Problem 23 - College Physics

Question

Calculate the time required for a sample of radioactive tritium to lose \(80.0 \%\) of its activity. (Tritium has a half-life of 12.3 years.)

Step-by-Step Solution

Answer

19.9 years

Step 1: Understanding the Problem

We need to calculate the time it takes for tritium to lose 80% of its radioactivity. This implies finding the time it takes for the radioactivity to drop to 20% (since 100% - 80% = 20%) of its original activity. We know that the half-life of tritium is 12.3 years.

Step 2: Using the Exponential Decay Formula

The formula for radioactive decay is given by:\[ N(t) = N_0 e^{-kt} \]Where:- \( N(t) \) is the remaining quantity of the substance at time \( t \).- \( N_0 \) is the initial quantity of the substance.- \( k \) is the decay constant.- \( t \) is the time elapsed.

Step 3: Calculating the Decay Constant

The decay constant \( k \) can be derived from the half-life \( T_{1/2} \) using the formula:\[ k = \frac{\ln(2)}{T_{1/2}} \]Substituting the given half-life of 12.3 years:\[ k = \frac{\ln(2)}{12.3} \approx 0.0564 \text{ per year} \]

Step 4: Setting Up the Equation

We want the activity to decrease to 20% of its original value. Therefore, we set \( N(t) = 0.2N_0 \) and solve for \( t \):\[ 0.2N_0 = N_0 e^{-kt} \]This simplifies to:\[ 0.2 = e^{-kt} \]

Step 5: Solving for Time, t

Taking the natural logarithm of both sides, we get:\[ \ln(0.2) = -kt \]Substituting \( k = 0.0564 \), solve for \( t \):\[ t = \frac{\ln(0.2)}{-0.0564} \approx 19.9 \text{ years} \]

Step 6: Conclusion

It takes approximately 19.9 years for tritium to lose 80% of its radioactivity.

Key Concepts

Exponential Decay

Radioactive decay, like that of tritium, is a classic example of exponential decay. In this process, the quantity of a substance decreases at a rate proportional to its current amount. This can be expressed mathematically with the formula:

• \( N(t) = N_0 e^{-kt} \)

Here,

• \( N(t) \) represents the quantity of radioactive substance remaining at time \( t \).
• \( N_0 \) is the initial quantity of the substance, or how much you started with.
• \( k \) is the decay constant, which is unique for each substance.

The function suggests that as time \( t \) increases, \( N(t) \) decreases exponentially. This type of behavior is common in natural processes, where the rate of change depends on the existing amount of substance.
Understanding exponential decay helps to predict how a substance diminishes over time, which is crucial for calculating how long it takes to reach a certain level of activity.

Half-Life Calculation

The concept of a half-life is fundamental in understanding radioactive decay. It is the time required for a quantity to reduce to half its initial amount. Tritium, for example, has a half-life of 12.3 years, meaning after 12.3 years, only half of the initial tritium remains.

• During the first half-life, the substance reduces to 50% of the original amount.
• By the second half-life, it's reduced to 25%, and so on.

To explore further, suppose we have 100 units of a radioactive isotope. After one half-life:

• 50 units remain. After another, only 25 units will.

This diminishes further with each subsequent half-life. Calculating how many half-lives have passed can help determine how much time has elapsed or when a future event (like a specific percentage decay) will occur. Such calculations are done using the exponential decay formula, plugging in values for the decay constant, and solving for time.

Decay Constant

The decay constant \( k \) plays a key role in radioactive decay. It's a value that defines the rate at which a particular substance decays. For every radioactive isotope, like tritium, \( k \) can be calculated using its half-life with the formula:

• \( k = \frac{\ln(2)}{T_{1/2}} \)

Where \( T_{1/2} \) represents the half-life of the substance.

• For tritium, with a half-life of 12.3 years, the decay constant is approximately 0.0564 per year.

Understanding the decay constant allows one to track how rapidly a substance will decay. It's essential for calculating the time required for a substance to decay to a certain activity level, like 20% in the tritium example. The decay constant gives insight into the stability of a substance; the larger it is, the more quickly the substance decays.