Radium-226, which undergoes alpha decay, has a half-life of 1600 yr. (a) How many alpha particles are emitted in 5.0 min by a 10.0-mg sample of ₂₂₆Ra?

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Step 1: Determine the number of moles of radium-226 in the 10.0 mg sample. Use the molar mass of radium-226, which is approximately 226 g/mol, to convert the mass to moles.

Step 2: Calculate the initial number of radium-226 atoms in the sample using Avogadro's number (6.022 x 10^23 atoms/mol).

Step 3: Use the half-life formula to find the decay constant (\( \lambda \)) for radium-226. The formula is \( \lambda = \frac{0.693}{\text{half-life}} \). Convert the half-life from years to minutes for consistency with the time period given.

Step 4: Apply the first-order decay equation \( N = N_0 e^{-\lambda t} \) to find the number of radium-226 atoms remaining after 5.0 minutes. Here, \( N_0 \) is the initial number of atoms, \( \lambda \) is the decay constant, and \( t \) is the time in minutes.

Step 5: Subtract the number of remaining radium-226 atoms from the initial number to find the number of atoms that decayed. Since each decay corresponds to the emission of one alpha particle, this number is also the number of alpha particles emitted.

Key Concepts

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

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, consisting of two protons and two neutrons. This process reduces the atomic number by two and the mass number by four, transforming the original element into a different element. Understanding alpha decay is crucial for calculating the number of emitted particles over a given time period.

Half-Life

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. For Radium-226, the half-life is 1600 years, meaning that after this period, half of the original amount will have decayed. This concept is essential for determining how much of the sample remains and how many decay events occur over a specified time frame.

Radioactive Decay Rate

The decay rate of a radioactive substance is the frequency at which its nuclei decay, typically expressed in terms of the number of decays per unit time. This rate can be calculated using the initial quantity of the substance, its half-life, and the time elapsed. In this question, it is necessary to convert the time from minutes to years to accurately apply the half-life in determining the number of alpha particles emitted.

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Radium-226, which undergoes alpha decay, has a half-life of 1600 yr. (a) How many alpha particles are emitted in 5.0 min by a 10.0-mg sample of 226Ra?

Key Concepts

Alpha Decay

Half-Life

Radioactive Decay Rate

Radium-226, which undergoes alpha decay, has a half-life of 1600 yr. (a) How many alpha particles are emitted in 5.0 min by a 10.0-mg sample of ₂₂₆Ra?