratio of atoms Rb-85/Rb-87 in natural rubidium is 2.591. Calculate the mass of Rb-87.

**Solution:**

Rb-85/Rb-87 is 2.591 to 1. For every 2.591 Rb-85 atoms, there is one Rb-87 atom. We will use 3.591 (coming from 2.591 + 1) to represent 100% of all naturally-occuring Rb atoms.The percent abundance of Rb-85 is 2.591 / 3.591 = 0.721526

The percent abundance of Rb-87 is 1 / 3.591 = 0.278474 (I actually did it by 1 minus 0.721526)

(84.9117) (0.721526) + (x) (0.278474) = 85.4678

x = 86.9086 amu

**Problem #11:** Beanium, a new 'element,' has been discovered. Beanium is known to have three naturally-occuring stable 'isotopes.' A sample of 77 'atoms' of beanium was tested and the following results were obtained:

Isotope Number in

SampleMass Black 55 10.26 g Brown 15 8.64 g White 7 8.67 g

Calculate the "atomic weight" of beanium.

**Solution:**

1) Calculate the percent abundance:

Bl: 55/77 = 0.7143

Br: 15/77 = 0.1948

W: 7/77 = 0.0909

2) Let's get the masses of one of each bean:

Bl: 10.26 / 55 = 0.1865 g

Br: 8.64 / 15 = 0.5760 g

W: 8.67 / 7 = 1.2386 g

3) The 'atomic mass' is a weighted average:

(0.1865) (0.7143) + (0.5760) (0.1948) + (1.2386) (0.0909) = 0.358 gThe 'isotopic' mixture of beans (black, brown and white) apparently weighs 0.358 g per bean atom. You can check this by doing this multiplication:

0.358 g times 77To a reasonable degree of precision, the answer to the above will equal the sum of the masses of the 77 beans involved.

**Problem #12:** A sample of boron with a relative atomic mass of 10.8 gives a mass spectrum with two peaks, one at m/z = 10, and one at m/z = 11. Calculate the ratio of the heights of two peaks

**Solution:**

1) Calculate the percent abundance:

(x) (10) + (1 - x) (11) = 10.810x + 11 - 11x = 10.8

x = 0.2

1 - x = 0.8

2) Calculate the peak ratio:

The ratio between 11 and 10 is 0.8 to 0.2.This is a 4:1 ratio. The peak at m/z = 11 is 4, the peak at m/z = 10 is 1.

**Problem #13:** Rhenium has two naturally occurring isotopes: Re-185 with a natural abundance of 37.40%, and Re-187 with a natural abundance of 62.60%. The sum of the masses of the two isotopes is 371.9087. What are the atomic weights of Re-185 and Re-187?

**Solution:**

1) Look up the average atomic weight of Rhenium:

186.207 amu

2) Set up the following:

(0.3740)(x) + (0.6260)(371.9087 - x) = 186.207where 'x' is the atomic weight of Re-185 and '371.9087 - x' is the atomic weight of Re-187.

3) Algebra ensues:

0.3740x + 232.8148 - 0.6260x = 186.207-0.2520x = -46.6078

x = 184.952 amu (the atomic weight of Re-185)

4) The atomic weight of Re-187 is:

371.9087 minus 184.952 = 186.957 amu

**Problem #14:** Calculate the percentage of each isotope of mass numbers 36, 37 and 39. The atomic weight of the element is 38.60 amu. Isotopes 36 and 37 are found in equal proportions.

**Solution:**

1) The first relevant equation:

(36 x a) + (37 x a) + (39 x b) = 38.60 amu [1]The above equation is the formula used to determine the amu. We know that 36 and 37 are present in equal abundances so we assign them the same unknown, that is to say, a.

39 has some other unknown abundance and we assign it the symbol b.

2) The second relevant equation:

2a + b = 1 [2]We know that the sum of the abundances must equal 1 (the decimal equivalent for 100%). This is how we obtain equation [2].

3) We now need to solve equation [1] for a and b, but that equation has two unknowns. We use equation [2] to reduce the number of unknowns:

2a + b = 1b = 1 - 2a [3]

4) We can now substitute equation [3] into [1] and solve:

(36 x a) + (37 x a) + (39 x b) = 38.60 amu(36 x a) + (37 x a) + (39 x (1-2a)) = 38.60 amu

a = 0.08

5) Now substitute value of a into [3]:

b = 1 - 2a

b = 1 - 2(0.08)

b = 0.84The precent abundances for 36 and 37 are 8% each and the percent abundance for 39 is 84%.

**Problem #15:** Chromium (atomic mass = 51.9961 amu) has four isotopes. Their masses are 49.94605 amu, 51.94051 amu, 52.94065 amu, and 53.93888 amu. The first two isotopes have a combined abundance of 87.87%, and the last isotope has an abundance of 2.365%. (a) What is the abundance of the third isotope? (b) Estimate the abundance of the first 2 isotopes.

**Solution:**

1) First question:

If chromium has four isotopes, they have to add up for a sum of 100%, so the abundance of the third one is 9.765%.

2) Second question:

Let the abundance of the 49.94605 amu isotope be equal to xTherefore the abundance of the 51.94051 amu is equal to 0.8787 - x (Notice how I am using decimal values, not percentages).

Equation to determine x is:

(x) (49.94605) + (0.8787-x) (51.94051) + (0.09765) (52.94065) + (0.002365) (53.93888) = 51.9961

Once x is known, 0.8787 - x is easily calculated.

**Problem #16:** Antimony, Sb, has two stable isotopes. Given that 43.8% of natural antimony is Sb-123 with the experimentally determined mass of 122.904 amu, what is the other stable isotope?

**Solution:**

The atomic weight of Sb from the periodic table is 121.760. Therefore, the "weighted average" of the abundance and mass of each of the two isotopes will be:

(0.438 * 122.904) + (0.562 * x) = 121.76053.832 + 0.562x = 121.760

0.562x = 67.928

x = 120.868

The other stable isotope is Sb-121

Calculate the average atomic weight when given isotopic weights and abundances