Cyclopropane rearranges to form propene:
CH2CH2CH2 --> CH2=CHCH3
by first-order kinetics. The rate constant is k = 2.74 x 10-3 s-1. The initial concentration of cyclopropane is 0.290 M. What will be the concentration of cyclopropane after 100 seconds?
To solve this problem, we can use the first-order rate equation:
ln([A]t/[A]0) = -kt
Where:
[A]t is the concentration of cyclopropane at time t
[A]0 is the initial concentration of cyclopropane
k is the rate constant
t is the time
We can rearrange the equation to solve for [A]t:
[A]t = [A]0 * e^(-kt)
Substituting the given values:
[A]0 = 0.290 M
k = 2.74 x 10^(-3) s^(-1)
t = 100 s
[A]t = 0.290 * e^(-2.74 x 10^(-3) * 100)
Calculating this expression:
[A]t ≈ 0.290 * e^(-0.274)
[A]t ≈ 0.290 * 0.760
[A]t ≈ 0.220 M
Therefore, the concentration of cyclopropane after 100 seconds will be approximately 0.220 M.
To determine the concentration of cyclopropane after a certain amount of time, we can use the first-order rate equation:
ln([Cyclopropane]t/[Cyclopropane]0) = -kt
Where:
[Cyclopropane]t = concentration of cyclopropane at time t
[Cyclopropane]0 = initial concentration of cyclopropane
k = rate constant
t = time
In this case, we are given the rate constant (k = 2.74 x 10-3 s-1), the initial concentration of cyclopropane ([Cyclopropane]0 = 0.290 M), and the time (t = 100 seconds). We can plug these values into the equation to find the concentration of cyclopropane after 100 seconds.
ln([Cyclopropane]100/0.290) = -(2.74 x 10-3 s-1)(100 s)
To solve for [Cyclopropane]100, we need to isolate it. First, we can simplify the equation by multiplying both sides by -1 to get rid of the negative sign:
-ln([Cyclopropane]100/0.290) = (2.74 x 10-3 s-1)(100 s)
Next, we take the exponential of both sides to eliminate the natural logarithm:
e^[-ln([Cyclopropane]100/0.290)] = e^[(2.74 x 10-3 s-1)(100 s)]
The exponential of a natural logarithm results in the original value inside the logarithm:
[Cyclopropane]100/0.290 = e^[(2.74 x 10-3 s-1)(100 s)]
Finally, we solve for [Cyclopropane]100 by multiplying both sides by 0.290:
[Cyclopropane]100 = 0.290 * e^[(2.74 x 10-3 s-1)(100 s)]
Calculating the right side of the equation will give us the concentration of cyclopropane after 100 seconds.
ln(No/N) = kt
No = 0.290
N = ?
k = given
t = given