Radioactive Decay: Calculating the Activity of Uranium-238

What is the number of disintegrations per second occur in 1g of Uranium-238?

A. 1.532 x 10⁴ s⁻¹ B. 1.323 x 10⁴ s⁻¹ C. 1.412 x 10⁴ s⁻¹ D. 1.235 x 10⁴ s⁻¹

Answer:

The number of disintegrations per second occurring in 1 gram of Uranium-238 is found to be D. 1.23 x 10^4 disintegrations/second (Bq).

Final answer: The number of disintegrations per second occurring in 1 gram of Uranium-238 is found to be D. 1.23 x 10^4 disintegrations/second (Bq). This calculation involves using Avogadro's number to convert grams of U-238 to atoms, finding the decay constant using the given half-life, and finally finding the activity.

Explanation: The problem is asking for the number of disintegrations per second occurring in 1g of Uranium-238, this is called the activity of the sample. The activity of a radioactive sample is determined by its decay constant (λ) and the number of atoms present (N). Decay constant is related to the half-life of the radioactive substance.

We first need to convert the mass of Uranium-238 (U-238) to the number of atoms using Avogadro's number. U-238 has an atomic mass of 238 g/mol. Therefore, 1 g of U-238 is 1/238 = 4.2 x 10^-3 mol. Using Avogadro's number, this equates to 4.2 x 10^-3 x 6.023x10^23 = 2.5 x 10^21 atoms.

Next, we need to find the decay constant. The decay constant is found using the formula λ = ln(2) / T_half. With T_half of U-238 being 4.5 x 10^9 years, λ = 0.693 / (4.5 x 10^9 x 3.15 x 10^7) = 4.9 x 10^-18 s^-1.

Finally, activity is given by the product of the number of atoms and the decay constant, A = λN. Substituting the values in, A = 4.9 x 10^-18 x 2.5 x 10^21 = 1.23 x 10^4 Bq (Becquerel). Becquerel is the SI unit of radioactivity, equivalent to one disintegration per second. Therefore, the number of disintegration per second in 1g of U-238 is D. 1.23 x 10^4, which corresponds to option D.