The half-life of protactinium-231 is 3.25 × 10 ⁴ years. How much time will 10.4 g of Pa-231 take to decay into 2.6 g?

Question

Quizplus · Accepted Answer

To solve the problem, we need to use the formula for radioactive decay:

N = N0e^(-λt)

Where:
N0 = initial amount (in g)
N = final amount (in g)
λ = decay constant (ln2/T1/2, where T1/2 is the half-life)
t = time (in years)

We can use the ratio of the initial and final amounts to find the fraction remaining:

N/N0 = (1/2)^(t/T1/2)

(2.6/10.4) = (1/2)^(t/3.25e14)

(1/4) = (1/2)^(t/3.25e14)

Taking the logarithm of both sides:

log(1/4) = (t/3.25e14) log(1/2)

-2 = (-0.693t/3.25e14)

t = (-2)(3.25e14)/(-0.693) = 6.5e14 seconds

Converting seconds to years:

T = 6.5e14 s ÷ (60 s/min)(60 min/hr)(24 hr/day)(365 day/yr) = 2.06 × 10^7 years

Rounding to the appropriate number of significant figures gives:

Answer: D) 6.5 × 10^14 years

The Half-Life of Protactinium-231 Is 3