Some watch dials are coated with a phosphor, like ZnS, and a polymer in which some of the 1H atoms have been replaced by 3H atoms, tritium. The phosphor emits light when struck by the beta particle from the tritium decay, causing the dials to glow in the dark. The half-life of tritium is 12.3 yr. If the light given off is assumed to be directly proportional to the amount of tritium, by how much will a dial be dimmed in a watch that is 50 yr old?

Verified step by step guidance

1. Identify the initial concept: The problem involves radioactive decay, specifically the decay of tritium (3H) with a half-life of 12.3 years.
2. Understand the relationship: The light emitted is directly proportional to the amount of tritium present, meaning as tritium decays, the light intensity decreases.
3. Use the half-life formula: The amount of a radioactive substance remaining after a certain time can be calculated using the formula: \( N(t) = N_0 \times (\frac{1}{2})^{\frac{t}{t_{1/2}}} \), where \( N(t) \) is the remaining quantity, \( N_0 \) is the initial quantity, \( t \) is the time elapsed, and \( t_{1/2} \) is the half-life.
4. Substitute the known values: Here, \( t = 50 \) years and \( t_{1/2} = 12.3 \) years. Substitute these values into the formula to find the fraction of tritium remaining after 50 years.
5. Calculate the dimming: The fraction of tritium remaining corresponds to the fraction of light intensity remaining. Subtract this fraction from 1 to find the fraction by which the dial has dimmed.

Key Concepts

Radioactive Decay and Half-Life

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. The half-life is the time required for half of the radioactive atoms in a sample to decay. For tritium, with a half-life of 12.3 years, after 50 years, multiple half-lives will have passed, significantly reducing the amount of tritium present.

Proportionality in Light Emission

The light emitted by the phosphor in the watch dial is directly proportional to the amount of tritium present. This means that as the quantity of tritium decreases due to radioactive decay, the intensity of the light emitted will also decrease in a predictable manner, allowing us to calculate the expected dimming of the dial over time.

Exponential Decay

Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. In the context of tritium decay, the amount of tritium remaining after a certain time can be calculated using the formula N(t) = N0 * (1/2)^(t/T), where N0 is the initial amount, t is the elapsed time, and T is the half-life. This concept is crucial for determining how much tritium—and thus light—remains after 50 years.