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Chapter 10: Problem 6

How is the initial activity rate of a radioactive substance related to its half-life?

Short Answer

Expert verified
The initial activity rate (A₀) of a radioactive substance is directly related to its half-life (T). The relationship can be derived by combining the initial activity rate equation and the decay constant's relation to half-life, forming the equation \( A₀ = \frac{ln(2)}{T} N₀ \).

Step by step solution

01

Understand Key Terms and Concepts

Radioactive decay is a spontaneous process where unstable nuclei lose energy by emitting radiation. The activity rate, represented by A (measured in becquerel (Bq)), is the number of decays occurring per unit of time. The half-life, denoted T (measured in seconds, minutes, hours, years, etc.), is the time it takes for a substance to decay by half. The decay constant, represented by the Greek letter λ (lambda), is a characteristic constant of each radioactive isotope and determines the probability per unit time that a nucleus will decay. The decay constant and half-life are related by the equation: \[ T = \frac{ln(2)}{\lambda} \]
02

Derive the Activity Rate Equation

The activity rate (A) of a radioactive substance is given as follows: \[ A = \lambda N \] Where N is the number of radioactive nuclei and λ is the decay constant.
03

Express the Activity Rate in Terms of Half-Life

Combining the equations for half-life and activity rate, we can express the activity rate in terms of half-life: First, solve the half-life equation for decay constant: \[ \lambda = \frac{ln(2)}{T} \] Now, substitute this decay constant into the activity rate equation: \[ A = \frac{ln(2)}{T} N \]
04

Determine the Initial Activity Rate

The initial activity rate (A₀) is the activity rate at the beginning when there are the most radioactive nuclei (N₀). In this case, the equation becomes: \[ A₀ = \frac{ln(2)}{T} N₀ \] This equation shows the relationship between the initial activity rate and the half-life (T) of a radioactive substance. In conclusion, the initial activity rate (A₀) of a radioactive substance is directly related to its half-life (T). The relationship can be derived by combining the initial activity rate equation and the decay constant's relation to half-life, forming the equation \( A₀ = \frac{ln(2)}{T} N₀ \).

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Most popular questions from this chapter

Show that the total energy released in the proton-proton chain is \(26.7 \mathrm{MeV},\) considering the overall effect in \(^{1} \mathrm{H}+^{1} \mathrm{H} \rightarrow^{2} \mathrm{H}+e^{+}+v_{\mathrm{e}},^{1} \mathrm{H}+^{2} \mathrm{H} \rightarrow^{3} \mathrm{He}+\gamma,\) and \(^{3} \mathrm{He}+^{3} \mathrm{He} \rightarrow^{4} \mathrm{He}+^{1} \mathrm{H}+^{1} \mathrm{H} .\) Be sure to include the annihilation energy.

For the reaction, \(n+^{3} \mathrm{He} \rightarrow^{4} \mathrm{He}+\gamma,\) find the amount of energy transfers to \(^{4} \mathrm{He}\) and \(\gamma\) (on the right side of the equation). Assume the reactants are initially at rest. (Hint: Use conservation of momentum principle.)

Explain why a bound system should have less mass than its components. Why is this not observed traditionally, say, for a building made of bricks?

If two nuclei are to fuse in a nuclear reaction, they must be moving fast enough so that the repulsive Coulomb force between them does not prevent them for getting within \(R \approx 10^{-14} \mathrm{m}\) of one another. At this distance or nearer, the attractive nuclear force can overcome the Coulomb force, and the nuclei are able to fuse. (a) Find a simple formula that can be used to estimate the minimum kinetic energy the nuclei must have if they are to fuse. To keep the calculation simple, assume the two nuclei are identical and moving toward one another with the same speed \(v\). (b) Use this minimum kinetic energy to estimate the minimum temperature a gas of the nuclei must have before a significant number of them will undergo fusion. Calculate this minimum temperature first for hydrogen and then for helium. (Hint: For fusion to occur, the minimum kinetic energy when the nuclei are far apart must be equal to the Coulomb potential energy when they are a distance \(R\) apart.)

Verify that the total number of nucleons, and total charge are conserved for each of the following fusion reactions in the proton-proton chain. (i) \(^{1} \mathrm{H}+^{1} \mathrm{H} \rightarrow^{2} \mathrm{H}+e^{+}+v_{\mathrm{e}},\) (ii) \(^{1} \mathrm{H}+^{2} \mathrm{H} \rightarrow^{3} \mathrm{He}+\gamma,\) and (iii) \(^{3} \mathrm{He}+^{3} \mathrm{He} \rightarrow^{4} \mathrm{He}+^{1} \mathrm{H}+^{1} \mathrm{H}\). (List the value of each of the conserved quantities before and after each of the reactions.)
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