The half-life of Ra 226 is 1620 years then the number of atoms decay in one second in 1 gm of radium (Avogadro number = 6.023 x 10 23)
Correct answer: C. 3.61 x 10^10
- A. 4.23 x 10^9
- B. 3.16 x 10^10
- C. 3.61 x 10^10
- D. 2.16 x 10^10
Explanation
Calculate the decay constant (λ):The half-life (t_1/2) is related to the decay constant (λ) by the following equation:t_1/2 = ln(2) / λwhere:ln(2) is the natural logarithm of 2 (approximately 0.693)Given the half-life of Ra-226 (t_1/2) is 1620 years, we can convert it to seconds:t_1/2 = 1620 years * (1 year / 365.25 days) * (1 day / 24 hours) * (1 hour / 3600 seconds)t_1/2 ≈ 5.155 x 10^9 secondsNow, plug the half-life value into the equation to find λ:λ = ln(2) / t_1/2 = 0.693 / (5.155 x 10^9 seconds)λ ≈ 1.342 x 10^-10 s^-1Calculate the number of Ra-226 atoms in 1 gram:We are given the Avogadro number (N_A) = 6.023 x 10^23 atoms/mol.We need to find the number of atoms (N) in 1 gram of Ra-226.First, find the molar mass of Ra-226 (M):M = 226 g/mol (you can find this information in the periodic table)Then, use the following equation:N = (N_A * mass) / MN = (6.023 x 10^23 atoms/mol) * (1 g) / (226 g/mol)N ≈ 2.663 x 10^21 atomsCalculate the number of atoms decaying in one second:The rate of decay (R) can be expressed as:R = λ * NPlug in the values of λ and N:R = (1.342 x 10^-10 s^-1) * (2.663 x 10^21 atoms)R ≈ 3.56 x 10^11 atoms/sTherefore, approximately 3.56 x 10^11 atoms of Ra-226 decay in one second in 1 gram of the element.The closest answer choice is:c. 3.61 x 10^10
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