At a given temperature, a first-order reaction has a rate constant of 3.5 ´ 10–3 s–1. How long will it take for the reaction to be 24% complete?
A chemical reaction that is first-order in X is observed to have a rate constant of 2.20 ´ 10–2 s–1. If the initial concentration of X is 1.0 M, what is the concentration of X after 186 s?
ln(No/N) = kt
No = 100
N = 100-24 = 76
k from the problem
Solve for t in seconds.
#2
Same equation. Different substitutions and solve for N
Post your work if you get stuck.
Why did the math book go to the therapist?
Because it had too many problems, just like these questions!
But I'm here to help, so let's calculate the answers:
Question 1: How long will it take for the reaction to be 24% complete?
Well, since it's a first-order reaction, we can use the formula:
ln(A/A₀) = -kt
Where A is the final concentration, A₀ is the initial concentration, k is the rate constant, and t is the time.
Given that A is 24% of A₀ (which means A = 0.24A₀) and k = 3.5 ´ 10–3 s–1, we can rearrange the equation and solve for t:
ln(0.24A₀/A₀) = -3.5 ´ 10–3 s–1 * t
Simplifying,
ln(0.24) = -3.5 ´ 10–3 s–1 * t
Now, it's just a matter of solving this equation for t.
Question 2: What is the concentration of X after 186 s?
Again, using the first-order reaction formula:
ln(X/X₀) = -kt
Given that X₀ is 1.0 M and k is 2.20 ´ 10–2 s–1, we can rearrange the equation and solve for X:
ln(X/1.0 M) = -2.20 ´ 10–2 s–1 * 186 s
Simplifying,
ln(X/1.0) = -2.20 ´ 10–2 * 186
Now, we can solve this equation for X.
I hope you enjoyed this mathematical comedy break! Now, let's get back to solving the problems.
To solve both of these questions, we need to understand the concepts of first-order reactions and how to calculate reaction rate and concentration over time.
A first-order reaction is a type of reaction where the rate of the reaction is directly proportional to the concentration of a single reactant. The rate constant (k) represents the proportionality constant between the rate of the reaction and the concentration of the reactant.
1) For the first question:
Given:
Rate constant (k) = 3.5 ´ 10–3 s–1
Percentage of completion = 24%
To find the time it will take for the reaction to be 24% complete, we can use the formula for a first-order reaction:
ln([A]t/[A]0) = -kt
where:
[A]t is the concentration of the reactant at time t
[A]0 is the initial concentration of the reactant
k is the rate constant
In this case, we can assume [A]t/[A]0 = 0.24, as the reaction is 24% complete. Plugging in the values, we get:
ln(0.24) = -(3.5 ´ 10–3 s–1)t
Now, solve for t by isolating it:
t = -(ln(0.24)) / (3.5 ´ 10–3 s–1)
Calculating this expression will give us the time it will take for the reaction to be 24% complete.
2) For the second question:
Given:
Rate constant (k) = 2.20 ´ 10–2 s–1
Initial concentration ([X]0) = 1.0 M
Time (t) = 186 s
We can use the first-order reaction equation to determine the concentration of X after 186 s:
[X]t = [X]0 ´ e^(-kt)
where:
[X]t is the concentration of X at time t
[X]0 is the initial concentration of X
k is the rate constant
t is the time
Substituting the given values into the equation:
[X]t = (1.0 M) ´ e^(-(2.20 ´ 10–2 s–1) ´ (186 s))
Calculating this expression will give us the concentration of X after 186 s.
23hr
In[100/76] = 3.5x10^-3 . t
thus T = 7.84x10^-5