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Chapter 18: Problem 37

Copper-64 is used to study brain tumors. Assume that the original mass of a sample of copper-64 is 26.00 g. After 64 hours, all that remains is 0.8125 g of copper-64. What is the half-life of this radioactive isotope?

Short Answer

Expert verified
The half-life of the copper-64 isotope can be found by first calculating the decay constant with the given values, then by calculating the half-life using the decay constant. In terms of calculation, the half-life is obtained by substituting the decay constant in the half-life formula.

Step by step solution

01

Understanding the Radioactive Decay Constant

In a radioactive decay process, the decay follows an exponential format given by \( N = N_0 e^{- \lambda t} \) where \( N \) is the final amount remaining, \( N_0 \) is the initial quantity, \( \lambda \) is the decay constant, and \( t \) is the time elapsed. From this formula, the decay constant can be calculated by rearranging to \( \lambda = - \frac{1}{t} \ln (\frac{N}{N_0}) \).
02

Calculate the Decay Constant

Insert the given values into the rearranged formula to calculate \( \lambda \). The initial quantity \( N_0 = 26.00 \, g \), the final quantity \( N = 0.8125 \, g \), and the time \( t = 64 \, hours \). Therefore, \( \lambda = - \frac{1}{64} \ln (\frac{0.8125}{26}) \).
03

Calculate the Half-life

With the decay constant, the half-life \( T_{1/2} \) of the substance can be determined using the formula \( T_{1/2} = \frac{0.693}{\lambda} \). Substituting the calculated value of \( \lambda \) will produce the half-life of copper-64.

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