The half-life of 238 (92) U undergoing a -decay is 4.5 × 10^9 years. What is the activity of 1g sample of 238 (92) U ?

To determine the activity of a 1 gram sample of Uranium-238 (U-238) undergoing beta decay, we need to understand a few concepts: half-life, activity, and the number of atoms in a given mass of a substance. Let's break this down step by step.

Understanding Half-Life and Activity

The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms in a sample to decay. For U-238, this half-life is approximately 4.5 billion years, or 4.5 × 10⁹ years. The activity of a radioactive sample is a measure of the number of decays per unit time and is typically expressed in becquerels (Bq), where 1 Bq equals 1 decay per second.

Calculating the Number of Atoms in 1 gram of U-238

First, we need to find out how many U-238 atoms are in 1 gram. The molar mass of U-238 is about 238 grams per mole. Using Avogadro's number (approximately 6.022 × 10²³ atoms/mole), we can calculate the number of atoms as follows:

• Number of moles in 1 gram of U-238 = mass (g) / molar mass (g/mole)
• Number of moles = 1 g / 238 g/mole ≈ 0.00420 moles
• Number of atoms = number of moles × Avogadro's number
• Number of atoms ≈ 0.00420 moles × 6.022 × 10²³ atoms/mole ≈ 2.53 × 10²¹ atoms

Calculating the Decay Constant

The decay constant (λ) can be derived from the half-life using the formula:

λ = ln(2) / half-life

Substituting the half-life of U-238:

• λ = 0.693 / (4.5 × 10⁹ years) ≈ 1.54 × 10^-10 years^-1

Finding the Activity of the Sample

Now, we can calculate the activity (A) using the formula:

A = λN

Where N is the number of atoms we calculated earlier:

• A = (1.54 × 10^-10 years^-1) × (2.53 × 10²¹ atoms)
• Converting years^-1 to seconds^-1 (1 year = 3.15 × 10⁷ seconds):
• λ = 1.54 × 10^-10 years^-1 × (1 / 3.15 × 10⁷) ≈ 4.88 × 10^-18 seconds^-1
• A ≈ (4.88 × 10^-18 seconds^-1) × (2.53 × 10²¹ atoms) ≈ 12.34 Bq

Final Result

Therefore, the activity of a 1 gram sample of Uranium-238 is approximately 12.34 becquerels. This means that, on average, about 12 decays occur every second in that 1 gram of U-238.