Potassium ion, K+, is present in most foods and is an essen-tial ...

Question

Potassium ion, K+, is present in most foods and is an essen-tial nutrient in the human body. Potassium-40, however, which has a natural abundance of 0.0117%, is radioactive with t1/2 = 1.25 x 10^9 years. What is the decay constant of 40K? How many 40K+ ions are present in 1.00 g of KCl? How many disintegration/s does 1.00 g of KCl undergo?

Accepted Answer

Step 1: Calculate the decay constant ($ \lambda $) using the half-life formula: $ \lambda = \frac{\ln(2)}{t_{1/2}} $. Substitute the given half-life of Potassium-40, $ t_{1/2} = 1.25 \times 10^9 $ years, into the formula.. Step 2: Determine the number of moles of KCl in 1.00 g. Use the molar mass of KCl (approximately 74.55 g/mol) to convert grams to moles: $ \text{moles of KCl} = \frac{1.00 \text{ g}}{74.55 \text{ g/mol}} $.. Step 3: Calculate the number of K+ ions in 1.00 g of KCl. Since each formula unit of KCl contains one K+ ion, use Avogadro's number ($ 6.022 \times 10^{23} $ ions/mol) to find the total number of K+ ions: $ \text{number of K+ ions} = \text{moles of KCl} \times 6.022 \times 10^{23} $.. Step 4: Determine the number of $ ^{40}K^+ $ ions. Use the natural abundance of Potassium-40 (0.0117%) to find the fraction of K+ ions that are $ ^{40}K^+ $: $ \text{number of } ^{40}K^+ \text{ ions} = \text{number of K+ ions} \times 0.000117 $.. Step 5: Calculate the disintegration rate (activity) in disintegrations per second. Use the decay constant and the number of $ ^{40}K^+ $ ions: $ \text{Activity} = \lambda \times \text{number of } ^{40}K^+ \text{ ions} $.

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Key Concepts

Radioactive Decay and Decay Constant

Molar Mass and Avogadro's Number

Disintegration Rate