Problem 18 - College Physics

Question

The common isotope of uranium, \(^{238} \mathrm{U},\) has a half-life of \(4.47 \times 10^{9}\) years, decaying to \(^{234} \mathrm{Th}\) by alpha emission. (a) What is the decay constant? (b) What mass of uranium is required for an activity of 1.00 curie? (c) How many alpha particles are emitted per second by 10.0 g of uranium?

Step-by-Step Solution

Answer

(a) \( \lambda = 1.55 \times 10^{-10} \text{ per year}\); (b) 1916 g; (c) 3.92 trillion alpha particles/second.

Step 1: Understanding the Decay Constant

To find the decay constant, we use the formula relating the half-life to the decay constant: \( \lambda = \frac{\ln(2)}{t_{1/2}} \). Here, \( t_{1/2} = 4.47 \times 10^9 \text{ years} \). Substituting this into the equation gives \( \lambda = \frac{\ln(2)}{4.47 \times 10^9} \). Calculating this value will give us the decay constant.

Step 2: Converting Half-life to Decay Constant

Calculate \( \ln(2) \approx 0.693 \). Substitute this value into the decay constant formula: \( \lambda = \frac{0.693}{4.47 \times 10^9} \approx 1.55 \times 10^{-10} \text{ per year} \). This is the decay constant of uranium-238.

Step 3: Finding Uranium Mass for Given Activity

The activity (A) is related to the decay constant and the number of atoms (N) by the formula \( A = \lambda N \). We are given \( A = 1.00 \text{ curie} = 3.70 \times 10^{10} \text{ disintegrations per second} \). Rearrange for N: \( N = \frac{A}{\lambda} \). Substitute \( \lambda = 1.55 \times 10^{-10}/(365.25 \times 24 \times 3600) \text{ per second} \) into the formula. Calculate \( N = \frac{3.70 \times 10^{10}}{1.55 \times 10^{-10}/(3.16 \times 10^7)} \approx 4.85 \times 10^{24} \).

Step 4: Calculating Mass from Number of Atoms

Knowing \( N \), we use Avogadro's number (\( 6.022 \times 10^{23} \text{ atoms/mol} \)) to convert atoms to moles of uranium: \( \text{moles} = \frac{4.85 \times 10^{24}}{6.022 \times 10^{23}} \approx 8.05 \text{ moles} \). The molar mass of uranium-238 is approximately 238 g/mol, so the mass is \( 238 \times 8.05 \approx 1916 \text{ g} \).

Step 5: Finding Alpha Emission Rate from 10 g Uranium

First, find the number of uranium atoms in 10 g: \( \text{moles} = \frac{10}{238} \approx 0.042 \text{ moles} \). Convert moles to atoms: \( \text{atoms} = 0.042 \times 6.022 \times 10^{23} \approx 2.53 \times 10^{22} \). Calculate the activity: \( \text{activity} = \lambda N = 1.55 \times 10^{-10} \times 2.53 \times 10^{22} \approx 3.92 \times 10^{12} \text{ disintegrations/second} \), implying that approximately 3.92 trillion alpha particles are emitted per second.

Key Concepts

Radioactive Decay

Radioactive decay is a process by which unstable atomic nuclei transform into more stable ones over time. This happens due to the emission of particles or radiation.

• The transformation is governed by the decay constant, which indicates how quickly a substance undergoes decay.
• It is an exponential process, leading to a decrease in the number of original radioactive atoms over time.
• Common decay processes include alpha decay, beta decay, and gamma decay.

Radioactive decay is often unpredictable for individual atoms but predictable in large numbers, behaving statistically with a probability defined by its decay constant.

Half-life Calculation

The half-life of a radioactive isotope is the time it takes for half of the radioactive nuclei in a sample to decay. It's a key concept for understanding the stability and longevity of isotopes.

• The formula for calculating the decay constant (\( \lambda \)) is \( \lambda = \frac{\ln(2)}{t_{1/2}} \).
• The half-life (\( t_{1/2} \)) provides insight into the duration for which a radioactive substance will remain active.
• Different isotopes have different half-lives, ranging from fractions of a second to billions of years.

Understanding the half-life is crucial for applications in nuclear physics, medicine, and archaeology.

Uranium-238

Uranium-238 (\(^{238}\text{U}\)) is a notable isotope of uranium, widely recognized in nuclear physics due to its abundance and long half-life.

• It has a half-life of about \(4.47 \times 10^9\) years, making it a prevalent long-lived radioactive element.
• This isotope naturally occurs in the Earth's crust and is commonly used in radiometric dating techniques, especially in dating rocks and geological formations.
• Uranium-238 undergoes a series of transformations, ultimately decaying into lead-206, but its decay chain starts with an alpha emission.

This element plays a crucial role in the study of radioactive decay and nuclear processes.

Alpha Emission

Alpha emission refers to the release of alpha particles from a radioactive nucleus. Alpha particles are composed of two protons and two neutrons, making them a helium nucleus.

• This type of decay results in a decrease of atomic number by two and mass number by four in the original element.
• The emitted alpha particles are relatively heavy and carry a positive charge, which causes them to be less penetrating compared to other types of radiation like beta particles or gamma rays.
• In the decay of uranium-238, alpha emission transforms it into thorium-234, initiating a decay series that eventually leads to stable lead.

Understanding alpha emission is vital in fields like radiation safety, nuclear chemistry, and environmental science.