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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.