Calculating the Half-Life of a Radioactive Isotope Using Its Decay Rate

When working with radioactive isotopes, understanding the decay rate and half-life of the isotope is crucial. The half-life is the time required for the activity to decrease to half of its initial value, and it is determined using mathematical modeling of the exponential decay process. This article will guide you through the steps to calculate the half-life given the decay rate and the principles of exponential decay.

Understanding Exponential Decay

Exponential decay is a key concept when dealing with radioactive isotopes. The rate at which a radioactive isotope decays can be described by the equation:

dN/dt -λN0e-λt

where:

λ is the decay constant, Nsub>0 is the initial number of radioactive atoms, t is time, and N(t) is the number of radioactive atoms at time t.

The Role of the Decay Constant

The decay constant λ is related to the half-life t1/2 by the formula:

λ -ln(1/2) / t1/2

Measuring the Decay Rate

The decay rate can be measured using a Geiger counter. The decay rate is the number of decays per unit time, which is directly related to the number of radioactive decays that occur in a given time period. By monitoring the decay rate over time, we can calculate the decay constant and, consequently, the half-life.

Using Logarithms to Determine the Half-Life

To determine the half-life, we can use the following steps:

Calculate the fraction of the radioactive atoms remaining after a given time using the exponential decay formula. Set up the equation for the half-life using the remaining fraction. Solve the equation for the half-life t1/2.

Example Calculation

Assume we know that 0.4% of an isotope decays every second. First, we need to determine the fraction of atoms remaining:

After 1 second, the fraction of undecayed atoms 1 - 0.004 0.996 After another second, the fraction surviving decay would be 0.996 × 0.996 of the original 0.9962 After 2 seconds, the fraction of undecayed atoms 0.9962 After n seconds, the fraction of undecayed atoms 0.996n By definition, the half-life is the time in which the fraction of undecayed atoms 0.5

To find n, we set up the equation:

0.996n 0.5

Taking the logarithm of both sides:

n ln(0.996) ln(0.5)

Solving for n:

n ln(0.5) / ln(0.996)

Using a calculator, we find:

n ≈ 172.93999 seconds

Therefore, the half-life of the isotope is approximately 172.94 seconds.

Alternative Calculation Method

According to the definition, the decay constant λ is given by:

λ 0.693 / t1/2

Given that the decay rate (disintegration constant) is 0.004, we can calculate the half-life:

t1/2 0.693 / 0.004 ≈ 172.93 seconds

This calculation confirms our previous result.

Conclusion

Calculating the half-life of a radioactive isotope is essential for radioactive dating and understanding the behavior of radioactive materials. By using the principles of exponential decay and the decay constant, we can accurately determine the half-life and other key parameters of a radioactive isotope. This process is fundamental in fields such as nuclear physics, radiology, and environmental science.