Question

Some watch dials are coated with a phosphor, like ZnS, and a polymer in which some of the \(^{1} \mathrm{H}\) atoms have been replaced by \(^{3} \mathrm{H}\) atoms, tritium.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?

Short answer

The remaining fraction of tritium after 50 years is approximately 6.06%, meaning that the dial is dimmed by about 93.94% after 50 years.

Step-by-step solution

Understand the problem and find the number of half-lives

In this problem, we want to find the amount of remaining tritium after 50 years, given that the half-life of the tritium is 12.3 years. To do this, we need to first determine the number of half-lives that have passed in 50 years. Let's divide the total time (50 years) by the half-life (12.3 years) to find the number of half-lives: Number of half-lives = Total time / Half-life = 50 years / 12.3 years

Calculate the number of half-lives

Now, we'll calculate the number of half-lives: Number of half-lives = 50 years / 12.3 years ≈ 4.07 Our result is approximately 4.07 half-lives. Since the decay happens continuously, we don't need to round this number to an integer.

Determine the remaining fraction of tritium after 50 years

To find the remaining fraction of tritium after 50 years, we'll use the decay formula for half-lives: Remaining fraction of tritium = \( (1/2) ^{\text{Number of half-lives}} \) Let's plug in the number of half-lives, which we found in Step 2: Remaining fraction of tritium = \( (1/2)^{4.07} \)

Calculate the remaining fraction of tritium

Now, we'll calculate the remaining fraction of tritium: Remaining fraction of tritium = \( (1/2)^{4.07} \) ≈ 0.0606 Our result is approximately 0.0606, which means that about 6.06% of the tritium is left after 50 years.

Find the dimming factor of the watch dial

Since the problem states that the light given off is directly proportional to the amount of tritium, we can use the remaining fraction of tritium to find out by how much the dial is dimmed. Dimming factor = 1 - Remaining fraction of tritium Let's plug in the remaining fraction of tritium, which we found in Step 4: Dimming factor = 1 - 0.0606

Calculate the dimming factor

Finally, let's calculate the dimming factor: Dimming factor = 1 - 0.0606 ≈ 0.9394 Our result is approximately 0.9394, which means that the dial is dimmed by about 93.94% after 50 years.

Key concepts

Half-Life

The half-life of a radioactive isotope, such as tritium, is the time it takes for half of the sample to undergo radioactive decay. It's a measure of the rate at which a substance loses its activity, and it's a constant value specific to each radioactive isotope. Understanding half-life is crucial for a variety of applications, from archaelogical dating to medical treatments.

In the context of our exercise, where we consider the dimming of a watch dial due to tritium decay, knowing the half-life allows us to predict how much the luminosity will decrease over time. Since the half-life of tritium is 12.3 years, in every 12.3-year period, half of the remaining tritium decays, and consequently, the light emitted by the watch dial is reduced by half. By calculating the number of half-lives that have passed during a certain timeframe, we can determine the fraction of tritium remaining and, by extension, the intensity of the glow.

Tritium Decay

Tritium (³H) is a radioactive isotope of hydrogen with a low-energy beta decay. When tritium decays, it transforms into helium-3 with the emission of a beta particle (an electron) and an antineutrino. The emitted beta particles then interact with phosphor materials, such as zinc sulfide (ZnS), in watch dials, causing them to emit light, which is what makes the watch dial glow in the dark.

The relationship between tritium decay and the luminosity of a watch dial is a direct one: as tritium decays, there are fewer beta particles to excite the phosphor, and thus, the dial's brightness diminishes. Over the span of its radioactive half-life, tritium's activity — and the associated light emission — decreases exponentially, following the principles of exponential decay.

Exponential Decay

Exponential decay is a process in which a quantity decreases at a rate proportional to its current value. This type of decay is often used to describe the decrease in the amount of a radioactive substance over time. The law of exponential decay is mathematically represented by the decay formula, which in our case, is the function that describes the remaining amount of tritium:

Remaining fraction of tritium = \( (1/2)^{\text{Number of half-lives}} \)

This exponential model implies that the decrease is steeper at the beginning and slows down as time progresses. It's important to note that the quantity never reaches zero in exponential decay; it approaches zero asymptotically. In practical terms for our exercise, this model helps us to precisely estimate the reduction in the glow of the watch dial over any given period based on the proportion of tritium that has decayed.