## 22.4 Half Life and Rate of Decay Radioactive decay follows a predictable pattern over time. This section introduces two essential concepts for understanding how quickly and to what extent a radioactive material changes: **half-life** and **rate of decay**. Half-life indicates the time required for half of a radioactive sample to transform, while the rate of decay describes how rapidly this process occurs. These concepts are quantified using the **decay constant** and **activity**, which enable measurement and comparison of different radioactive substances. ### Half-life ($T_{1/2}$) The half-life of a radioactive material is defined as the time required for half of the radioactive nuclei in a given sample to decay. It is a characteristic property of each radioactive substance and does not depend on the initial amount of material present. **Example**: If a sample initially contains 100 nuclei, after one half-life, 50 nuclei remain; after a second half-life, 25 nuclei remain, and so on. The half-life of cobalt-60 is 5.27 years, as illustrated in Figure 22.8. ### Rate Law for Nuclear Decay The rate at which a radioactive substance decays is directly proportional to the number of radioactive nuclei present at any given time. This relationship is expressed mathematically as: $$ \frac{\Delta N}{\Delta t} \propto -N $$ The negative sign indicates that the number of nuclei ($N$) decreases with time. ### Decay Constant ($\lambda$) The proportionality factor in the rate law of radioactive decay is called the **decay constant** ($\lambda$). It represents the fraction of radioactive nuclei that decay per unit time. The rate law including the decay constant is written as: $$ \frac{\Delta N}{\Delta t} = -\lambda N $$ This can also be expressed as: $$ \lambda = \frac{1}{N} \times \frac{\Delta N}{\Delta t} $$ The decay constant is related to the half-life by the expression: $$T_{1/2} = \frac{0.693}{\lambda}$$ The half-life can vary from fractions of a second to billions of years. Table 22.3 provides examples of half-lives for various radioactive elements. | **Element** | **Decays into** | **Half-life** | | :------------ | :-------------- | :---------------- | | Carbon-14 | Nitrogen-14 | 5,730 years | | Aluminum-26 | Magnesium-26 | 740,000 years | | Iodine-129 | Xenon-129 | 17 million years | | Uranium-235 | Lead-207 | 704 million years | | Uranium-238 | Thorium-234 | 4.5 billion years | | Potassium-40 | Argon-40 | 1.3 billion years | | Rubidium-87 | Strontium-87 | 49 billion years | | Plutonium-239 | Uranium-235 | 24,110 years | | Cesium-137 | Barium-130 | 30 years | | Strontium-90 | Yttrium-90 | 28.8 years | | Radon-222 | Polonium-218 | 3.82 days | *Table 22.3: Half-lives of some elements.* ### Activity ($A$) The ratio $\Delta N / \Delta t$ is defined as the **activity** of a radioactive sample. It represents the number of disintegrations per unit time. The formula for activity is: $$ A = \frac{\Delta N}{\Delta t} $$ Relating activity to the decay constant and number of nuclei: $$ A = -\lambda N $$ #### Units of Activity **Becquerel (Bq)**: The SI unit of activity. One becquerel equals one disintegration per second. **Curie (Ci)**: A commonly used unit. One curie equals an activity of $3.70 \times 10^{10}$ decays per second. **Conversion**: $1 \ \mathrm{Ci} = 3.70 \times 10^{10} \ \mathrm{Bq}$. This demonstrates that the becquerel is a much smaller unit than the curie. **Examples of Activity**: - Radium in a watch dial: $4 \times 10^{4} \ \mathrm{Bq}$ - Radiotherapy dose range: $4 \times 10^{13} \ \mathrm{Bq}$ ### Background Radiation Radiation is naturally present in the environment everywhere on Earth. This **background radiation** originates from cosmic rays, natural radioactive materials in the ground, and other sources. Humans are constantly exposed to background radiation at low levels, which are typically considered safe for biological organisms. Harmful effects from radiation require significantly higher doses than those encountered in normal background conditions.