In a first-order decomposition reaction, 50.0% of a compound decomposes in 19.5 min.
(a) What is the rate constant of the reaction? (round to four decimal places)
k = .03554 min−1
(b) How long does it take for 82.0% of the compound to decompose? (round to one decimal place)
t = ? min
In a first-order decomposition reaction, 50.0% of a compound decomposes in 19.5 min.
(a) What is the rate constant of the reaction? (round to four decimal places)
k = .03554 min−1
That's right but didn't you round to 5 decimal places? four significant figures.
(b) How long does it take for 82.0% of the compound to decompose? (round to one decimal place)
t = ? min
ln(No/N) = kt
No = 100% to start
N = 18% (100% - 82% = 18%)
k = from above
Solve for t in minutes. Post your work if you get stuck.
To answer (b), we need to use the first-order reaction equation:
ln([A]t/[A]0) = -kt
Where:
- [A]t is the concentration of the compound at time t
- [A]0 is the initial concentration of the compound
- k is the rate constant
- t is the time
Since we want to find the time it takes for 82% of the compound to decompose, we can set [A]t as 0.82[A]0.
ln(0.82[A]0/[A]0) = -kt
Simplifying the equation:
ln(0.82) = -kt
We know the rate constant (k) is 0.03554 min^(-1) (from part (a)), so we can rearrange the equation to solve for t:
t = ln(0.82) / k
Calculating the value:
t = ln(0.82) / 0.03554
t ≈ 11.8 min
Therefore, it takes approximately 11.8 minutes for 82.0% of the compound to decompose.
To find the rate constant of a first-order decomposition reaction, we can use the equation:
ln([A]t/[A]0) = -kt
where [A]t is the concentration of the compound at time t, [A]0 is the initial concentration of the compound, k is the rate constant, and t is the time.
(a) We are given that 50.0% of the compound decomposes in 19.5 min. This means that [A]t/[A]0 = 0.5. Plugging these values into the equation, we have:
ln(0.5) = -k * 19.5
Rearranging the equation to solve for the rate constant k, we have:
k = -ln(0.5) / 19.5
Evaluating this expression, we find that k ≈ 0.03554 min^−1 (round to four decimal places).
(b) To find how long it takes for 82.0% of the compound to decompose, we need to solve for time (t) in the same equation:
ln([A]t/[A]0) = -kt
We can rearrange this equation to solve for t:
t = -ln([A]t/[A]0) / k
Plugging in [A]t/[A]0 = 0.82 (82.0% decomposed) and the value of k we found earlier, we have:
t = -ln(0.82) / 0.03554
Evaluating this expression, we find that t ≈ 35.3 min (round to one decimal place).