Final answer:
To find the time for 99% decay of Po-210 following first-order kinetics with a rate constant of 1.833 yr⁻¹, we use the first-order decay equation and solve for time after calculating the natural logarithm of the remaining fraction of the substance.
Step-by-step explanation:
The decay of polonium-210 (Po-210) follows first-order kinetics, and we can use the first-order rate equation to calculate the time required for a given percentage of decay. Since the rate constant (k) is given as 1.833 yr⁻¹, and we are looking for the time (t) required for 99% of the polonium to decay, we will use the equation:
ln(N_t/N_0) = -kt
where:
- N_t is the remaining amount of substance at time t
- N_0 is the initial amount of substance
- k is the rate constant
- t is the time
For a 99% decay, 1% of the substance remains, so N_t/N_0 is 0.01. We plug the values into the equation to solve for t:
ln(0.01) = -1.833 yr⁻¹ * t
t = ln(0.01) / -1.833 yr⁻¹
After calculating, we find the time t required for 99% decay of Po-210.
It's important to be aware that first-order decay is characterized by a rate of disintegration that is directly proportional to the amount of the substance present at any given time.