The half-life for the radioactive decay of C-14 is 5715 years and is independent of the initial concentration. If a sample of C-14 initially contains 1.5 mmol of C-14, how many millimoles are left after 2725 years?

검증된 단계별 안내

Identify the type of decay process: This is a first-order decay process, as indicated by the constant half-life.

Use the first-order decay formula: \( N_t = N_0 \times e^{-kt} \), where \( N_t \) is the amount remaining, \( N_0 \) is the initial amount, \( k \) is the decay constant, and \( t \) is the time elapsed.

Calculate the decay constant \( k \) using the half-life formula: \( k = \frac{\ln(2)}{\text{half-life}} \). Substitute the given half-life of 5715 years into the formula.

Substitute the values into the first-order decay formula: Use \( N_0 = 1.5 \) mmol, \( t = 2725 \) years, and the calculated \( k \) to find \( N_t \).

Solve the equation to find the remaining amount of C-14 in millimoles after 2725 years.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Half-life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For C-14, this period is 5715 years, meaning that after this time, only half of the original amount remains. This concept is crucial for understanding the decay process and calculating the remaining quantity of a radioactive substance over time.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, the amount of C-14 decreases exponentially over time, which can be mathematically represented by the equation N(t) = N0 * (1/2)^(t/T), where N0 is the initial amount, t is the elapsed time, and T is the half-life.

Radioactive Decay Formula

The radioactive decay formula allows us to calculate the remaining quantity of a radioactive isotope after a certain period. For C-14, using the initial amount and the half-life, we can determine how much remains after a specific time by applying the formula N(t) = N0 * (1/2)^(t/T), where t is the time elapsed and T is the half-life of the isotope.