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Chapter 19: Problem 63

Define "third-life" in a similar way to "half-life", and determine the "third- life" for a nuclide that has a half-life of \(31.4\) years.

Short Answer

Expert verified
Third-life is the time required for a radioactive substance to decay to one-third of its initial mass. We derived a relationship between half-life and third-life using the radioactive decay formula and found that the third-life of a nuclide with a half-life of 31.4 years is approximately 21.56 years.

Step by step solution

01

Define Third-life

Third-life is the time required for a radioactive substance to decay to one-third of its initial mass. This is similar to the concept of half-life, where we consider the time required for the substance to decay to half of its initial mass.
02

Derive the relationship between half-life and third-life

In this step, we will derive the relationship between half-life (t鈧�) and third-life (t鈧�) using the radioactive decay formula: \[N_t = N_0 e^{-\lambda t}\] Where: - \(N_t\) is the remaining mass after time t - \(N_0\) is the initial mass - \(\lambda\) is the decay constant - t is the time elapsed We know that for half-life, the remaining mass (\(N_t\)) will be half of the initial mass, so we can write: \[\frac{1}{2} N_0 = N_0 e^{-\lambda t_h}\] For third-life, the remaining mass will be one-third of the initial mass: \[\frac{1}{3} N_0 = N_0 e^{-\lambda t_3}\] We can determine a relationship between the half-life and third-life by dividing these two equations: \[\frac{\frac{1}{2} N_0}{\frac{1}{3} N_0} = \frac{e^{-\lambda t_h}}{e^{-\lambda t_3}}\] This simplifies to: \[\frac{3}{2} = e^{-\lambda (t_h-t_3)}\] Now, we'll use the given half-life to find the third-life.
03

Calculate the decay constant

To calculate the decay constant (\(\lambda\)), we'll use the half-life formula: \[\frac{1}{2} = e^{-\lambda t_h}\] Given the half-life of 31.4 years, we can solve for \(\lambda\): \[0.5 = e^{-\lambda (31.4)}\] Taking the natural log of both sides: \[\ln(0.5) = -\lambda (31.4)\] Solving for the decay constant: \[\lambda = \frac{\ln(0.5)}{-31.4}\]
04

Calculate the third-life

Using the relationship derived in Step 2: \[\frac{3}{2} = e^{-\lambda (t_h-t_3)}\] Substitute the known values of \(\lambda\) and t鈧�: \[\frac{3}{2} = e^{\frac{\ln(0.5)}{-31.4} (31.4-t_3)}\] Taking the natural log of both sides: \[\ln(\frac{3}{2}) = \frac{\ln(0.5)}{-31.4} (31.4-t_3)\] Solving for the third-life (t鈧�): \[t_3=31.4-\frac{-31.4\ln(\frac{3}{2})}{\ln(0.5)}\] Calculating the third-life: \[t_3 \approx 21.56 \text{ years}\]
05

Interpret the results

The third-life of the nuclide with a half-life of 31.4 years is approximately 21.56 years. This means that after 21.56 years, one-third of the initial mass of this nuclide will have decayed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radioactive Decay
Radioactive decay is a fundamental process by which an unstable atomic nucleus loses energy by emitting radiation. During this process, the original nucleus, referred to as the parent, transforms into a nucleus of a different element, the daughter, via the emission of alpha particles, beta particles, or gamma rays. This transformation occurs spontaneously and without any external influence.

This decay phenomenon is random but can be characterized statistically through decay constants and half-lives, offering a predictable pattern for large quantities of identical nuclides. Importantly, radioactive decay underlies several applications, such as radiometric dating, medical diagnostics and treatments, and the generation of power in nuclear reactors.
Half-Life
The concept of half-life is central to understanding radioactive decay. The half-life of a radioactive nuclide is the time required for half of the atoms in a sample to undergo decay. It serves as a measure of how quickly or slowly a nuclide decays.

Half-life is a constant for each radioactive isotope and is not affected by external factors such as temperature or pressure. Due to its exponential nature, no matter how many times half-life has passed, there will always be some fraction of the original substance remaining, unless an infinite amount of time has elapsed. The exercise above demonstrates the calculation of a specific type of half-life, termed 'third-life,' which is the period needed for a substance to decay to one-third of its initial mass.
Decay Constant
The decay constant \(\lambda\) is a probability factor that characterizes the rate of decay of a radioactive substance. It represents the fraction of atoms that decay in a given time unit and is inversely related to the half-life of the nuclide; a large decay constant indicates a short half-life and vice versa.

In the provided exercise, the decay constant is derived from the known half-life using a natural logarithm, illustrating the mathematical relationship between these two parameters. This constant also appears in the exponential term of the radioactive decay equation, which shows the number of remaining nuclei at any given time relative to the initial quantity. Simply put, the decay constant is a vital element for predicting how rapidly the decay process unfolds and plays a critical role in applications ranging from carbon dating to the safety assessment of nuclear waste.

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