Problem 21 - University Physics with Modern 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) \(1.55 \times 10^{-10} \text{ years}^{-1}\), (b) 9.43 g, (c) \(3.92 \times 10^{12} \text{ particles per second}\).

Step 1: Calculate Decay Constant

The decay constant, denoted by \( \lambda \), is calculated using the formula for the half-life: \[ \lambda = \frac{\ln(2)}{T_{1/2}} \]where \(T_{1/2} = 4.47 \times 10^{9} \) years. Plug in the values:\[ \lambda = \frac{0.693}{4.47 \times 10^{9}} \text{ years}^{-1} \]Calculate \( \lambda \):\[ \lambda \approx 1.55 \times 10^{-10} \text{ years}^{-1} \].

Step 2: Convert Curie to Disintegrations Per Second

1 Curie (Ci) is defined as 3.7 × 10^{10} disintegrations per second. For an activity \(A\) of 1.00 Curie:\[ A = 3.7 \times 10^{10} \text{ disintegrations per second} \].

Step 3: Calculate Mass for Given Activity

We can calculate the number of atoms using the relationship between activity and decay constant:\[ A = \lambda N \]where \(N\) is the number of undecayed atoms. Solve for \(N\):\[ N = \frac{A}{\lambda} \]Substitute \(A = 3.7 \times 10^{10} \) disintegrations per second and \( \lambda = 1.55 \times 10^{-10} \) years^{-1}:\[ N = \frac{3.7 \times 10^{10}}{1.55 \times 10^{-10}} \text{ atoms} \approx 2.39 \times 10^{20} \text{ atoms} \].Next, convert atoms to mass. The molar mass of \(^{238} \mathrm{U}\) is 238 g/mol, and Avogadro's number is \(6.022 \times 10^{23}\) atoms/mol:\[ \text{Mass} = \frac{N \times \text{Molar mass of } ^{238} \mathrm{U}}{\text{Avogadro's number}} \]\[ \text{Mass} = \frac{2.39 \times 10^{20} \times 238 \text{ g/mol}}{6.022 \times 10^{23} \text{ atoms/mol}} \approx 9.43 \text{ g} \].

Step 4: Calculate Alpha Particles Emitted per Second

For 10.0 g of uranium, first find the number of uranium atoms:\[ N = \frac{10.0 \text{ g} \times 6.022 \times 10^{23} \text{ atoms/mol}}{238 \text{ g/mol}} \approx 2.53 \times 10^{22} \text{ atoms} \].The emission rate (activity) is:\[ A = \lambda N \].Substitute the values:\[ A = 1.55 \times 10^{-10} \times 2.53 \times 10^{22} \text{ atoms} = 3.92 \times 10^{12} \text{ alpha particles per second} \].

Key Concepts

Half-life

The concept of half-life is central to understanding radioactive decay. It describes the time it takes for half of the radioactive atoms in a sample to decay. In the case of uranium-238, the half-life is an impressive 4.47 billion years. This duration illustrates how stable some radioactive materials can be over millions of generations.
Half-life helps us understand not only the decay process but also the stability of an atom. Importantly, the half-life is unaffected by external conditions such as temperature or physical state.
Knowing the half-life of a radioactive isotope allows us to calculate the decay constant, which is pivotal for various calculations in nuclear physics.

Alpha Emission

Alpha emission is one type of radioactive decay where an atomic nucleus emits an alpha particle and transforms into a new element. An alpha particle is composed of two protons and two neutrons, the same as a helium nucleus.
When uranium-238 undergoes alpha emission, it loses these particles to become thorium-234. This process reduces the atomic number by 2 and the mass number by 4.
Understanding alpha emission is crucial for grasping how elements change over time. It's a natural step in the life of heavy elements, contributing to the diversity of atomic species we observe in nature.

Decay Constant

The decay constant, represented by the Greek letter \( \lambda \), is a unique value for each radioactive isotope that determines the rate at which it decays. It connects directly to the half-life through the formula \(\lambda = \frac{\ln(2)}{T_{1/2}}\), where \(T_{1/2}\) is the half-life of the isotope.
In the example of uranium-238, the decay constant is approximately 1.55 x 10^{-10} years^{-1}.
The decay constant is essential for calculating activities, the number of atoms in a sample, and the rate of decay, making it a cornerstone in the mathematics of nuclear decay.

Radioactivity Measurement

Radioactivity is typically measured using the activity of a sample, which signifies the number of decays per second.
In this context, the curie is a common unit, representing 3.7 x 10^{10} disintegrations per second. Knowing how to convert a curie into a usable form of disintegrations per second allows scientists to work with decay data effectively.
Activity measurement is vital for various applications, from medical treatments to understanding environmental radiation levels. It helps us assess the risk and potential benefits of radioactive substances in different situations.

Atomic Mass Conversion

Converting atomic mass to other units is an important part of working with radioactive materials. The atomic mass measures the mass of an atom relative to a defined standard, usually carbon-12.
In nuclear physics problems, it's often necessary to convert the number of atoms to mass. This is done by using the molar mass of the element and Avogadro's number, \(6.022 \times 10^{23}\) atoms per mole.
For uranium-238, with a molar mass of 238 grams per mole, converting atoms to grams involves specific calculations that relate the sample's number of atoms to total mass, allowing precise measurements in decay calculations.