Practice problems for these concepts can be found at:

Nuclear Chemistry Review Questions for AP Chemistry

A radioactive isotope may be unstable, but it is impossible to predict when a certain atom will decay. However, if a statistically large enough sample is examined, some trends become obvious. The radioactive decay follows first-order kinetics. If the number of radioactive atoms in a sample is monitored, it can be determined that it takes a certain amount of time for half the sample to decay; it takes the same amount of time for half the remaining sample to decay; and so on. The amount of time it takes for half the sample to decay is called the half-life of the isotope and is given the symbol *t*_{1/2}. The table below shows the percentage of radioactive isotope remaining versus half-life.

As a general rule, the amount of radioactivity at the end of 10 half-lives drops below the level of detection and the sample is said to be "safe."

Half-lives may be very short, 4.2 × 10^{–6} seconds for Po-213, or very long, 4.5 × 10^{9} years for U-238. The long half-lives of some waste products is a major problem with nuclear fission reactors. Remember, it takes 10 half-lives for the sample to be safe.

If only multiples of half-lives are considered, the calculations are very straightforward. For example, I-131 is used in the treatment of thyroid cancer and has a *t*_{1/2} of eight days. How long would it take to decay to 25% of its original amount? Looking at the chart, you see that 25% decay would occur at two half-lives or 16 days. However, since radioactive decay is not a linear process, you cannot use the chart to predict how much would still be radioactive at the end of 12 days or at some time (or amount) that is not associated with a multiple of a half-life. To solve these types of problems, one must use the mathematical relationships associated with first-order kinetics that were presented in the Kinetics chapter. In general, two equations are used:

In these equations, the ln is the natural logarithm; *A _{t}* is the amount of isotope radioactive at some time

*t*;

*A*is the amount initially radioactive; and

_{o}*k*is the rate constant for the decay. If you know initial and final amounts and are looking for the half-life, you would use equation (1) to solve for the rate constant and then use equation (2) to solve for

*t*

_{1/2}.

For example: What is the half-life of a radioisotope that takes 15 min to decay to 90% of its original activity?

Using equation (1):

If one knows the half-life and amount remaining radioactive, equation (2) can be used to calculate the rate constant k and equation (1) can then be used to solve for the time. This is the basis of C-14 dating, which is used to determine the age of objects that were once alive.

For example, suppose a wooden tool is discovered and its C-14 activity is determined to have decreased to 65% of the original. How old is the object?

The half-life of C-14 is 5730 yr. Substituting this into equation (2):

Substituting this rate constant into equation (1):

### Mass–Energy Relationships

Whenever a nuclear decay or reaction takes place, energy is released. This energy may be in the form of heat and light, gamma radiation, or kinetic energy of the expelled particle and recoil of the remaining particle. This energy results from the conversion of a very small amount of matter into energy. (Remember that in nuclear reactions there is no conservation of matter, as in ordinary chemical reactions.) The amount of energy that is produced can be calculated by using Einstein's equation *E = mc ^{2}*, where

*E*is the energy produced,

*m*is the mass converted into energy (the mass defect), and c is the speed of light. The amount of matter that is converted into energy is normally very small, but when it is multiplied by the speed of light (a very large number) squared, the amount of energy produced is very large.

For example: When 1 mol of U-238 decays to Th-234, 5 × 10^{–6} Kg of matter is converted to energy (the mass defect). To calculate the amount of energy released:

if the mass is in kilograms, the answer will be in joules.

Practice problems for these concepts can be found at:

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