A sample contains radioactive atoms of two types, A and B. Initially there are five times as many A atoms as there are B atoms. Two hours later, the numbers of the two atoms are equal. The half-life of A is 0.50 hour. What is the half-life of B?

검증된 단계별 안내

Define the decay law for radioactive atoms: The number of atoms remaining at time \( t \) is given by \( N(t) = N_0 e^{-\lambda t} \), where \( N_0 \) is the initial number of atoms, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.

Relate the decay constant \( \lambda \) to the half-life \( T_{1/2} \) using the formula \( \lambda = \frac{\ln(2)}{T_{1/2}} \). For atom A, \( \lambda_A = \frac{\ln(2)}{0.50} \). For atom B, \( \lambda_B = \frac{\ln(2)}{T_{1/2,B}} \), where \( T_{1/2,B} \) is the unknown half-life of B.

Set up the initial condition: Let the initial number of B atoms be \( N_{B0} \). Then the initial number of A atoms is \( N_{A0} = 5N_{B0} \), as given in the problem.

Write the decay equations for the number of atoms after 2 hours: \( N_A(2) = N_{A0} e^{-\lambda_A \cdot 2} \) and \( N_B(2) = N_{B0} e^{-\lambda_B \cdot 2} \). Since the numbers of A and B atoms are equal after 2 hours, set \( N_A(2) = N_B(2) \).

Substitute the expressions for \( N_A(2) \) and \( N_B(2) \) into the equality: \( 5N_{B0} e^{-\lambda_A \cdot 2} = N_{B0} e^{-\lambda_B \cdot 2} \). Cancel \( N_{B0} \) and solve for \( \lambda_B \) in terms of \( \lambda_A \). Finally, use \( \lambda_B = \frac{\ln(2)}{T_{1/2,B}} \) to find \( T_{1/2,B} \).

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주요 개념

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Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This decay occurs at a characteristic rate for each radioactive isotope, defined by its half-life, which is the time required for half of the radioactive atoms in a sample to decay. Understanding this concept is crucial for analyzing the changes in the number of radioactive atoms over time.

Half-Life

The half-life of a radioactive substance is the time it takes for half of the original amount of the substance to decay. For example, if a sample has a half-life of 0.50 hours, after this time, only half of the original atoms will remain. This concept is essential for calculating the remaining quantities of radioactive isotopes after a given period, as seen in the problem involving atoms A and B.

Exponential Decay

Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, the number of atoms decreases exponentially over time, which can be mathematically represented by the equation N(t) = N0 * (1/2)^(t/T), where N0 is the initial quantity, t is time, and T is the half-life. This concept is vital for determining the relationship between the quantities of atoms A and B after a specified time.

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