It takes 42.0 min for the concentration of a reactant in a first-order reaction to drop from 0.45 M to 0.32 M at 25C. How long will it take for the reaction to be 90% complete?
For a first order reaction
ln (No/N) = kt
ln (0.45/0.32) = k(42min)
Solve for k, then
ln (No/N) = kt
ln (100/90) = k from above*t
Solve for t. Post your work if you get stuck.
To determine how long it will take for the reaction to be 90% complete, we need to find the reaction time at which the concentration of the reactant drops to 10% of its initial value.
First, we calculate the reaction rate constant (k) using the first-order integrated rate equation:
ln([A]t/[A]0) = -kt [Equation 1]
Where:
[A]t is the concentration at time t
[A]0 is the initial concentration at t = 0
k is the reaction rate constant
t is the time
We can rearrange Equation 1 to solve for k:
k = -ln([A]t/[A]0) / t
Now, we substitute the given values:
[A]t = 0.32 M
[A]0 = 0.45 M
t = 42.0 min
k = -ln(0.32/0.45) / 42.0
Next, we can use the rate constant (k) to calculate the time it takes for the reaction to be 90% complete.
Let's call this time t90.
To do this, we can rearrange Equation 1 and solve for t90:
ln([A]t90/[A]0) = -k * t90
Now we substitute the given values:
[A]t90 = 0.10 * [A]0 (since the reaction is 90% complete and the concentration has dropped to 10% of the initial value)
[A]0 = 0.45 M
k = -ln(0.32/0.45) / 42.0
ln((0.10 * 0.45) / 0.45) = -(-ln(0.32/0.45) / 42.0) * t90
ln(0.10) = ln(0.32/0.45) / 42.0 * t90
Now, we can solve for t90:
t90 = ln(0.10) / ((-ln(0.32/0.45)) / 42.0)
t90 = 152.9 min (rounded to the nearest tenth)
Therefore, it will take approximately 152.9 minutes for the reaction to be 90% complete.
To determine how long it will take for the reaction to be 90% complete, we need to use the concept of half-life in first-order reactions.
The half-life (t1/2) of a first-order reaction can be calculated using the equation:
t1/2 = (0.693 / k)
Where 'k' is the rate constant for the reaction.
Given that it takes 42.0 min for the concentration to drop from 0.45 M to 0.32 M, we can determine the value of 'k' using the following equation:
ln(0.32 M / 0.45 M) = -k * 42.0 min
To find 'k', we need to rearrange this equation:
k = -ln(0.32 M / 0.45 M) / 42.0 min
Calculating this expression gives us the value of 'k'. Next, we can use this rate constant to determine the half-life of the reaction using the formula:
t1/2 = (0.693 / k)
With the calculated half-life, we can find the time required for the reaction to be 90% complete. From the given information, we know that the initial concentration is 0.45M, and to reach 90% completion, the concentration should be 0.45 M * (1 - 0.90) = 0.045M.
Now, we can use the half-life to determine the time for the reaction to be 90% complete.
Let me calculate all these values for you.