Question

A sample of sodium-24 chloride contains \(0.050 \mathrm{mg}\) of \(\mathrm{Na}-24\) to study the sodium balance of an animal. After \(24.9 \mathrm{~h}, 0.016 \mathrm{mg}\) of \(\mathrm{Na}-24\) is left. What is the half- life of \(\mathrm{Na}-24 ?\)

Short answer

Answer: The half-life of sodium-24 is approximately 14.97 hours.

Step-by-step solution

Write down the radioactive decay formula.

The radioactive decay formula is given by: \(N(t) = N_0 \times (\frac{1}{2})^{\frac{t}{T}}\) Where: \(N(t)\) is the remaining amount of the substance after a given time \(t\), \(N_0\) is the initial amount of the substance, \(T\) is the half-life, and \(t\) is the time elapsed.

Plug in values into the radioactive decay formula.

We are given \(N_0 = 0.050mg\), \(N(t) = 0.016mg\), and \(t = 24.9h\). We will plug these values into the radioactive decay formula: \(0.016 = 0.050 \times (\frac{1}{2})^{\frac{24.9}{T}}\)

Solve for the half-life (T).

First, divide both sides by \(0.050\): \((\frac{1}{2})^{\frac{24.9}{T}} = \frac{0.016}{0.050}\) Then, take the logarithm of both sides (base 0.5) to isolate the exponent: \(\log_{\frac{1}{2}} (\frac{1}{2})^{\frac{24.9}{T}} = \log_{\frac{1}{2}} (\frac{0.016}{0.050})\) Now, we can simplify by applying the logarithm rules: \(\frac{24.9}{T} = \log_{\frac{1}{2}} (\frac{0.016}{0.050})\) Next, solve for \(T\): \(T=\frac{24.9}{\log_{\frac{1}{2}} (\frac{0.016}{0.050})}\)

Calculate the half-life (T).

Now we will calculate the half-life of sodium-24 using the expression found in step 3: \(T=\frac{24.9}{\log_{\frac{1}{2}} (\frac{0.016}{0.050})} \approx 14.97h\) The half-life of sodium-24 is approximately \(14.97\) hours.

Key concepts

half-life

Half-life is a fundamental concept in radioactive decay describing how long it takes for half of a radioactive substance to decay into another form. This period is constant for any given substance and doesn't depend on the initial amount. It allows scientists and researchers to predict the decay pattern of a substance over time.

In practical terms, knowing the half-life helps in understanding how long a radioactive isotope will remain active or potentially hazardous. For instance, when treating or diagnosing a disease using a radioactive tracer like sodium-24 (Na-24), understanding the half-life ensures effective and safe application in medical practices.

• Half-life is symbolized by "T" in equations.
• It provides a clear timeline to predict when a material will reduce to a certain amount.
• This concept is vital for fields like archaeology, medicine, and nuclear physics.

The half-life equation is simple but powerful in application, making it an essential part of radioactive decay analysis.

sodium-24

Sodium-24 is a radioactive isotope of sodium used in a variety of scientific and medical applications. Its primary application lies in the study of sodium management in biological systems, like in our given exercise dealing with an animal's sodium balance.

Sodium-24 emits beta particles and gamma rays, which are detectable using specific equipment. This property makes it an effective tracer in diagnostic tests and in understanding physiological processes. In labs and medical studies, small, controlled amounts of Na-24 are used, given its radioactive nature.

• Sodium-24 has a relatively short half-life, approximately 15 hours, meaning it decays swiftly compared to other isotopes.
• It safely exits biological systems before its radioactivity could pose any health hazard.
• Its decay helps researchers track the distribution of sodium in the body.

Understanding the role and behavior of sodium-24 in tests is crucial, particularly in medicine, due to its efficiency in providing results before decaying entirely.

radioactive decay formula

The radioactive decay formula is an essential tool for calculating how much of a radioactive substance remains after a given period. It is represented mathematically as:\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T}} \]This formula helps in predicting the remaining quantity of a substance, such as sodium-24, after a specific time has elapsed.

The components of the formula include:

• \( N(t) \): the amount of the substance left after time \( t \)
• \( N_0 \): the initial quantity of the substance
• \( T \): the half-life of the substance
• \( t \): the time over which decay is observed

To use this formula effectively, substitute the known values for the variables, as demonstrated in solving for the half-life of sodium-24. By rearranging and solving the equation, scientists can determine unknowns like the half-life \( T \) through logarithmic calculations.

In our example, using this formula allowed the determination that sodium-24 has a half-life of approximately 14.97 hours, illustrating the formula's utility in real-world applications.