Question

How many milligrams of a 15.0 mg sample of radium-22 6 remain after 6396 years? The half-life of radium-226 is 1599 years.

Short answer

There will be 0.9375 mg of radium-226 remaining after 6396 years.

Step-by-step solution

Understand the half-life formula

The half-life formula relates the remaining quantity of a substance to its initial quantity, the half-life, and the time elapsed. The formula is \[ N(t) = N_0 \times \frac{1}{2}^{\frac{t}{T}} \] where \( N(t) \) is the remaining quantity after time \( t \), \( N_0 \) is the initial quantity, \( T \) is the half-life, and \( t \) is the elapsed time.

Identify the known variables

Given: \( N_0 = 15.0 \text{ mg} \), \( T = 1599 \text{ years} \), \( t = 6396 \text{ years} \). Plug these values into the formula.

Plug in the values

Substitute the known values into the half-life formula: \[ N(6396) = 15.0 \times \frac{1}{2}^{\frac{6396}{1599}} \].

Perform the exponent calculation

Calculate the exponent: \[ \frac{6396}{1599} = 4 \]. Thus, \[ N(6396) = 15.0 \times \frac{1}{2}^4 \].

Calculate the remaining quantity

Compute the remaining quantity: \[ \frac{1}{2}^4 = \frac{1}{16} \]. So, \[ N(6396) = 15.0 \times \frac{1}{16} = 0.9375 \text{ mg} \].

Key concepts

radioactive decay

Radioactive decay is a natural process in which an unstable atomic nucleus loses energy by emitting particles or radiation. This leads to the transformation of the original (parent) atom into a different atom (daughter). The rate of decay is specific to each radioactive isotope and is fundamentally probabilistic, meaning we can predict the half-life but not the exact moment an individual atom will decay.This phenomenon occurs in a variety of forms such as alpha decay, beta decay, and gamma decay, depending on the type of particle or energy emitted. The key aspect of radioactive decay is that it follows an exponential decay pattern, which means the quantity of the substance decreases at a rate proportional to its current amount.

half-life formula

The half-life formula is essential for calculating how much of a radioactive substance remains after a given period. The formula is: \[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T}} \] Here:

• \( N(t) \) is the remaining quantity of the substance after time \( t \).
\( N_0 \) is the initial quantity of the substance.
• \( T \) is the half-life of the substance, i.e., the time taken for half the amount of the substance to decay.
• \( t \) is the elapsed time.

Using the formula, you can solve problems related to the decay of radioactive materials easily. For example, if you start with a sample of radium-226 and want to know how much is left after 6396 years (given its half-life of 1599 years), you can substitute these values into your formula:\[ N(6396) = 15.0 \times \left( \frac{1}{2} \right)^\frac{6396}{1599} \] Calculations result in approximately 0.9375 mg left of the original 15.0 mg sample after 6396 years.

exponential decay

Exponential decay describes a process where the quantity of something decreases at a rate proportional to its current value. This type of decay is common in various natural processes, including radioactive decay.In the context of radioactive materials, exponential decay means that a substance will reduce by a consistent percentage over equal time periods, not by a constant amount. This is best illustrated by the half-life concept, where, after each half-life period, half of the remaining substance decays.Mathematically, exponential decay can be expressed as:\[ N(t) = N_0 \times e^{-\lambda t} \] where:

li> \( N(t) \) is the remaining quantity.
• \( N_0 \) is the initial quantity.
• \( \lambda \) is the decay constant, related to the half-life.
• \( t \) is the time elapsed.

This formula is closely related to the half-life formula, offering another way to understand the time-dependent nature of radioactive decay. Exponential decay is a critical concept in fields ranging from nuclear physics to environmental science, helping us understand a wide range of time-dependent decay phenomena.