Ch 43: Nuclear Physics

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 43: Nuclear Physics

Problem 21

Chapter 43, Problem 21

The common isotope of uranium, $to the 238th power U$ , has a half-life of $4.47 \times 10^{9}$ years, decaying to $to the 234th power T h$ 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?

Verified step by step guidance

Step 1: To find the decay constant (λ) in part (a), use the relationship between the half-life (T₁/₂) and the decay constant: λ = ln(2) / T₁/₂. Substitute the given half-life of uranium-238, T₁/₂ = 4.47 × 10⁹ years, into the formula. Ensure the units of time are consistent (e.g., convert years to seconds if needed).

Step 2: For part (b), use the relationship between activity (A), decay constant (λ), and the number of nuclei (N): A = λN. Rearrange to find N = A / λ. Then, use the relationship between the number of nuclei and the mass of uranium: N = (m / M) × Nₐ, where m is the mass of uranium, M is the molar mass of uranium-238 (238 g/mol), and Nₐ is Avogadro's number (6.022 × 10²³ nuclei/mol). Combine these equations to solve for the required mass m.

Step 3: For part (c), calculate the number of nuclei (N) in 10.0 g of uranium using the formula N = (m / M) × Nₐ, where m = 10.0 g, M = 238 g/mol, and Nₐ = 6.022 × 10²³ nuclei/mol. This gives the total number of uranium nuclei in the sample.

Step 4: Use the activity formula A = λN to calculate the activity of the 10.0 g uranium sample. Substitute the decay constant (λ) from part (a) and the number of nuclei (N) from step 3 into the formula to find the activity in disintegrations per second (Becquerels).

Step 5: Since each decay corresponds to the emission of one alpha particle, the number of alpha particles emitted per second is equal to the activity calculated in step 4. Express the result in terms of alpha particles per second.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Decay Constant

The decay constant (λ) is a probability rate at which a radioactive isotope decays. It is related to the half-life (t½) of the isotope by the formula λ = ln(2) / t½. For uranium-238, knowing the half-life allows us to calculate the decay constant, which is essential for determining the rate of decay and the activity of the sample.

Radioactive Activity

Radioactive activity is the rate at which a sample of radioactive material decays, measured in disintegrations per second. The unit of activity is the curie (Ci), where 1 Ci equals 3.7 x 10^10 disintegrations per second. Understanding how to relate mass to activity is crucial for calculating how much uranium is needed to achieve a specific activity level.

Alpha Emission

Alpha emission is a type of radioactive decay in which an atomic nucleus emits an alpha particle, consisting of two protons and two neutrons. This process reduces the mass number of the original nucleus by four and the atomic number by two, transforming the element into a different isotope. Knowing the nature of alpha decay is important for calculating the number of alpha particles emitted from a given mass of uranium.

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