Atomic Mass Calculations

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Atomic Mass Calculations from Percent Abundance

From http://www.webelements.com/webelements/properties/text/definitions/atomic-weight.html

Atomic Mass

"An atomic weight (relative atomic mass) of an element from a specified source is the ratio of the average mass per atom of the element to 1/12 of the mass of ¹²C" in its nuclear and electronic ground state. (Refs. 1 and 2)

A sample of any element consists of one or more isotopes of that element. Each isotope is a different weight. The relative amounts of each isotope for any element represents the isotope distribution for that element. The atomic weight is the average of the isotope weights weighted for the isotope distribution and expressed on the ¹²C scale as mentioned above.

Calculations

From-http://learning.mgccc.cc.ms.us/pk/sciencedocs/isotopeatomicw.htm

Atomic Mass =

[(%abundance of isotope) (mass of isotope)] + [(%abundance of isotope) (mass of isotope)] + [….]

% abundance of EACH isotope is converted to decimal ex: 19.9% ÷ 100% = 0.199

The equation continues on[….] based on the number of isotopes in the problem.

Ex: The natural abundance for boron isotopes is: 19.9% ¹⁰B (10.013 amu) and 80.1% ¹¹B (11.009amu). Calculate the atomic weight of boron.

Atomic Mass =	[(0.199)(10.013)] + [(0.801)(11.009)]
=	[1.992587] + [8.818209]
=	10.810796 so, the atomic weight of B = 10.811

If you look in the periodic table you will be able to check that our answer is correct!

Determining the percent abundance of each isotope from atomic mass.

The atomic mass of lithium is 6.94, the naturally occurring isotopes are ⁶Li = 6.015121 amu, and ⁷Li = 7.016003 amu.

Determine the percent abundance of each isotope.

Aw = [(%abundance of isotope) (mass of isotope)] + [(%abundance of isotope) (mass of isotope)] + [….]

6.94 =[(% ⁶Li)(6.015121)] + [(%⁷Li)(7.016003)]

Since I don’t know what the percentage are, I will have to use variables.

100% of Lithium is determined by these two naturally occurring isotopes.

We will let ⁶Li = x and ⁷ Li = 1-x; we use 1 – x instead of 100 – x because the small number is easier to work with.

(in other words we reduced 100% to decimal form 1.00)

Now let’s plug our variables in:	6.94 =[(% ⁶Li)(6.015121)] + [(%⁷Li)(7.016003)]
	6.94 = [(x)(6.015121)] +[(1-x)(7.016003)]
	6.94 = 6.015121x + 7.016003 – 7.016003x
Combine like terms:	6.94 -7.016003 = (6.015121x - 7.016003x)
	-0.076003 = -1.000882 x
Solve for x:	-0.076003	= x
	-1.000882

X = 0.075936,	therefore ⁶Li = 0.075936 x 100% = 7.59%
1-X = 1 -0.075936 = 0.924064	, therefore ⁷Li = 0.924064 x 100% = 92.41%

Try these problems

Problems	Highlight to reveal solutions
1.Calculate the weight of silicon using the following data for the percent natural abundance and mass of each isotope: 92.23% ²⁸Si (27.9769 amu); 4.67% ²⁹ Si (28.9765); 3.10% ³⁰Si (29.9738 amu).	Atomic mass= [(0.9223)(27.9769)] + [(0.0467)(28.97650)] + [(0.031) (29.9738)] Aw = 25.803 + 1.353 + 0.929 Aw for Silicon = 28.085
2. Thallium has two stable isotopes, ²⁰³Tl and ²⁰⁵Tl. Knowing that the atomic weight of thallium is 204.4, which isotope is the more abundant of the two?	²⁰³Tl % = x ²⁰⁵Tl % = 1 – x 204.4 = [(x)(203)] + [(1- x)(205)] 204.4 = 203x + 205 – 205x -0.6 = -2x
	-0.6	= x
	-2
	X= 0.3 ²⁰³Tl = 0.3 x 100% = 30 % ²⁰⁵Tl = 1 –x = 1 – 0.3 = 0.7 x 100% = 70% Therefore ²⁰⁵Tl = 70% is more abundant.
3. Verify that the atomic mass of magnesium is 24.31, given the following :²⁴Mg= 23.985042amu, 78.99% ²⁵Mg= 24.985837 amu, 10.00% ²⁶Mg= 25.982593, 11.01%	Atomic mass= [(0.7899)(23.985042)] + [(0.1)(24.985837)] + [(0.1101)(25.982593)] Atomic mass = 18.946 + 2.499 + 2.861
4.Copper exists as two isotopes: ⁶³Cu (62.9298 amu) and ⁶⁵Cu (64.9278 amu). What are the percent abundances of the isotopes?	Since the overall atomic weight for copper is not given in the problem, you must look it up in the periodic table to work this solution. Atomic mass for Cu = 63.546 ⁶³Cu % = x ⁶⁵Cu % = 1 – x 63.546 = [(x)(62.9298)] + [(1-x)(64.9278)] 63.546 = 62.9298x + 64.9278 – 64.9278x -1.3818 = -1.998x
	-1.3818	= x
	-1.998
	X = 0.6916 ⁶³Cu = 0.6916 x 100% = 69.16% ⁶⁵Cu = 1 – x = 1 – 0.6916 = 0.3084 x 100% = 30.84%

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