Problem 36 - Chemistry The Science in Context

Question

The decay rate constant of sodium- \(24,\) a tracer used in blood studies, is \(4.6 \times 10^{-2} \mathrm{h}^{-1} .\) What is the value of its half-life?

Step-by-Step Solution

Answer

Answer: The half-life of sodium-24 is approximately 15.04 hours.

Step 1: Write down the given values

We are given the decay rate constant, \(k = 4.6 \times 10^{-2} \mathrm{h}^{-1}\).

Step 2: Write down the formula to find the half-life

We will use the formula: \(t_{1/2} = \frac{ln(2)}{k}\).

Step 3: Plug in the given values into the formula

Now substitute the given value of \(k\) into the formula: \(t_{1/2} = \frac{ln(2)}{4.6 \times 10^{-2} \mathrm{h}^{-1}}\).

Step 4: Calculate the half-life

Using a calculator, we find the half-life: \(t_{1/2} = \frac{ln(2)}{4.6 \times 10^{-2} \mathrm{h}^{-1}} \approx 15.04 \mathrm{h}\).

Step 5: State the answer

The half-life of sodium-24 is approximately 15.04 hours.

Key Concepts

Decay Rate Constant

The decay rate constant, often represented as \( k \), plays a crucial role in determining how quickly a radioactive substance breaks down. This value indicates the proportion of a sample that decays per unit of time. In our case, for sodium-24, the decay rate constant is \( 4.6 \times 10^{-2} \mathrm{h}^{-1} \).

Understanding the decay rate constant helps us understand the speed of radioactive decay. The units of \( k \) are typically inverse time, such as \( \mathrm{h}^{-1} \) or \( \mathrm{s}^{-1} \), indicating how much of the substance diminishes per hour or second, respectively.

Knowing the value of \( k \) is essential when calculating the half-life of a material, as it is part of the standard formula \( t_{1/2} = \frac{\ln(2)}{k} \). This formula shows that the half-life and the decay rate constant are inversely related—if \( k \) is larger, the half-life is shorter, meaning the substance decays faster.

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This process transforms the nucleus into a different element or a different state of the same element.

At its core, radioactive decay strives for nuclear stability. During this process, particles or electromagnetic waves, such as alpha, beta particles, or gamma rays, are emitted. These emissions can cause the original element to change into a new one or leave it in a less energetic or less 'radioactive' state.

Importantly for students, understanding radioactive decay involves knowing that the rate at which a material decays can be predicted statistically but not precisely for individual atoms. It depends on the material's decay rate constant and its half-life. Sodium-24, for example, decays as a tracer, which makes it valuable in scientific studies like tracking blood flow because of its predictable decay rate.

Sodium-24

Sodium-24 is a radioactive isotope of sodium, commonly used in medical and scientific applications due to its short half-life and predictable decay pattern. It is particularly useful as a tracer in blood studies, allowing researchers to track the movement and rate of blood flow through the human body.

As a radionuclide, sodium-24 emits beta particles and gamma rays during its decay. This property enables its detection through specific imaging techniques, aiding in diagnostic and research outcomes.

Despite its utility, the radioactive nature of sodium-24 requires critical handling and management. Knowing its half-life, calculated at approximately 15.04 hours, informs researchers and medical professionals of the time frame they have available for conducting experiments or treatments effectively before the isotope decays to half its initial value.