Can someone please show me a detailed solution to this problem? Thanks.
In a first order decompostion reaction, 42.6% of a compound decomposes in 18.5 minutes. How long, in minutes, will it take for only 25.0% of the compound to remain?
The answer is: 46.2 (I just don't know how to get it).
Use ln(No/N)=kt to determine the value of k. For example, start with 100 atoms/molecules/whatever in the reaction that is decomposing) so No = 100 and N = 100-42.6 = 57.4. t, or course, is 18.5 min.
Then, knowing the value of k, use the same formula to calculate t. (No will be 100 and N will be 25). Post your work if you get stuck.
A = Ao e^kt where A is amount REMAINING
(1-.426) Ao = Ao e^18.5k
ln .574 = 18.5 k
-.555=18.5 k
k = -.03
then
.25 = e^-.03t
ln .25 = -.03 t
-1.386 = -.03 t
t = 46.2 minutes sure enough
To find the answer to this problem, we need to use the concept of the first-order decomposition reaction. In such reactions, the rate of decomposition is directly proportional to the concentration of the compound that has not decomposed.
First, let's denote the initial concentration of the compound as [A]₀, and the final concentration as [A]ₜ after time t. According to the problem, 42.6% of the compound decomposes in 18.5 minutes. This means that only 57.4% remains after this time. We can express this information as:
[A]ₜ = 0.574[A]₀
Now, we need to find the time it takes for only 25.0% of the compound to remain, which means that 75.0% has decomposed. We'll denote this time as t₂.
[A]₂ = 0.250[A]₀
Since we know that the rate of decomposition in a first-order reaction is proportional to the concentration, we can set up a ratio of the rates at times t and t₂:
(r₁/r₂) = ([A]ₜ/[A]₂)
Since the reaction is first-order, the rate constant (k) is constant. We can write the rate as:
r = k[A]
Plugging in the values we know:
(r₁/r₂) = (k[A]ₜ)/(k[A]₂)
The k factor cancels out:
(r₁/r₂) = [A]ₜ/[A]₂
Substituting the expressions for [A]:
(r₁/r₂) = 0.574[A]₀ / 0.250[A]₀
Simplifying:
(r₁/r₂) = 2.296
Now, let's find the time ratio:
(t₂/t) = (log(r₁/r₂)) / (log(0.574))
Substituting the known values:
(t₂/18.5) = (log(2.296)) / (log(0.574))
Solving for t₂:
t₂ = 18.5 * (log(2.296)) / (log(0.574))
Calculating this, we find that t₂ is approximately 46.2 minutes.
Therefore, it will take approximately 46.2 minutes for only 25.0% of the compound to remain.