Ch 42: Nuclear Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 42: Nuclear Physics

Problem 21

Chapter 42, Problem 21

A sample of 1.0 x 10¹⁰ atoms that decay by alpha emission has a half-life of 100 min. How many alpha particles are emitted between t = 50 min and t = 200 min?

Verified step by step guidance

Step 1: Understand the concept of radioactive decay and half-life. The half-life is the time it takes for half of the radioactive atoms in a sample to decay. Alpha emission refers to the release of alpha particles during this decay process.

Step 2: Use the formula for the number of remaining atoms after a given time: $ N(t) = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} $, where $ N_0 $ is the initial number of atoms, $ T_{1/2} $ is the half-life, and $ t $ is the elapsed time.

Step 3: Calculate the number of atoms remaining at $ t = 50 $ min using the formula. Substitute $ N_0 = 1.0 \times 10^{10} $, $ T_{1/2} = 100 $ min, and $ t = 50 $ min into the equation.

Step 4: Calculate the number of atoms remaining at $ t = 200 $ min using the same formula. Substitute $ N_0 = 1.0 \times 10^{10} $, $ T_{1/2} = 100 $ min, and $ t = 200 $ min into the equation.

Step 5: Subtract the number of atoms remaining at $ t = 200 $ min from the number of atoms remaining at $ t = 50 $ min to find the total number of atoms that decayed during this time interval. Since each decay corresponds to the emission of one alpha particle, this value represents the number of alpha particles emitted.

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Key Concepts

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

Half-life

Half-life is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life of 100 minutes means that after 100 minutes, half of the original 1.0 x 10^10 atoms will have decayed, leaving 5.0 x 10^9 atoms. This concept is crucial for determining how many atoms remain at any given time and how many have decayed.

Alpha decay

Alpha decay 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 atom by four and the atomic number by two, resulting in a different element. Understanding alpha decay is essential for calculating the number of alpha particles emitted during the decay process.

Exponential decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In radioactive decay, the number of undecayed atoms decreases exponentially over time, which can be modeled mathematically. This concept helps in calculating the number of decayed atoms over specific time intervals, such as between 50 and 200 minutes in this question.

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A sample of 1.0 x 1010 atoms that decay by alpha emission has a half-life of 100 min. How many alpha particles are emitted between t = 50 min and t = 200 min?

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Key Concepts

Half-life

Alpha decay

Exponential decay

A sample of 1.0 x 10¹⁰ atoms that decay by alpha emission has a half-life of 100 min. How many alpha particles are emitted between t = 50 min and t = 200 min?