A Physics Primer 5: Atomic and Molecular Mass

Every atom has a mass measured in daltons (also called unified atomic mass units, abbreviated as amu or just u¹). Protons and neutrons each have about the same mass equal to about one dalton². Therefore, the mass of an atom is the sum of its protons and neutrons, called its mass number³. Most hydrogen nuclei consist of a lone proton, so the atomic mass of most hydrogen atoms is 1 dalton. Most helium nuclei have two protons and two neutrons, so the atomic mass of most helium atoms is 4 daltons. Most iron nuclei have 26 protons and 30 neutrons, so the atomic mass of most iron atoms is 56 daltons.

While the atoms of any one element all have the same number of protons, the number of neutrons can differ. Each possible number of neutrons in a particular element’s nucleus is called an isotope of that element⁴. Hydrogen has three isotopes:

• regular hydrogen with no neutrons (1 dalton)
• deuterium having one neutron (2 daltons)
• tritium with two neutrons (3 daltons)⁴

The isotopes of hydrogen have individual names, hydrogen, deuterium, and tritium. When distinguishing among isotopes of all other elements, it is common practice to write the mass number after the name of the element. Oxygen has three isotopes:

• oxygen 16 with eight neutrons (16 daltons)
• oxygen 17 with nine neutrons (17 daltons)
• oxygen 18 with ten neutrons (18 daltons)⁵

The two most prominent isotopes of uranium are uranium 238 and uranium 235. Uranium 238, with 146 neutrons, is by far the more common isotope. But only uranium 235, with 143 neutrons in its nucleus, can undergo fission and produce nuclear energy⁶.

Periodic tables list the average atomic mass of each element. This is a weighted average of the masses of all the isotopes for that element as they appear in nature. If an element has three isotopes with atomic masses A, B, and C, and isotope A appears p% of the time, isotope B appears q% of the time, and isotope C appears r% of the time (p + q + r = 100%), then the average atomic mass of the element is⁷:

pA/100 + qB/100 + rC/100

For example, chlorine has two isotopes: 76% of all chlorine atoms and ions in nature are chlorine 35, but 24% are chlorine 37. The average atomic mass of chlorine is:

(.76)(35 daltons) + (.24)(37 daltons)

which gives an average atomic weight of 35.48 daltons for chlorine. (Actually, the real average atomic weight is 35.45 daltons, the difference being due to imprecision in the atomic mass values that I used.

As atoms have their masses, so do molecules. The mass of a molecule is the sum of the masses of the atoms that make it up⁸.

The most common type of water molecule has a mass of 18 daltons: its lone oxygen atom is 16 amu, and the two hydrogen atoms contribute 1 dalton each.

The most common carbon dioxide molecule has a mass of 44 daltons the carbon atom has a mass of 12 amu, and the two oxygen atoms contribute 16 daltons each. If these molecules have heavier isotopes within them, the mass will vary accordingly.

Ionic compounds like table salt also have molecular mass, although they are made up of ions, not molecules. Here what is important is the ratio of ions to each other and the average atomic weight of the ions in the compound. If two ions appear in a compound, ion A and ion B, then consider the following. Ion A has an average mass of m_A and ion B has the average mass of m_B. For every x number of ion A in the compound, there is a number of y ions in the compound (i.e. the ratio of ion A to ion B is x:y). The molecular mass of the compound is then:

xm_A + ym_B

expressed in daltons⁹.

For example, the chemical calcium chloride. This chemical consists of calcium and chlorine ions at a ratio of 1:2 (there are twice as many chlorine ions as calcium ions). If the chemical was taken from nature (as opposed to being produced in the laboratory from unusual isotopes), then the average calcium nucleus has a mass of 40.078 daltons and the average chlorine nucleus has a mass of 35.453 daltons. The molecular mass of calcium chloride is the average weight of a calcium ion added to twice the average weight of a chlorine ion:

40.078 + (2 × 35.453)

which is equal to 110.984 daltons, the molecular mass of calcium chloride.

A very important quantity in chemistry is the mole. We define the mole of a homogeneous substance, a substance consisting solely of one element or one chemical compound, such as pure oxygen or pure water. A mole of such a substance has as much mass in grams as the substance’s average molecular mass measured in daltons¹⁰. One mole of hydrogen has the mass of about one gram. A mole of helium has the mass of about four grams. A mole of water has a mass about 18 grams, and a mole of calcium chloride has a mass of 110.984 grams. The number of grams per mole of a particular substance is called the molar mass of that substance. If the substance is homogeneous, its molar mass in grams per mole will be numerically equal to the average molecular mass of the individual molecules expressed in daltons.

Footnotes

1. Princeton University website. Unified atomic mass unit. To view, click here. Note that the article states that the abbreviation “amu” for atomic mass unit has been deprecated in favor of the abbreviation “u” or the unit Dalton (abbreviation: Da).
2. Texas A & M University, Chemistry website, First Year Chemistry Program, Educator Resources, [Isotopes] Introduction. To view, click here. As you can see on the website, protons and neutrons have nearly but not exactly the same mass. Neutrons have slightly more mass than protons. But the differences are very slight, so for many purposes one dalton is a very good approximation for the mass of both the proton and the neutron, although it is insufficient for exact calculations. Technically, one dalton is defined as exactly ¹/₁₂ of the mass of a carbon-12 nucleus, one that has six protons and six neutrons (see reference in previous footnote). Also note that the mass of the nucleus can be less than the sum of masses of its protons and neutrons — the difference called the mass defect (that is why the mass of a carbon-12 nucleus is exactly 12 daltons but the sum of the masses of six protons and six neutrons is 12.09565 daltons). To learn more, see the Purdue University Department of Chemistry website, Nuclear Binding Energy. For a list of precise masses of various isotopes and the proportion they occur in nature, see the Scientific Instruments Services website, Exact Masses of the Elements and Isotopic Abundance.
3. ChemistryUnderstood.com website. How to Find Mass Number. To view, click here.
4. University of Colorado at Boulder website, Physics 2000. Isotopes. To view, click here
5. Thomas Jefferson National Accelerator Facility website. Isotopes of the Element Oxygen. To view, click here.
6. World Nuclear Association website. What is Uranium? How Does it Work? To view, click here.
7. Helmenstine, Anne Marie, Atomic Weight Calculation: Calculating Atomic Weight of an Element with Isotopes. About.com Chemistry website. To view, click here.
8. Miriam Webster online dictionary. Molecular Mass. To view, click here.
9. I didn’t find this formula anywhere, I derived it on my own. But I tried it the Biological Magnetic Resonance Data Bank Molecular Mass Calculator, and it works. Actually, there is a mental short cut you can use. If you know the chemical formula of the ionic compound (we discuss chemical formulas in the next post), you can treat the formula as if it described a molecule. The chemical formula for calcium chloride is CaCl₂. If this was a molecule, it would have a mass anywhere from 110 to 122 daltons. Based on the natural frequency of isotopes, the average calcium chloride “molecule” would have a mass of 110.984 daltons. But there is no such molecule, only ions. We take this as the molecular mass of calcium chloride nonetheless.
10. Frostburg State University General Chemistry Online website. Moles confuse me — why are they used? To view, click here. Note that the web page first defines the mole in terms of Avogadro’s number: 6.02 × 10²³ molecules. But I was taught to think of a mole as a measure of mass with as many grams as one of its molecules has in daltons, and I find it easier to deal with in this way.