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Chapter 38: Problem 42

You measure the activity of a radioactive sample at \(2.4 \mathrm{MBq}\). Thirty minutes later, the activity level is 1.9 MBq. Find the material's half-life.

Short Answer

Expert verified
The half-life of the radioactive sample is calculated using the derived decay constant \(k\), by substituting into the formula for half-life \(T_{1/2}\) as \(T_{1/2} = \frac{ln(2)}{k}\).

Step by step solution

01

Identify given quantities

Identify the given quantities in the problem: the initial activity level \(N_0 = 2.4 MBq\), the activity level after some time \(N = 1.9 MBq\), and the time \(t = 0.5\) hours
02

Set up the decay equation

Use the given quantities to set up the decay equation \(N = N_0e^{kt}\). In this case, \(1.9 = 2.4e^{0.5k}\).
03

Solve for the decay constant k

Solve the equation for the decay constant \(k\). This involves isolating \(k\) on one side of the equation, which can be done in several steps. First, divide both sides of the equation by 2.4 to get \(\frac{1.9}{2.4} = e^{0.5k}\). Then, take the natural logarithm of both sides to get \(ln\left(\frac{1.9}{2.4}\right) = 0.5k\). Divide again by 0.5 to finally solve for \(k\).
04

Calculate half-life

Use the decay constant \(k\) to calculate the half-life of the radioactive material. The formula for the half-life \(T_{1/2}\) is derived from the decay equation, and is given by \(T_{1/2} = \frac{ln(2)}{k}\).

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Most popular questions from this chapter

A \(^{235} \mathrm{U}\) nucleus undergoes neutron-induced fission, yielding "Iti Cs, three neutrons, and another nucleus, What's that nucleus?

Why might future archaeologists have problems dating samples from the second half of the 20th century?

Passage Problems In \(1972,\) a worker at a nuclear fuel plant in France discovered that uranium from a mine in Oklo, in the African Republic of Gabon, had less U-235 than the normal 0.7 \(\%\) - a quantity known from meteorites and Moon rocks to be constant throughout the solar system. Further analysis showed the presence of isotopes that would result from the decay of fission products. Scientists drew the remarkable conclusion that a natural nuclear fission reaction had occurred some 2 billion years ago, lasting for about 100,000 years. Water, mixing with rich uranium ore, provided the moderator that made the chain reaction possible. More significantly, U-235's 700-million-year half-life means that 2 billion years ago there was a higher abundance of \(U-235\) in natural uranium. At the time of the Oklo fission reaction, the actual amount of U-235 present was a. about the same as today. b. about twice as much as today. c. about four times as much as today. d. about eight times as much as today.

Write the symbol for the germanium isotope with 44 neutrons.

On an energy-release-per-unit-mass basis, by approximately what factor do nuclear reactions exceed chemical reactions?
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