A small building has become accidentally contaminated with...

A small building has become accidentally contaminated with radioactivity. The longest-lived material in the building is strontium- 90 $\left(\begin{array}{c}{90} \\ {38}\end{array} \mathrm{Sr} \text { has an atomic mass } 89.9077 \mathrm{u},\right.$ and its half-life is 29.1 yr. It is particularly dangerous because it substitutes for calcium in bones.) Assume the building initially contained 5.00 $\mathrm{kg}$ of this substance uniformly distributed throughout the building and the safe level is defined as less than 10.0 decays/min (which is small compared to background radiation). How long will the building be unsafe?

Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This naturally random process leads to a decrease in the amount of radioactive material over time following a well-defined statistical pattern. It is foundational in nuclear physics and helps explain how radioactive substances transform and diminish, which is key for assessing long-term safety issues.

Half-life

The half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a characteristic property of each radioactive isotope and serves as a critical parameter in predicting the behavior of radioactive materials over time. In safety assessments, knowing the half-life allows one to estimate how long a material remains hazardous.

Decay Constant

The decay constant is a parameter that quantifies the probability of decay per unit time for a radioactive atom. It is mathematically related to the half-life (? = ln(2)/half-life) and appears in the exponential decay equation. This constant helps in calculating the rate at which a radioactive substance decreases and is essential for modeling the decay process.

Activity

Activity refers to the number of radioactive decays that occur per unit time, commonly measured in decays per minute or becquerels. It is a direct indicator of the intensity of radioactivity in a substance and is calculated by multiplying the decay constant with the number of radioactive nuclei. Monitoring the activity level is crucial for ensuring that environments remain within safe radiation limits.

Exponential Decay Equation

The exponential decay equation (N(t) = N0 * exp(-?t)) models the reduction of radioactive material over time, where N0 is the initial number of atoms and ? is the decay constant. This equation is central to predicting the remaining quantity of a radioactive substance at any given time and is instrumental in determining safety timelines in radiation management.

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