Problem 38 - Chemistry The Central Science In Si Units

Question

How much time is required for a 5.00-g sample of \({ }^{233} \mathrm{~Pa}\) to decay to \(0.625 \mathrm{~g}\) if the half-life for the beta decay of \({ }^{233} \mathrm{~Pa}\) is 27.4 days?

Step-by-Step Solution

Answer

82.2 days

Step 1: Understand the concept of half-life

The half-life of a substance is the time it takes for half of the substance to decay. For \({}^{233}\text{Pa}\), the half-life is 27.4 days, meaning every 27.4 days, half of the remaining \({}^{233}\text{Pa}\) transforms into another substance through decay.

Step 2: Determine the number of half-lives required

Use the formula for decay: \[ N = N_0 imes \left(\frac{1}{2}\right)^n \] where \(N\) is the final amount, \(N_0\) is the initial amount, and \(n\) is the number of half-lives. Plug in \(N_0 = 5.00\text{ g}\) and \(N = 0.625\text{ g}\): \[ 0.625 = 5.00 \times \left(\frac{1}{2}\right)^n \]

Step 3: Solve for the number of half-lives

Divide both sides by 5.00: \[ \frac{0.625}{5.00} = \left(\frac{1}{2}\right)^n \] Simplify the left side: \[ 0.125 = \left(\frac{1}{2}\right)^n \] Express 0.125 as a power of 0.5. Since \(0.125=(0.5)^3\), set \(n=3\).

Step 4: Calculate the total time for the decay process

Multiply the number of half-lives by the length of one half-life to find the total time: \[ \text{Total time} = n \times \text{half-life} = 3 \times 27.4 = 82.2 \text{ days} \]

Step 5: Conclusion

The time required for a 5.00-g sample of \({}^{233}\text{Pa}\) to decay to 0.625 g is 82.2 days.

Key Concepts

Beta Decay

Beta decay is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted from an atomic nucleus. It occurs in two forms: beta-minus and beta-plus decay. In beta-minus decay, a neutron transforms into a proton, emitting an electron and an antineutrino. In beta-plus decay, a proton becomes a neutron, emitting a positron and a neutrino. This process changes the element into a different element.

The resulting atom has a different atomic number but the same mass number. This means that while the number of protons changes, the total number of particles in the nucleus remains constant. Beta decay is crucial in processes such as radiometric dating, where understanding decay rates helps determine the age of objects.

For most natural elements undergoing this type of decay, the process happens because the nucleus has an imbalance of protons and neutrons, seeking stability by transforming some of its components.

Radiometric Dating

Radiometric dating is a technique used to date materials such as rocks or carbon. It is based on measuring the abundance of a radioactive isotope and its decay products. As an element undergoes radioactive decay, it changes into another element at a predictable rate, known as its half-life.

This method relies on principles of beta decay. Consider ²³³Pa, which undergoes beta decay with a known half-life. By calculating how long it takes for half of the sample to decay, scientists can estimate the material's age. This method is powerful, as it allows age determination over a vast timeline from thousands to billions of years.

• Useful for dating ancient artifacts, geological formations, and even archaeological finds.
• Requires precise measurement and understanding of decay products.
• Commonly used isotopes include carbon-14, uranium-238, and potassium-40.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions and properties of atomic nuclei. It deals with radioactivity, nuclear processes, and properties and behavior of elements on a subatomic level. This branch of chemistry is essential for understanding not only natural nuclear phenomena but also applications such as energy production and medical treatments.

Key concepts within nuclear chemistry include understanding half-lives, decay types (such as beta decay), and the changes these processes cause in elements. For example, when ²³³Pa undergoes beta decay, it helps us learn about the transformation of elements, the release of energy, and the balance of particles within a nucleus.

Nuclear chemistry has real-world applications:

• Nuclear Power: Harnessing energy from nuclear reactions to produce electricity.
• Medicine: Using radioactive isotopes for diagnostic imaging and cancer treatments.
• Environmental Science: Studying radioactive materials in the environment to monitor pollution and effects on ecosystems.