Question

A certain radioactive nuclide has a half-life of 80.9 years. How long does it take for $87.5\mathrm{\%}$ of a sample of this substance to decay?

Short answer

It takes approximately 287.7 years for 87.5% of the radioactive substance to decay, with the given half-life of 80.9 years.

Step-by-step solution

Determine the Percentage Remaining

To find the remaining percentage of the substance after 87.5% has decayed, we need to subtract the decayed percentage from the total (100%). Percentage remaining = 100% - 87.5% Percentage remaining = 12.5% So, N(t) = 12.5%.

Set Up the Decay Equation

With the remaining percentage from step 1, we can set up the decay equation as: 12.5 = 100 * 0.5^(t/80.9)

Solve for Time (t)

To find the time (t), we first divide both sides of the equation by 100: 0.125 = 0.5^(t/80.9) Now, we take the natural logarithm (ln) of both sides: ln(0.125) = ln(0.5^(t/80.9)) Then, we can use the power rule of logarithms which states that ln(a^b) = b*ln(a): ln(0.125) = (t/80.9) * ln(0.5) Now, we will isolate the variable t by dividing both sides by ln(0.5): t/80.9 = ln(0.125) / ln(0.5) Finally, we multiply both sides by the half-life (80.9) to get the time: t = 80.9 * (ln(0.125) / ln(0.5)) Calculate the result: t ≈ 287.7 So, it takes approximately 287.7 years for 87.5% of the radioactive substance to decay.

Key concepts

Half-Life

The concept of half-life is central to understanding radioactive decay. Half-life is the time required for half of a radioactive substance to decay into a stable substance. During this time, the amount of the original radioactive material decreases by 50%.

It is an important tool to measure the age and stability of a substance. For example, in the given exercise, the radioactive nuclide has a half-life of 80.9 years. This means that after 80.9 years, only half of the original amount will be left.

• Half-life is a constant value and does not depend on the quantity of the substance.
• Once half-life is known, the rate of decay can be predicted.

Understanding half-life helps us predict how long a radioactive substance can last and when it will reach a safer state.

Decay Equation

The decay equation is a mathematical representation of how a radioactive substance decreases over time. In the context of the exercise, the equation helps find out how long it takes for 87.5% of the substance to decay. It uses exponential decay, a key concept in understanding radioactive processes.

• Formula used: $$N(t)=N_0\times {0.5}^{t/T}$$
• Where: $N(t)$ is the quantity remaining, $N_0$ is the initial quantity, $t$ is the time elapsed, $T$ is the half-life.

In the exercise, we set up the equation with 12.5% remaining after 87.5% has decayed. Initially, 100% is present, so:

• Equation becomes: $$0.125={0.5}^{t/80.9}$$

This decay equation provides a systematic way of solving for time using a known half-life.

Logarithmic Functions

Logarithmic functions help in solving exponential decay equations by transforming them into linear ones. They are crucial for isolating variables in equations where the unknown is in the exponent.To solve the decay equation in the exercise, the natural logarithm ($\ln$) of both sides is taken:

• $$\ln (0.125)=\ln ({0.5}^{t/80.9})$$

Using the power rule $\ln (a^b)=b\cdot \ln (a)$, the expression simplifies to:

• $$t/80.9=\ln (0.125)/\ln (0.5)$$

Finally, by multiplying by the half-life, we solve for $t$:

• $$t=80.9\cdot (\ln (0.125)/\ln (0.5))$$

Logarithmic functions simplify the procedure, allowing us to solve real-world decay problems effectively.