42.1.3 Atomic Mass

You learned in Chapter 18 that atomic masses are specified in terms of the atomic mass unit u, defined such that the atomic mass of the isotope $^{12} C$ is exactly 12 u. The conversion to SI units is

1 u = 1.6605 \times 10^{- 27} kg

Alternatively, we can use Einstein’s $E_{0} = m c^{2}$ to express masses in terms of their energy equivalent. The energy equivalent of 1 u of mass is

\begin{array}{l} E_{0} & = & (1.6605 \times 10^{- 27} kg) {(2.9979 \times 10^{8} m/s)}^{2} \\ = & 1.4924 \times 10^{- 10} J = 931.49 MeV \end{array}

(42.1)

Thus the atomic mass unit can be written

1 u = 931.49 MeV / c^{2}

It may seem unusual, but the units $MeV / c^{2}$ are units of mass.

NOTE We’re using more significant figures than usual. Many nuclear calculations look for the small difference between two masses that are almost the same. Those two masses must be calculated or specified to four or five significant figures if their difference is to be meaningful.

TABLE 42.2 shows the atomic masses of the electron, the nucleons, and three important light elements. Appendix C contains a more complete list. Notice that the mass of a hydrogen atom is the sum of the masses of a proton and an electron. But a quick calculation shows that the mass of a helium atom (2 protons, 2 neutrons, and 2 electrons) is 0.03038 u less than the sum of the masses of its constituents. The difference is due to the binding energy of the nucleus, a topic we’ll look at in Section 42.2.

TABLE 42.2 Some atomic masses

Particle	Symbol	Mass (u)	Mass $(MeV / c^{2})$
Electron	e	0.00055	0.51
Proton	p	1.00728	938.28
Neutron	n	1.00866	939.57
Hydrogen	$^{1} H$	1.00783	938.79
Deuterium	$^{2} H$	2.01410	1876.12
Helium	$^{4} He$	4.00260	3728.40

The isotope $^{2} H$ is a hydrogen atom in which the nucleus is not simply a proton but a proton and a neutron. Although the isotope is a form of hydrogen, it is called deuterium. The natural abundance of deuterium is 0.015%, or about 1 out of every 6700 hydrogen atoms. Water made with deuterium (sometimes written $D_{2} O$ rather than $H_{2} O$ ) is called heavy water.

NOTE Don’t let the name “deuterium” cause you to think this is a different element. Deuterium is an isotope of hydrogen. Chemically, it behaves just like ordinary hydrogen.

The chemical atomic mass shown on the periodic table of the elements is the weighted average of the atomic masses of all naturally occurring isotopes. For example, chlorine has two stable isotopes: $^{35} Cl$ , with $m = 34.97 u,$ is 75.8% abundant and $^{37} Cl$ , at 36.97 u, is 24.2% abundant. The average, weighted by abundance, is $0.758 \times 34.97 + 0.242 \times 36.97 = 35.45$ . This is the value shown on the periodic table and is the correct value for most chemical calculations, but it is not the mass of any particular isotope of chlorine.

NOTE The atomic masses of the proton and the neutron are both $\approx 1 u$ . Consequently, the value of the mass number $A$ is approximately the atomic mass in u. The approximation $m \approx A u$ is sufficient in many contexts, such as when we’re calculating the masses of atoms in the kinetic theory of gases, but in nuclear physics calculations, we almost always need the more accurate mass values that you find in Table 42.2 or Appendix C.