The half-life for the radioactive decay of C-14 is 5730 years and is independent of the initial concentration. How long does it take for 25% of the C-14 atoms in a sample of C-14 to decay?

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Identify that the problem involves first-order kinetics, as radioactive decay follows first-order kinetics.

Use the formula for first-order decay: \( N_t = N_0 e^{-kt} \), where \( N_t \) is the remaining quantity of substance, \( N_0 \) is the initial quantity, \( k \) is the rate constant, and \( t \) is time.

Recognize that 25% of the C-14 has decayed, meaning 75% remains. Therefore, \( N_t = 0.75N_0 \).

Use the relationship between half-life and rate constant for first-order reactions: \( k = \frac{0.693}{t_{1/2}} \), where \( t_{1/2} \) is the half-life.

Substitute \( k \) and \( N_t = 0.75N_0 \) into the first-order decay equation and solve for \( t \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 5730 years, meaning that after this time, only half of the original amount of C-14 remains. This concept is crucial for understanding the rate of decay and predicting how long it will take for a certain percentage of the substance to decay.

Radioactive Decay

Radioactive decay is a stochastic process by which unstable atomic nuclei lose energy by emitting radiation. This process occurs at a constant rate, characterized by the half-life, and is independent of external conditions such as temperature or pressure. Understanding this concept helps in calculating the remaining quantity of a radioactive isotope over time.

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 amount of C-14 decreases exponentially over time, which means that after each half-life, the remaining quantity is halved. This concept is essential for determining how long it takes for a specific fraction of the substance to decay.

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