Chapter 38: Problem 52

You're assessing the safety of an airport bomb-detection system in which neutron activation of the stable nitrogen isotope \(\frac{15}{7} \mathrm{N}\) turns it into unstable \(^{\text {is }}\) N. The \(N\) - 16 decays by beta emission with 7.13-s half-life. How long after activation will the \(\mathrm{N}-16\) activity have dropped by a factor of 1 million?

Short Answer

Expert verified

By inserting the given and calculated values into the equation, the time period it takes for the \(\mathrm{N}-16\) activity to drop by a factor of 1 million can be obtained.

Step by step solution

Understand the Decay Law

In radioactive decay, the number of radioactive atoms follows the exponential decay law which states that the activity is proportional to the number of atoms present. Mathematically, this can be written as \(N = N_0 e^{−λt}\) where N is the final quantity of the substance, \(N_0\) is the initial quantity, λ is the decay constant, and t is time.

Calculate Decay Constant

The decay constant can be calculated from the half-life, \(T_{1/2}\), using the formula: \(λ = log(2) / T_{1/2}\) . In this case, the half-life is 7.13 seconds, so the decay constant \(λ = log(2) / 7.13\).

Using Decay Law to solve for Time

We know from the problem that the isotope's activity has dropped by a factor of 1 million. That means \(N/N_0 = 1 / 1,000,000\). Substituting into the decay law gives us \(1/1,000,000 = e^{−λt}\). Solving for time (t) gives the equation \(t = - log(1/1,000,000) / λ\). Substituting the value of λ from step 2, we can find t.

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Short Answer

Step by step solution

Understand the Decay Law

Calculate Decay Constant

Using Decay Law to solve for Time

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