13.4. MASS-ENERGY AND NUCLEAR BINDING ENERGY

Einstein showed from his theory of special relativity that it is necessary to treat mass as another form of energy. Before the advent of this theory of special relativity it was presumed that mass and energy were conserved separately in a reaction. However, Einstein showed that mass is another form of energy and one can convert mass-energy into other forms of energy, say kinetic energy and vice-versa.

Einstein gave the famous mass-energy equivalence relation

E = mc² (13.6)

Here the energy equivalent of mass m is related by the above equation and c is the velocity of light in vacuum and is approximately equal to 3×10⁸ m s⁻¹.

This chunk discusses the concept of mass-energy equivalence introduced by Albert Einstein. According to his theory of special relativity, mass and energy are not separate entities; they can be converted into each other. The equation E = mc² defines this relationship, where 'E' is the energy, 'm' is the mass, and 'c' is the speed of light squared, highlighting how even a small amount of mass can be converted into a large amount of energy due to the c² factor being a huge number. This is a paradigm shift from earlier beliefs that mass and energy were conserved independently.

Consider a small battery that powers a flashlight. When you turn off the flashlight, the battery has stored energy that gets converted into light energy. In a similar way, when mass is converted in nuclear processes, like in stars, it is transformed into vast amounts of energy—the same principle allows for energy generation in atomic bombs.

Example 13.2 Calculate the energy equivalent of 1 g of substance.

Solution Energy, E = 10⁻³ × (3 × 10⁸)² J E = 10⁻³ × 9 × 10¹⁶ = 9 × 10¹³ J Thus, if one gram of matter is converted to energy, there is a release of enormous amount of energy.

This chunk provides an example of calculating the energy equivalent of a small mass (1 gram) using Einstein's equation. Since 1 gram is a very small mass, when calculated through the mass-energy equivalence formula, it shows that even a tiny amount of mass can yield an enormous amount of energy—approximately 90 trillion joules. This immense energy output illustrates why nuclear reactions can release so much energy compared to chemical reactions.

Think of a log of wood. When burned, it releases a certain amount of energy. However, if you could convert that wood into pure energy following Einstein's equation, you would unleash energy equivalent to the power of several atomic bombs from just a small log. This helps to show the potential of mass-energy conversion in nuclear reactions.

In Section 13.2 we have seen that the nucleus is made up of neutrons and protons. Therefore it may be expected that the mass of the nucleus is equal to the total mass of its individual protons and neutrons. However, the nuclear mass M is found to be always less than this. For example, let us consider 16O; a nucleus which has 8 neutrons and 8 protons.

Mass of 8 neutrons = 8 × 1.00866 u Mass of 8 protons = 8 × 1.00727 u Mass of 8 electrons = 8 × 0.00055 u

Therefore the expected mass of 16O nucleus = 8 × 2.01593 u = 16.12744 u.

The atomic mass of 16O found from mass spectroscopy experiments is seen to be 15.99493 u. Subtracting the mass of 8 electrons (8 × 0.00055 u) from this, we get the experimental mass of 16O nucleus to be 15.99053 u. Thus, we find that the mass of the 16O nucleus is less than the total mass of its constituents by 0.13691u.

This chunk discusses nuclear binding energy, emphasizing that the actual mass of a nucleus is always less than the sum of the individual masses of its protons, neutrons, and electrons. This difference in mass is called the mass defect. In the case of Oxygen-16, the expected mass did not account for the binding energy within the nucleus which holds the nucleons together. This binding energy accounts for the missing mass and is a crucial factor in understanding nuclear stability and energy requirements in reactions.

Imagine a group of kids holding hands in a circle. Individually, they have certain weights when standing alone, but when they stand together as a tight circle, they form a 'team' that effectively weighs less than the sum of their individual weights due to a sort of 'mutual strength'—this represents binding energy. The closer they hold hands (or the stronger the binding), the less 'weight' they effectively have, just like nucleons in a nucleus.