# Nuclear Reactions Example Problem

## Calculate the Energy Change in a Nuclear Reaction

The energy change in a nuclear reaction is considerably greater than that of a normal chemical reaction. This change can be calculated using Einstein's equation:

ΔE = Δmc2

where ΔE is the change in energy, Δm is the change in mass (mass of products - mass of reactants), and c is the speed of light (3.00 x 108 m/s).

As written, this relation gives the energy change in joules and the mass change in kilograms.

Usually small quantities of a sample decay, and the energy change is very large, so it's more common to get an energy change in kilojoules (kJ) corresponding to a mass change in grams. Using the relations

1 kJ = 103 J and 1 kg = 103 g

Einstein's equation may be rewritten

ΔE (in kJ) = 9.00 x 1010 Δm (in grams)

For example, to calculate the ΔE in kJ for the radioactive decay of radium:

22688Ra --> 22286Rn + 42He

when one mole of radium decays, we first calculate Äm for the reaction and then obtain ΔE using the equation.

Δm = mass of 1 mol 42He + mass of 1 mol 22286Rn - mass of 1 mol 22688Ra

Δm = 4.0015 g + 221.9703 g - 225.9771 g

Δm = -0.0053 g

Note that Δm may be an extremely small quantity, so it is important to know the masses of products and reactants with a high degree of accuracy in order to know the mass difference to two significant figures.

ΔE (in kJ) = 9.00 x 1010 x (-0.0053)

ΔE = -4.8 x 108kJ

ΔE in kJ when one gram of radium (one mole weighs 226 g) decays would be:

ΔE = 1 g Ra x (-4.8 x 108 kJ)/226 g Ra

ΔE = -2.1 x 106 kJ