Question

It took 75 y for \(10.0 \mathrm{~g}\) of a radioactive isotope to decay to \(1.25 \mathrm{~g}\). What is the half-life of this isotope?

Short answer

The half-life of the isotope is 25 years.

Step-by-step solution

Understand the Problem

We have a radioactive isotope and need to find its half-life. We know the initial amount of isotope is 10.0 g and it has decayed to 1.25 g after 75 years.

Identify the Formula

The formula to find the half-life \( t_{1/2} \) of a decaying isotope is \( N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \), where \( N \) is the remaining amount, \( N_0 \) is the initial amount, and \( t \) is the time elapsed.

Substitute Known Values

Substitute the known values into the formula: \( 1.25 = 10.0 \left( \frac{1}{2} \right)^{\frac{75}{t_{1/2}}} \).

Solve for \( \frac{75}{t_{1/2}} \)

Rearrange the formula to solve for \( \frac{75}{t_{1/2}} \) by dividing both sides by 10.0: \( 0.125 = \left( \frac{1}{2} \right)^{\frac{75}{t_{1/2}}} \).

Use Logarithms to Solve for \( t_{1/2} \)

Take the logarithm of both sides to solve for \( \frac{75}{t_{1/2}} \): \[ \log_{10}(0.125) = \frac{75}{t_{1/2}} \log_{10}(0.5) \] Solve for \( \frac{75}{t_{1/2}} \) using \( \log_{10}(0.125) \approx -0.903 \) and \( \log_{10}(0.5) \approx -0.301 \).

Calculate \( t_{1/2} \)

Rearrange the formula to find \( t_{1/2} \):\[ \frac{75}{t_{1/2}} = \frac{-0.903}{-0.301} \]This simplifies to \( 3 \). Therefore, \( t_{1/2} = \frac{75}{3} = 25 \) years.

Key concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. Over time, this process changes the number of protons and neutrons in the nucleus, leading to the transformation of the element itself. This gradual transformation results in the parent isotope decreasing in mass, while a daughter isotope forms.

In the given problem, the initial mass of the radioactive isotope is 10.0 g and decays to 1.25 g over 75 years. This transformation is a perfect example of radioactive decay in action, where isotopes decay at a consistent rate over time.

• It involves random processes at the atomic level, but it is predictable statistically.
• Radioactive decay is crucial in understanding the concept of half-life, a term that denotes the time required for half of the original isotope to convert into another form.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. This is frequently observed in natural phenomena, including radioactive decay.

The formula used to calculate quantities undergoing exponential decay follows this pattern: \[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \] Where:

• \( N \) = the amount of substance remaining.
• \( N_0 \) = the initial amount.
• \( t \) = time elapsed.
• \( t_{1/2} \) = half-life, the time it takes for half of the material to decay.

The exercise is based on this exponential decay pattern. It displays how the initial amount of 10.0 g decreases to 1.25 g over time. Understanding this process is integral to solving decay problems, as it maps out how fast or slow a material decreases, which in turn helps to predict the half-life.

Logarithms in Chemistry

Logarithms are an essential mathematical tool used in solving exponential decay problems in chemistry. They allow us to rewrite exponential equations, making it easier to solve for unknowns like the half-life.

In our problem, we reach a point where we need to isolate an exponent to solve for the variable \( t_{1/2} \). Taking logarithms on both sides of the equation allows us to transform multiplicative processes into additive ones:
\[ \log_{10}(0.125) = \frac{75}{t_{1/2}} \log_{10}(0.5) \] Here, logarithms help us to efficiently calculate the half-life without painstaking trial and error, converting complex exponential relationships into more manageable arithmetic ones.

• Logarithms simplify real-world chemistry problems, especially dealing with decay rates and half-life determinations.
• They provide a more straightforward calculation method when handling exponential functions, crucial in scientific inquiries.

Understanding how to use logarithms to break down exponential relationships is key to tackling a wide array of chemistry problems involving concepts like radioactive decay.