Skip to content

Mass and energy

Radiation and the atom · level 8, lesson 1 of 6

A question to start

Curiosity's nuclear battery gives out about 2,000 W of heat, year after year, with nothing added. Its plutonium gets very slightly lighter as it does. By how much?

The idea

In 1905, Albert Einstein showed that mass and energy are two forms of the same thing. Energy \(E\) and mass \(m\) are linked by

\[E = mc^2,\]

where \(c = 2.998 \times 10^8\) m/s is the speed of light. Since \(c^2\) is huge, a tiny mass is a vast energy: 1 gram is \(9.0 \times 10^{13}\) J.

Any system that gives out energy loses mass, \(m = E/c^2\). In chemistry the loss is far too small to measure. In nuclear reactions it is large enough to weigh.

Nuclear physicists measure mass in atomic mass units (u) and energy in mega-electronvolts (MeV; 1 MeV \(= 1.602 \times 10^{-13}\) J). By \(E = mc^2\), NIST gives

\[1 \text{ u} = 931.494 \text{ MeV}.\]
1 gE = mc²9 × 10¹³ J1 GW for a whole daya gram

Worked examples

1. One gram. \(E = 0.001 \times (2.998\times10^8)^2 = 9.0\times10^{13}\) J: about what a large power station makes in a day.

2. One atomic mass unit. \(1.6605\times10^{-27} \text{ kg} \times (2.998\times10^8)^2 = 1.492\times10^{-10}\) J. Dividing by \(1.602\times10^{-13}\) J per MeV gives 931.5 MeV.

3. The nuclear battery. In a year it gives out \(2{,}000 \times 3.156\times10^7 = 6.3\times10^{10}\) J. The mass lost is \(6.3\times10^{10} / (2.998\times10^8)^2 = 7\times10^{-7}\) kg: less than a milligram.

Try it

  1. Find the energy in 1 kg of mass.
  2. A nuclear reaction releases 17.6 MeV. How much mass is lost, in u?
  3. Why do chemical reactions not seem to change mass?
  4. A reactor gives out 10 MW of heat for a year. Find the mass it loses.
Answers
  1. \(1 \times (2.998\times10^8)^2 = 9.0\times10^{16}\) J.
  2. \(17.6 / 931.5 = 0.0189\) u.
  3. They release so little energy that the mass lost, \(E/c^2\), is far too small to measure.
  4. \(10^7 \times 3.156\times10^7 / (9.0\times10^{16}) \approx 3.5\times10^{-3}\) kg, about 3.5 g.

Common mistake

"Only nuclear reactions turn mass into energy." Every energy release lowers the mass, even a burning match. It is just that nuclear reactions release a million times more per atom, enough to measure.

Where it leads