Question

The half-life of strontium-90 is 29.1 years. Determine the rate constant for the decay of strontium-90 in units of \(s^{-1}\) [The SI unit of time is the second.]

Short answer

The rate constant is approximately \( 7.55 \times 10^{-10} \text{ s}^{-1} \).

Step-by-step solution

Understand the Half-life Formula

The half-life of a radioactive isotope is the time it takes for half of it to decay. The formula relating half-life \( t_{1/2} \) and the rate constant \( k \) is \( t_{1/2} = \frac{0.693}{k} \). This formula comes from the integrated rate law for first-order decay.

Convert Units of Half-life

The given half-life for strontium-90 is 29.1 years. Since we need the rate constant in seconds, convert years to seconds. \( 1 \text{ year} = 365.25 \times 24 \times 60 \times 60 \text{ seconds} \). Thus, \( 29.1 \text{ years} = 29.1 \times 365.25 \times 24 \times 60 \times 60 \text{ seconds} \).

Perform the Conversion Calculation

Calculate the conversion from years to seconds: \( 29.1 \times 365.25 \times 24 \times 60 \times 60 = 918,018,000 \text{ seconds} \). Hence, the half-life in seconds is 918,018,000 seconds.

Calculate the Rate Constant

Use the half-life formula \( t_{1/2} = \frac{0.693}{k} \) and solve for \( k \). Rearrange it to find \( k = \frac{0.693}{t_{1/2}} \). Substitute \( t_{1/2} = 918,018,000 \text{ seconds} \) into the formula to find \( k = \frac{0.693}{918,018,000} \approx 7.55 \times 10^{-10} \text{ s}^{-1} \).

Key concepts

Strontium-90

Strontium-90 is a radioactive isotope that plays a significant role in various scientific fields. It is particularly noted for being a byproduct of nuclear fission from uranium and plutonium, which is why it is often found in nuclear waste and fallout from nuclear weapons tests. This isotope is part of the strontium family, which consists of a variety of isotopes, both stable and unstable.
Strontium-90 is notable for its beta decay process, where it transforms into yttrium-90, and its radioactive nature makes it a health risk if incorporated into bones, resembling calcium due to its chemical properties. Understanding the decay properties of strontium-90 is essential in many areas, such as nuclear physics, environmental science, and health science, where it helps scientists manage its potential risks.

Half-Life Calculation

The concept of half-life is crucial in understanding the stability of radioactive elements. It refers to the time required for half of the radioactive atoms in a sample to decay. The half-life is a fixed property for any given isotope, providing a measure of its radioactive decay rate. Understanding the half-life of a substance, like strontium-90, aids in calculating how long it will remain active in the environment or within the human body. For example, strontium-90 has a half-life of 29.1 years, meaning after 29.1 years, half of a given amount of strontium-90 would have decayed. This process continues over multiple half-lives, gradually reducing the isotope to negligible amounts.
For scientists, half-life calculations help in planning storage and disposal strategies for radioactive material, ensuring safety and compliance with regulatory frameworks.

First-Order Decay Process

Radioactive decay often follows first-order kinetics, a process where the rate of decay is directly proportional to the amount of substance present. This pattern is crucial for understanding the behavior of radioactive isotopes such as strontium-90.In a first-order decay process, the relationship between the half-life and the decay constant (\( k \)) can be expressed via the equation:\[ t_{1/2} = \frac{0.693}{k} \]This equation indicates that the half-life is independent of the initial concentration of the substance. This direct relationship provides a straightforward way to find the decay rate constant if the half-life is known and vice versa. For strontium-90, using its half-life of 29.1 years, we can calculate the decay constant to predict how quickly it transforms into another element over time.

Unit Conversion

Unit conversion is an essential skill in chemistry and physics since it allows scientists to express measurements in the units necessary for specific calculations. When working with radioactive isotopes, such as strontium-90, unit conversion ensures that rate constants are compared accurately and consistently.For example, while the half-life of strontium-90 is given in years, calculations often require converting this time into seconds to be compatible with other units in scientific formulas. Here's how you can convert years into seconds:

• 1 year = 365.25 days (accounting for leap years)
• 1 day = 24 hours
• 1 hour = 60 minutes
• 1 minute = 60 seconds

So, to find the seconds in 29.1 years:\[ 29.1 \times 365.25 \times 24 \times 60 \times 60 = 918,018,000 \text{ seconds} \]
This conversion allows us to use the decay formula efficiently, ultimately yielding the decay rate constant in the common scientific unit of seconds inverse (\( s^{-1} \)). Understanding unit conversions ensures precision and uniformity in scientific calculations.