What is the number of disintegrations per second that occur in 1g of Uranium-238, given its half-life and Avogadro's number?
The activity (decays per second) of 1g of Uranium-238, given its half-life and Avogadro's number, is closest to option D (1.235 x 10⁴ s⁻¹). This is calculated by finding the number of uranium nuclei present, the decay constant given the half-life, and finally calculating activity.
Explanation:
To calculate the disintegration per second (activity) of ²³⁸ U, we should first convert the mass of the ²³⁸ U into moles. Given that the molar mass of ²³⁸ U is about 238 g/mol, one gram is approximately 0.0042 mol. Utilizing Avogadro's number, we know that there are 6.022 x 10²³ nuclei per mole, so 0.0042 mol corresponds to roughly 2.53 x 10²¹ uranium nuclei.
Given a half-life of 4.5 x 10⁹ years, the decay constant (λ) can be calculated using the equation λ = ln(2) / t₁/₂, resulting in a λ of about 0.693 / (4.5 x 10⁹ x 3.1536x10⁷) s⁻¹ (since we convert the half-life from years to seconds).
Uranium-238 decays via alpha decay, meaning that each decay event results in one atom of ²³⁸ U decaying to form one atom of ²³⁴ Th. The activity can thus be calculated using the formula A = λN, where N is the number of ²³⁸ U nuclei. This yields an activity rate of approximately 1.238 x 10⁴ decays per second. Therefore, the closest option to this calculation appears to be D.