# Average Atomic Weights

Submitted by ChemPRIME Staff on Wed, 12/08/2010 - 23:58

There are 21 elements with only one isotopeOne of two or more samples of an element whose atoms differ in the number of neutrons found in the nucleus., so all their atomsThe smallest particle of an element that can be involved in chemical combination with another element; an atom consists of protons and neutrons in a tiny, very dense nucleus, surrounded by electrons, which occupy most of its volume. have identical masses. All other elements have two or more isotopes, so their atoms have at least two different masses. But all elements obey the law of definite proportionsThe statement that the mass ratio of elements in a given substance is always the same. Also called the law of constant composition. when they combine with other elements, so they behave as if they had just one kind of atom with a definite mass. In order to solve this dilemma, we define the atomic weightThe average mass of the naturally occurring isotopes of an element, taking into account the different natural abundances of the isotopes. Expressed relative to the value of exactly 12 for carbon-12; also called atomic mass. as the weighted average mass of all naturally occurring (occasionally radioactiveDescribes a substance that gives off radiation&dash;alpha particles, beta particles, or gamma rays&dash;by the disintegration of its nucleus.) isotopes of the element.

A weighted average is defined as

Atomic Weight =

$\left(\tfrac{\%\text{ abundance isotope 1}}{100}\right)\times \left(\text{mass of isotope 1}\right)~ ~ ~ +$

$\left(\tfrac{\%\text{ abundance isotope 2}}{100}\right)\times \left(\text{mass of isotope 2}\right)~ ~ ~ + ~ ~ ...$

Similar terms would be added for all the isotopes. The calculation is analogous to the method used to calculate grade point averages in most colleges:

GPA =

$\left(\tfrac{\text{Credit Hours Course 1}}{\text{total credit hours}}\right)\times \left(\text{Grade in Course 1}\right)~ ~ ~ +$

$\left(\tfrac{\text{Credit Hours Course 2}}{\text{total credit hours}}\right)\times \left(\text{Grade in Course 2}\right)~ ~ ~ + ~ ~ ...$

## Some Conventions

The term "Average Atomic Weight" or simply "Atomic Weight" is commonly used to refer to what is properly called a "relative atomic massThe average mass of the naturally occurring isotopes of an element, taking into account the different natural abundances of the isotopes. Expressed relative to the value of exactly 12 for carbon-12; also called atomic weight.". Atomic Weights are technically dimensionless, because they cannot be determined as absolute values. They were historically calculated from mass ratios (early chemists could say that magnesium atoms have atoms of mass 24.305/15.999 times as heavy as oxygen atoms, because that is the mass ratio of magnesium to oxygen in MgO). Now atomic weights are calculated from the position of peaks in a mass spectrumThe record of masses detected in a mass spectrometer presented as a plot of number of particles with a given mass against atomic mass.. While the peak positions may be labeled in amuAbbreviation for atomic mass unit, a unit for expressing the relative masses of atoms. An atom of carbon-12 has a mass of 12 amu. 1 amu corresponds to 1.662x10-24 g., that is only possible if the mass spectrometerAn instrument that measures the mass of charged particles by accelerating them through electric and magnetic fields and detecting the angles through which their paths are deflected. is calibrated with a standard, whose mass can only be known relative to another, and it is also technically dimensionless. To solve this dilemma, we define an amu as 1/12 the mass of a 126C atom. 126C can then be used to calibrate a mass spectrometer. For convenience, we often use "token" dimensions of amu/average atom for atomic weight, or g/mol for molar massThe mass of a mole of substance; the same as molecular weight for molecular substances..

The calculation of an atomic weight includes "naturally occurring isotopes", which are defined by the Commission on Isotopic Aundances and Atomic Weights of IUPAC (IUPAC/CIAAW) to include radioactive isotopes with half lives greater than 1 x 1010 years. Thus thorium, protactinium, and uranium are assigned atomic weights of 232.0, 231.0, and 238.0, but no other radioactive elements have isotopes with long enough lifetimes to be assigned atomic weights.

## Example

Naturally occurring lead is found to consist of four isotopes:

1.40% 20482Pb whose isotopic weight is 203.973.

24.10% 20682Pb whose isotopic weight is 205.974.

22.10% 20782Pb whose isotopic weight is 206.976.

52.40% 20882Pb whose isotopic weight is 207.977.

Calculate the atomic weight of an average naturally occurring sample of lead.

SolutionA mixture of one or more substances dissolved in a solvent to give a homogeneous mixture. Suppose that you had 1 mol lead. This would contain 1.40% ($\tfrac{1.40}{100}$ × 1 mol) 20482Pb whose molar mass is 203.973 g mol–1. The mass of 20482Pb would be

$\text{m}_{\text{204}}=n_{\text{204}}\times \text{ }M_{\text{204}}=\left( \frac{\text{1}\text{.40}}{\text{100}}\times \text{ 1 mol} \right)\text{ (203}\text{.973 g mol}^{\text{-1}}\text{)}=\text{2}\text{.86 g}$

Similarly for the other isotopes

\begin{align} & \text{m}_{\text{206}}=n_{\text{206}}\times \text{ }M_{\text{206}}=\left( \frac{\text{24}\text{.10}}{\text{100}}\times \text{ 1 mol} \right)\text{ (205}\text{.974 g mol}^{\text{-1}}\text{)}=\text{49}\text{.64 g} \ & \text{m}_{\text{207}}=n_{\text{207}}\times \text{ }M_{\text{207}}=\left( \frac{\text{22}\text{.10}}{\text{100}}\times \text{ 1 mol} \right)\text{ (206}\text{.976 g mol}^{\text{-1}}\text{)}=\text{45}\text{.74 g} \ & \text{m}_{\text{208}}=n_{\text{208}}\times \text{ }M_{\text{208}}=\left( \frac{\text{52}\text{.40}}{\text{100}}\times \text{ 1 mol} \right)\text{ (207}\text{.977 g mol}^{\text{-1}}\text{)}=\text{108}\text{.98 g} \ \end{align}

Upon summing all four results, the mass of 1 mol of the mixtureA combination of two or more substances in which the substances retain their chemical identity. of isotopes is to be found

2.86 g + 49.64 g + 45.74 g + 108.98 g = 207.22 g

Thus the atomic weight of lead is 207.2 g/mol, as mentioned earlier in the discussion.

An important corollary to the existence of isotopes should be emphasized at this point. When highly accurate results are obtained, atomic weights may vary slightly depending on where a sample of an element was obtained. For this reason, the IUPAC CIAAW has recently redefined the atomic weights of 10 elements having two or more isotopes [1]. The percentages of different isotopes often depends on the source of the element. For example, oxygen in Antarctic precipitationThe formation of a solid within a solution, often by the combination of cations and anions to form an insoluble ionic compound. has an atomic weight of 15.99903, but oxygen in marine N2O has an atomic weight of 15.9997. "Fractionation" of the isotopes results from slight differences rates of chemical and physical processes caused by small differences in their masses. The difference can be more dramatic when an isotope is derived from transmutation. For example, lead produced by decay of uranium contains a much larger percentage of 20682Pb than the 24.1 percent given in the example for the average sample. Consequently the atomic weight of lead found in uranium ores is less than 207.2 and is much closer to 205.974, the isotopic weight of 20682Pb.

After the discovery of isotopes of the elements by J.J. Thompson in 1913 [2], it was suggested that the scale of relative masses of the atoms (the atomic weights) should use as a reference the mass of an atom of a particular isotope of one of the elements. The standard that was eventually chosen was 126C, and it was assigned an atomic-weight value of exactly 12.000 000. Thus the atomic weights given in the Table of Atomic Weights are the ratios of weighted averages (calculated as in the Example) of the masses of atoms of all isotopes of each naturally occurring element to the mass of a single 126C atom. Since carbon consists of two isotopes, 98.99% 126C isotopic weight 12.000 and 1.11% 136C of isotopic weight 13.003, the average atomic weight of carbon is

$\frac{\text{98}\text{.89}}{\text{100}\text{.00}}\text{ }\times \text{ 12}\text{.000 + }\frac{\text{1}\text{.11}}{\text{100}\text{.00}}\text{ }\times \text{ 13}\text{.003}=\text{12}\text{.011}$

for example.

## Conventional Atomic Weights and "Intervals"

Deviations from average isotopic composition are usually not large, and so the Conventional Atomic Weight Values were defined by the IUPAC/CIAAW for the ements showing the most variation in abundances. They can be used for nearly all chemical calculations. But at the same time, Atomic Weights were redefined for those elements as ranges, or "intervals", for any workA mechanical process in which energy is transferred to or from an object, changing the state of motion of the object. where small differences may be important[3]. The table below gives typical values.

Element Name Symbol Conventional
Atomic Weight
Atomic Weight
Boron B 10.81 [10.806; 10.821]
Carbon C 12.011 [12.0096; 12.0116]
Chlorine Cl 35.45 [35.446; 35.457]
Hydrogen H 1.008 [1.00784; 1.00811]
Lithium Li 6.94 [6.938; 6.997]
Nitrogen N 14.007 [14.00643; 14.00728]
Oxygen O 15.999 [15.99903; 15.99971]
Silicon Si 28.085 [28.084; 28.086]
Sulfur S 32.06 [32.059; 32.076]
Thallium Tl 204.38 [204.382; 204.385]

In the study of nuclear reactions, however, one must be concerned about isotopic weights. This is discussed further in the section on Nuclear Chemistry.

## SI Definition of the Mole

The SI definition of the mole also depends on the isotope 126C and can now be stated. One mole is defined as the amount of substanceA material that is either an element or that has a fixed ratio of elements in its chemical formula. of a system which contains as many elementary entities as there are atoms in exactly 0.012 kg of 126C. The elementary entities may be atoms, molecules, ions, electrons, or other microscopic particles. This definition of the mole makes the mass of 1 mole of an element in grams numerically equal to the average mass of the atoms in grams. This official definition of the mole makes possible a more accurate determination of the Avogadro constant than was reported earlier. The currently accepted value is NA = 6.02214179 × 1023 mol–1. This is accurate to 0.00000001 percent and contains five more significant figures than 6.022 × 1023 mol–1, the number used to define the mole previously. It is very seldom, however, that more than four significant digits are needed in the Avogadro constant. The value 6.022 × 1023 mol–1 will certainly suffice for most calculations needed.

## References

1. The atomic weight has been recently redefined. See [1]
2. http://en.wikipedia.org/wiki/Isotope
3. [2]