Question

In the context of diagnosing liver problems, consider a 5.6 mg sample of radioactive gold-198 with a half-life of 2.7 days. Assuming first-order kinetics, determine the remaining mass of the gold-198 after 1.0 day.

Answer 1

After 1.0 day, approximately 4.41 mg of the initial 5.6 mg sample of gold-198 remains, calculated using the formula for exponential decay based on its half-life of 2.7 days.

Step-by-step explanation:

To determine the remaining mass of gold-198 after 1.0 day given an initial mass of 5.6 mg and a half-life of 2.7 days, we use the formula for exponential decay based on first-order kinetics. The calculation involves the use of the following formula:

Remaining mass = Initial mass × (1/2)(time elapsed/half-life)

Substituting the given values:

Remaining mass = 5.6 mg × (1/2)(1.0 day / 2.7 days)

Now, we solve for the exponent:

Exponent = 1.0 day / 2.7 days = 0.37037 (approximately)

Remaining mass = 5.6 mg × (1/2)0.37037

Using a calculator, we find that (1/2)0.37037 is approximately 0.7878, so:

Remaining mass ≈ 5.6 mg × 0.7878

Remaining mass ≈ 4.4117 mg

Therefore, after 1.0 day, approximately 4.41 mg of gold-198 remains.