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watch dials are coated with a phosphor, like ZnS, and a polymer in which some of the \({ }^{1} \mathrm{H}\) atoms have been replaced by \({ }^{3} \mathrm{H}\) atoms, tritium.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

After 50 years, the watch dial coated with a phosphor like ZnS and tritium will be dimmed by approximately 94.3%, as the remaining tritium fraction is about 5.7%.

Step-by-step solution

Identify the given information and the formula to use

We know the half-life of tritium is 12.3 years and the watch is 50 years old. We also know that the light emitted is directly proportional to the amount of tritium. We can use the half-life formula to find the remaining percentage of tritium: \[ RemainingFraction = \frac{1}{2^{\frac{Time}{Half-Life}}}\]

Insert the known values into the formula

Now we shall plug the values into the formula. \[ RemainingFraction = \frac{1}{2^{\frac{50}{12.3}}}\]

Calculate the remaining tritium fraction

Next, we need to compute the remaining fraction of tritium: \[ RemainingFraction = \frac{1}{2^{\frac{50}{12.3}}} \approx 0.057\] This means that after 50 years, approximately 5.7% of the tritium remains.

Determine the dimming percentage

Since the light emitted is directly proportional to the amount of tritium, we can say that the remaining light emitted after 50 years is also 5.7%. Thus, to calculate how much the dial will be dimmed after 50 years, we need to subtract the remaining light percentage from 100%: \[ DimmingPercentage = 100\% - 5.7\% = 94.3\%\]

Interpret the result

After 50 years, the watch dial will be dimmed by approximately 94.3%.

Key concepts

Half-life

The concept of half-life is integral to understanding radioactive decay. Half-life is the time required for a quantity to reduce to half its initial value. This applies to any substance undergoing exponential decay, such as a radioactive isotope. Here, we focus on tritium, a radioactive isotope of hydrogen.
Half-life is a predictable and measurable property. It helps us determine how long it takes for half the isotopes in a sample to decay. For tritium, its half-life is 12.3 years. This means that if you start with a certain amount of tritium, only half will remain after 12.3 years. This characteristic allows us to calculate the remaining amount of material after a given period.
The decay process is exponential. Therefore, using the half-life, you can project the decay over multiple periods. The formula used for calculations in these scenarios is:

• \[ RemainingFraction = \frac{1}{2^{\frac{Time}{Half-Life}}} \]

This equation helps predict the fraction of a substance that remains after any given time, crucial in applications like nuclear medicine, geology, and understanding radiation effects over time.

Tritium Decay

Tritium decay is a type of radioactive decay involving tritium, also known as hydrogen-3. Tritium is a weakly radioactive isotope of hydrogen commonly found in a variety of applications. In watches, tritium is used for illuminating dials.
Tritium decay occurs through a process called beta decay. In this process, tritium atoms transform into helium-3, releasing a beta particle (electron). The beta particles emitted in tritium decay possess enough energy to excite phosphorescent materials, causing them to emit visible light. This feature is why tritium is valuable in watch dials and other luminescent devices.
The decay of tritium is predictable and can be calculated using its half-life. After each half-life (12.3 years), the remaining tritium amount is halved. This nature of decay helps determine how much of the isotope remains over time and, consequently, how much light continues to be emitted from the phosphorescent material in the watch dial.

Phosphorescence

Phosphorescence is the process that allows materials to emit light after being energized. It is a slow release of stored energy as light, longer than fluorescence. This glow continues even after the energy source is removed.
In watch dials, phosphorescent materials like zinc sulfide (ZnS) are used in conjunction with radioactive isotopes such as tritium. When tritium decays, it releases beta particles that strike the phosphorescent material, causing it to emit a soft glow. This glow can be seen in the dark, helping users to read the time without external light sources.
The duration and intensity of phosphorescence are influenced by the decay rate of the radioactive isotope and properties of the phosphorescent material. As the isotopes decay over time, causing reduced interaction with the phosphorescent material, the glow diminishes.

Proportionality in Radioactive Processes

In radioactive processes, proportionality plays a crucial role, particularly relating to decay and emitted energy. For watches using tritium, the light output from the phosphorescent coating is directly proportional to the amount of remaining tritium.
This concept of proportionality implies that as the number of radioactive decays decreases—due to tritium decaying over time—so too does the light intensity emitted by the phosphorescent material. The relationship is linear, meaning a decrease in tritium results in a corresponding decrease in light emission.

• For example, if initially 100% tritium leads to 100% light emission, after one half-life (12.3 years), approximately 50% of tritium remains, leading to around 50% light emission.

The proportionality concept is used to calculate how much a watch dial dims over the years. Knowing that dimming is proportionate to tritium decay allows precise predictions of brightness at any future time.