A certain first-order reaction A to B is 25% complete in 42 min at 25 ° C. What is the half-life of the reaction?
ln No/N = kt
ln 100/75 = k*42. Solve for k. Post your work if you get stuck.
To find the half-life of a first-order reaction, we can use the formula:
t₁/₂ = 0.693 / k
where t₁/₂ represents the half-life, and k is the rate constant for the reaction.
First, we need to determine the rate constant (k) using the information given. We know that the reaction is 25% complete in 42 minutes, which means the concentration of A has decreased to 75% of its initial value.
The fraction of A remaining after time t is given by:
[A] / [A₀] = 1 - 0.25 = 0.75
To find the rate constant (k), we can use the integrated rate law for a first-order reaction:
ln([A] / [A₀]) = -kt
Substituting the values we have:
ln(0.75) = -k * 42 min
Now, we can solve for k:
k = -ln(0.75) / 42 min
Calculating this expression gives us the value of k.
Finally, we can substitute the calculated value of k into the formula for the half-life (t₁/₂ = 0.693 / k) to find the half-life of the reaction.
To find the half-life of the reaction, we can use the concept of the reaction being 25% complete in 42 minutes.
The rate of a first-order reaction is given by the equation:
Rate = k[A]
Where:
- k is the rate constant
- [A] is the concentration of reactant A
Since the reaction is first-order, the rate of the reaction is proportional to the concentration of reactant A.
We know that the reaction is 25% complete, which means that the concentration of A has decreased to 75% of its initial concentration (100% - 25% = 75%).
Using the equation for a first-order reaction, we can set up the following relationship:
[At/ A0] = e^(-kt)
Where:
- [At] is the concentration of A at time t
- [A0] is the initial concentration of A
- k is the rate constant
- t is the time at which we want to determine the concentration of A
Substituting the values we know:
[At/ A0] = 0.75
t = 42 minutes
We can rearrange the equation to solve for the rate constant (k):
0.75 = e^(-42k)
Now, we can solve for k:
ln(0.75) = -42k
Finally, we can use the rate constant (k) to calculate the half-life (t1/2) using the equation:
t1/2 = (0.693 / k)
Let's plug in the values and calculate the half-life: