The first order rate constant for the decomposition of a certain antibiotic in water at 25 degrees Celsius is 1.30 years^-1.
(1). If a 5.0 * 10^-3 moldm^-3 solution of this antibiotic is stored at 25 degrees Celsius for 30 days, what is the concentration of the antibiotic?
(2). How long will it take for the concentration of the antibiotic to reach 1.0 * 10^-3moldm^-3?
(1). To determine the concentration of the antibiotic after 30 days, we can use the first-order rate equation:
ln(Cf/Ci) = -kt
Where Cf is the final concentration, Ci is the initial concentration, k is the rate constant, and t is the time.
We can rearrange the equation to solve for Cf:
Cf = Ci * e^(-kt)
In this case, Ci = 5.0 * 10^-3 moldm^-3, k = 1.30 years^-1, and t = 30 days.
Converting 30 days to years:
t = 30 days * (1 year/365 days) ≈ 0.082 years
Plugging in the values:
Cf = (5.0 * 10^-3 moldm^-3) * e^(-1.30 years^-1 * 0.082 years)
Cf ≈ 3.11 * 10^-3 moldm^-3
Therefore, the concentration of the antibiotic after 30 days is approximately 3.11 * 10^-3 moldm^-3.
(2). To determine the time required for the concentration of the antibiotic to reach 1.0 * 10^-3 moldm^-3, we can rearrange the first-order rate equation:
ln(Cf/Ci) = -kt
In this case, Cf = 1.0 * 10^-3 moldm^-3, Ci = 5.0 * 10^-3 moldm^-3, and k = 1.30 years^-1. We want to solve for t.
Plugging in the values:
ln((1.0 * 10^-3 moldm^-3)/(5.0 * 10^-3 moldm^-3)) = -1.30 years^-1 * t
Simplifying:
ln(0.2) = -1.30 years^-1 * t
Solving for t:
t = ln(0.2)/(-1.30 years^-1)
t ≈ 2.78 years
Therefore, it will take approximately 2.78 years for the concentration of the antibiotic to reach 1.0 * 10^-3 moldm^-3.