Question

The half-life of radium is 1690 years. If 10 grams is present now, how much will be present in 50 years?

Short answer

About 9.79 grams of radium will be present in 50 years.

Step-by-step solution

- Understand the Half-Life Concept

The half-life of a substance is the time it takes for half of the substance to decay. Radium has a half-life of 1690 years, meaning every 1690 years, half of the radium decays.

- Use the Exponential Decay Formula

The amount of a radioactive substance remaining after a certain time can be calculated using the formula: \(N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}\), where \(N(t)\) is the amount remaining at time \(t\), \(N_0\) is the initial amount, \(T_{1/2}\) is the half-life, and \(t\) is the time elapsed.

- Identify Known Values

From the problem, \(N_0 = 10\) grams, \(T_{1/2} = 1690\) years, and \(t = 50\) years.

- Substitute Values into the Formula

Substitute the known values into the formula: \(N(50) = 10 \left( \frac{1}{2} \right)^{\frac{50}{1690}}\).

- Calculate the Exponent

Calculate the exponent: \(\frac{50}{1690} \approx 0.0295857988\).

- Evaluate the Expression

Raise \(\left( \frac{1}{2} \right)\) to the calculated exponent: \(\left( \frac{1}{2} \right)^{0.0295857988} \approx 0.979\).

- Multiply by the Initial Amount

Multiply the result by the initial amount: \(N(50) = 10 \times 0.979 \approx 9.79\) grams.

Key concepts

exponential decay

Radioactive decay is a fascinating process that follows a specific mathematical pattern known as exponential decay. This means that the quantity of a radioactive substance decreases at a rate proportional to its current value. In simpler terms, instead of losing a fixed amount over time, the substance loses a consistent percentage.

To illustrate, if you start with 10 grams of a substance, after each half-life period, you will have half of the remaining amount. This creates a decay curve that drops quickly at first and then more slowly over time. This pattern is what we call exponential decay, and it helps us predict how much of a radioactive substance will remain after a given period.

radioactive decay

Radioactive decay happens when unstable atomic nuclei lose energy by emitting radiation. This process changes the nucleus into a different element or a different isotope of the same element. Every radioactive substance, such as radium, has a characteristic