Question

During World War II, tritium \(\left({ }^{3} \mathrm{H}\right)\) was a component of fluorescent watch dials and hands. Assume you have such a watch that was made in January 1944 . If \(17 \%\) or more of the original tritium was needed to read the dial in dark places, until what year could you read the time at night? (For \({ }^{3} \mathrm{H}, t_{1 / 2}=12.3 \mathrm{yr}\).)

Short answer

The tritium watch made in January 1944 could be read in dark places until approximately 1951, as it would have about 17% of its original tritium left by then. This is calculated using the half-life formula for radioactive decay and the given half-life of tritium, which is 12.3 years.

Step-by-step solution

Write the formula for half-life

The formula for half-life is: \[N_t = N_0 \cdot (1/2)^{t/t_{1/2}}\] Where: - \(N_t\) is the amount of tritium left after time t - \(N_0\) is the initial amount of tritium - \(t_{1/2}\) is the tritium half-life, which is given as 12.3 years - \(t\) is the time elapsed

Find the necessary remaining tritium percentage

We are given that at least 17% of the original tritium is needed to read the watch at night. So, let's find how much tritium is left in the watch after t years. Using the given data, we have: \[\frac{N_t}{N_0} = 1 - \frac{17}{100}\]

Plug in the values and find the elapsed time t

Now, we plug in the values in the half-life formula: \[\frac{N_t}{N_0} = (1/2)^{t/t_{1/2}}\] So, we have: \[\frac{83}{100} = (1/2)^{t/12.3}\]

Solve the equation for t

To solve the equation for time t, we need to isolate t. Taking the natural logarithm of both sides helps us do that: \[t = 12.3 \cdot \frac{\ln(83/100)}{\ln(1/2)}\]

Calculate the time t

Finally, we can calculate the value of t: \[t \approx 7.1\: \mathrm{years}\]

Calculate the final year

Now, we need to add the calculated time to the initial year of January 1944: Final year = 1944 +7.1 ≈ 1951.1 So, the watch could be read in dark places until approximately 1951.

Key concepts

Tritium Decay

Before diving into complex equations, let's talk about tritium decay. Tritium, also known as hydrogen-3, is a radioactive isotope of hydrogen characterized by a half-life of about 12.3 years. What does this mean? Essentially, after 12.3 years, half of any sample of tritium will have decayed into another element, helium-3, through a process known as beta decay. This decay process is crucial for devices like radioactive-lit watches because the emitted particles energize the luminescent material, allowing the watch face to glow in the dark. However, as time goes on and the tritium decays, the watch will gradually lose its ability to glow until it's no longer readable in the dark.

Understanding the half-life of tritium is essential for estimating how long these devices will continue to function effectively. It is this half-life that we use when calculating when the glow from the tritium in a watch will diminish to a point where it's no longer able to be seen in low light conditions.

Exponential Decay

The process of tritium decay is an example of exponential decay. Exponential decay describes the decrease in a quantity by a consistent percentage over a period. Unlike linear decay, which decreases by a fixed amount, exponential decay occurs at a rate proportional to the remaining amount. This means that no matter how much of the substance you start with, whether it's a single gram or a kilogram, it will always take the same amount of time for half of it to decay – that's the half-life.

In the context of a radioactive element like tritium, this decay pattern is critical for predicting the exact time at which the glow from the watch would no longer be visible. By using the concept of exponential decay, we can accurately calculate this time and understand how substances diminish over time in a variety of fields, from pharmacology to environmental science.

Half-Life Formula

How does one actually calculate the time when a substance will fall to a certain percentage of its original amount? The answer lies in the half-life formula, which is a straightforward mathematical representation of exponential decay. Here's the basic form of the half-life equation:

\begin{align*}N_t &= N_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\end{align*}This formula tells you the remaining amount (\(N_t\)) after a certain amount of time (\(t\)) if you know the initial amount (\(N_0\)) and the half-life (\(t_{1/2}\)). The half-life (\(t_{1/2}\)) is the time required for half the substance to decay, which, for tritium, is 12.3 years. When you know these values, you can figure out not just when the radioactive material will be undetectable, but also how much there will be at any point in time until then.

In practice, you'll rearrange the formula to solve for the variable you're looking for – usually time – which involves some logarithmic calculations. But once you’ve got it set up, you can answer practical questions, like how long a tritium-powered watch face will stay illuminated.

Logarithmic Calculations

When working with exponential decay and the half-life formula, logarithmic calculations are the key to finding the answers you need. Why? Because exponential and logarithmic functions are mathematical inverses of each other. In the context of decay, you often know the remaining percentage of a substance and need to find out the time that has passed. To isolate the time variable in the half-life equation, use natural logarithms, which are logarithms to the base e, approximately equal to 2.71828.

To solve an equation like \(\frac{N_t}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\), you need to apply the natural logarithm to both sides. This will allow you to express the time \(t\) as a multiple of the half-life \(t_{1/2}\): \(t = t_{1/2} \cdot \frac{\ln(N_t/N_0)}{\ln(1/2)}\).

The result gives you a direct way to calculate elapsed time based on how much of the original substance remains. Logarithmic calculations may seem daunting at first, but understanding them can unlock a world of practical and theoretical applications across various scientific disciplines.