It takes 4 h 39 min for a 2.00-mg sample of radium-230 to decay to 0.25 mg. What is the half-life of radium-230?

Verified step by step guidance

1. Identify the initial mass \( m_0 = 2.00 \text{ mg} \) and the final mass \( m = 0.25 \text{ mg} \).
2. Note the time taken for this decay is 4 hours and 39 minutes, which needs to be converted to minutes: \( 4 \times 60 + 39 = 279 \text{ minutes} \).
3. Use the first-order decay formula: \( m = m_0 \times \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \( t \) is the time elapsed and \( t_{1/2} \) is the half-life.
4. Rearrange the formula to solve for the half-life \( t_{1/2} \): \( t_{1/2} = \frac{t}{\log_2\left(\frac{m_0}{m}\right)} \).
5. Substitute the known values into the equation to find \( t_{1/2} \).

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process occurs at a predictable rate for each radioactive isotope, characterized by its half-life, which is the time required for half of the radioactive atoms in a sample to decay.

Half-Life

Half-life is a specific measure of the time it takes for half of a given quantity of a radioactive substance to decay. It is a constant for each isotope and is crucial for calculating the remaining quantity of the substance over time, as well as for understanding the decay process in nuclear chemistry.

Exponential Decay Formula

The exponential decay formula describes how the quantity of a radioactive substance decreases over time. It is expressed as N(t) = N0 * (1/2)^(t/T), where N(t) is the remaining quantity at time t, N0 is the initial quantity, and T is the half-life. This formula allows for the calculation of remaining mass after a certain period, which is essential for solving decay problems.