Physics · Ch 13 — Nuclei

Mass – Energy

13.4.1

Mass – Energy

The Mass-Energy Equivalence

Before Einstein’s theory of special relativity, physicists treated mass and energy as two completely separate quantities. In any reaction, it was believed that mass was conserved (the total mass before equals the total mass after) and energy was conserved (the total energy before equals the total energy after) — two independent laws.

Einstein’s revolutionary insight was that mass is not separate from energy. Instead, mass is a highly concentrated form of energy. The two are interconvertible: you can turn mass into other forms of energy (like heat, light, or kinetic energy), and you can turn those forms of energy back into mass. This means the old separate conservation laws must be merged into a single, deeper law: the conservation of mass-energy.

The quantitative relationship between mass and energy is given by the most famous equation in physics:

E = mc^2

Here:

$E$ is the energy equivalent of a mass $m$ .
$m$ is the mass that is converted into energy (or vice-versa).
$c$ is the speed of light in a vacuum, approximately $3 \times 10^8 \text{ m s}^{-1}$ .

The key point is the factor $c^2$ , which is an enormous number ( $9 \times 10^{16} \text{ m}^2\text{/s}^2$ ). This means that even a tiny amount of mass is equivalent to a colossal amount of energy.

The Energy Equivalent of One Gram

To make this concrete, consider the energy released if you could completely convert just 1 gram of matter into energy.

Given: $m = 1 \text{ g} = 10^{-3} \text{ kg}$ , $c = 3 \times 10^8 \text{ m/s}$ .

Calculation:

E = m c^2 = (10^{-3} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2

E = 10^{-3} \times 9 \times 10^{16} \text{ J}

E = 9 \times 10^{13} \text{ J}

This is an immense amount of energy. To put it in perspective, $9 \times 10^{13}$ joules is roughly the energy released by a 20-kiloton atomic bomb. This staggering number is why nuclear reactions, which involve changes in mass far smaller than a gram, can release such enormous amounts of energy.

Watch out

The equation $E = mc^2$ does not mean that a stationary object of mass $m$ has a kinetic energy of $mc^2$ . It means that mass itself is a form of energy. The total energy of an object at rest is its rest-mass energy, $E_0 = m_0 c^2$ , where $m_0$ is its rest mass. When the object is moving, its total energy is higher, given by $E = \gamma m_0 c^2$ , where $\gamma$ is the Lorentz factor.

Experimental Verification and the Conservation of Mass-Energy

The mass-energy equivalence is not just a theoretical idea; it has been verified experimentally countless times. The most direct and precise verifications come from the study of nuclear reactions.

In any nuclear reaction (e.g., the fission of a uranium nucleus, the fusion of hydrogen into helium, or the collision of particles in an accelerator), the total mass of the products is not equal to the total mass of the reactants. There is a difference, called the mass defect ( $\Delta m$ ).

The law of conservation of energy is then applied in its modern, unified form:

Important

The total initial energy (including the rest-mass energy of all particles) equals the total final energy (including the rest-mass energy of all particles). …

Start your 14-day free trial to unlock the full solution →

← Previous · 13.4Mass-energy and Nuclear Binding Energy Next · 13.4.1 →Example 13.2 — Calculate the energy equivalent of 1g of substance.