A decomposition reaction has a rate constant of 0.0019 yr -1.
(
a) What is the half-life of the reaction?
______ yr
(b) How long does it take for [reactant] to reach 12.5% of its original value?
______ yr
a)
k = 0.693/t1/2
Substitute and solve for t1/2
b)
ln(No/N) = kt
If you let No = 100, then N = 12.5; substitute k and solve for t in years.
.0013167
(a) Well, decomposition reactions can be quite messy, but let's solve this one together! Remember, the half-life of a reaction is the time it takes for half of the reactant to decompose. So, we can use the formula:
t(1/2) = ln(2) / k
Here, k is the rate constant, given as 0.0019 yr -1. Let's plug that in:
t(1/2) = ln(2) / 0.0019
t(1/2) ≈ 364.05 years
So, the half-life of the reaction is approximately 364.05 years.
(b) Ah, so you want to know when the reactant reaches 12.5% of its original value. Clever question! To find the time it takes, let's figure out the proportion of reactant remaining after a certain period:
[R]t = [R]0 * e^(-kt)
Here, [R]t is the concentration of reactant at time t, [R]0 is the initial concentration, and k is the rate constant. Considering [R]t as 12.5% of [R]0, we can rewrite the equation as:
0.125[R]0 = [R]0 * e^(-kt)
Now, we need to solve for t. Let's start:
0.125 = e^(-kt)
To isolate t, we can take the natural logarithm of both sides:
ln(0.125) = -kt
Now, let's rearrange to solve for t:
t = -ln(0.125) / k
Plugging in the given rate constant, we have:
t ≈ -ln(0.125) / 0.0019
t ≈ 717.75 years
So, it would take approximately 717.75 years for the reactant to reach 12.5% of its original value. Good luck keeping track of those decimal places though!
To find the half-life of a reaction, we need to use the formula:
t₁/₂ = (0.693 / k)
where t₁/₂ is the half-life and k is the rate constant.
(a) Let's substitute the given rate constant into the formula:
t₁/₂ = (0.693 / 0.0019)
Calculating this expression, we find:
t₁/₂ ≈ 364.74 years
Therefore, the half-life of the reaction is approximately 364.74 years.
(b) To find the time it takes for the reactant to reach 12.5% of its original value, we can use the formula for the concentration of a reactant in a first-order reaction:
[reactant] = [reactant]₀ * e^(-kt)
where [reactant] is the concentration at a given time, [reactant]₀ is the original concentration, k is the rate constant, and t is the time.
We want to find the time when [reactant] = 12.5% of [reactant]₀, which can be written as:
0.125 = e^(-kt)
Taking the natural logarithm of both sides:
ln(0.125) = -kt
Solving for t:
t = ln(0.125) / -k
Substituting the given rate constant into the formula:
t = ln(0.125) / -0.0019
Evaluating this expression gives:
t ≈ 1355.62 years
Therefore, it takes approximately 1355.62 years for the reactant to reach 12.5% of its original value.