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Chapter 29: Problem 39

Calculate the activity of \(1.0 \mathrm{g}\) of radium- 226 in Ci.

Short Answer

Expert verified
Answer: The activity of 1.0 g of radium-226 is approximately 0.9886 Ci.

Step by step solution

01

1. Determine the number of moles of radium-226 and convert it to atoms

First, we must determine the number of moles of radium-226 (Ra-226) in 1.0 g. To do this, we will use the molar mass of Ra-226, which is 226.0254 g/mol: Number of moles = mass / molar mass Number of moles of Ra-226 = 1.0 g / 226.0254 g/mol ≈ 0.004424 mol Now we can convert the number of moles to atoms using Avogadro's number (6.022 x 10^23 atoms/mol): Number of atoms of Ra-226 = 0.004424 mol x 6.022 x 10^23 atoms/mol ≈ 2.661 x 10^21 atoms
02

2. Calculate the decay constant for radium-226

To find the activity, we need the decay constant, which can be calculated using the half-life of radium-226. The half-life of radium-226 is 1600 years. We will first convert the half-life into seconds: Half-life in seconds = 1600 years x 365.25 days/year x 24 hours/day x 60 minutes/hour x 60 seconds/minute ≈ 5.049 x 10^10 seconds Now, we can calculate the decay constant (λ) using the formula \(\lambda = \frac{ln(2)}{T_{1/2}}\): λ ≈ ln(2) / (5.049 x 10^10 seconds) ≈ 1.373 × 10^-11 s^-1
03

3. Determine the activity of radium-226 in becquerels (Bq)

Now we can calculate the activity in becquerels (Bq), which is equal to the number of decays per second. The activity (A) can be calculated using the formula A = λN, where N is the number of atoms and λ is the decay constant: A ≈ (1.373 × 10^-11 s^-1)(2.661 x 10^21 atoms) ≈ 3.657 x 10^10 Bq
04

4. Convert the activity from Bq to Curies (Ci)

Finally, we need to convert the activity from becquerels (Bq) to Curies (Ci). The conversion factor is 1 Ci = 3.7 x 10^10 Bq, so we divide the activity in Bq by 3.7 x 10^10 Bq/Ci: Activity in Ci ≈ 3.657 x 10^10 Bq / (3.7 x 10^10 Bq/Ci) ≈ 0.9886 Ci So, the activity of 1.0 g of radium-226 is approximately 0.9886 Ci.

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Most popular questions from this chapter

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Suppose that a radioactive sample contains equal numbers of two radioactive nuclides \(A\) and \(B\) at \(t=0 .\) A has a half-life of \(3.0 \mathrm{h},\) while \(\mathrm{B}\) has a half-life of \(12.0 \mathrm{h}\) Find the ratio of the decay rates or activities \(R_{\mathrm{A}} / R_{\mathrm{B}}\) at (a) \(t=0,\) (b) \(t=12.0 \mathrm{h},\) and \((\mathrm{c}) t=24.0 \mathrm{h}\).
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