Chapter 18: Problem 53

Uranium-238 decays through alpha decay with a half-life of \(4.46 \times 10^{9}\) y. How long would it take for seven-eighths of a sample of uranium- \(-238\) to decay?.

Short Answer

Expert verified

The total time for seven-eighths of the Uranium-238 to decay is approximately \(13.38 \times 10^{9}\) years.

Step by step solution

Understand the Half-life Concept

Half-life is the time required for a quantity to reduce to half its initial amount. In this case, every \(4.46 \times 10^{9}\) years, the amount of uranium-238 present halves.

Determine the Fraction Remaining

It is given that seven-eighths of the sample has decayed. This means that only one-eighth of the sample remains. We need to determine how many half-life periods are needed for only one-eighth (or \(0.125\)) of the sample to remain, using the formula for exponential decay \(N = N_0 \times (0.5)^{t/T}\), where \(N\) is the final amount, \(N_0\) is the initial amount, \(t\) is time, and \(T\) is the half-life.

Solve For the Time

You can rearrange the formula to solve for time: \(t = T \times (\log(N/N_0) / \log(0.5))\). Substituting the given values and calculating the result gives a total time for seven-eighths of the sample to decay.

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Uranium-238 decays through alpha decay with a half-life of \(4.46 \times 10^{9}\) y. How long would it take for seven-eighths of a sample of uranium- \(-238\) to decay?.

Short Answer

Step by step solution

Understand the Half-life Concept

Determine the Fraction Remaining

Solve For the Time

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