The decomposition reaction, X → products, has rate constant k = 0.00707 mol L−1 s−1 at a particular temperature. How long will it take for the decomposition to be 90% complete, if the initial concentration of X is 1.65 mol L−1 ?
A look at the units will tell you this is a zero order reaction. So
[A] = [A]o - kt
Ao is 1.65 M so A
A is 90% complete so (A) = 1.65*0.1 = ?
You know k; solve for t
To find out how long it will take for the decomposition to be 90% complete, we need to use the concept of reaction kinetics.
The rate of a reaction is typically described by the differential rate law, which relates the rate of the reaction to the concentrations of the reactants. For a first-order reaction like the decomposition reaction in this question, the rate law is:
rate = k * [X]
Where rate is the rate of the reaction, k is the rate constant, and [X] is the concentration of the reactant X.
In this case, the rate constant (k) is given as 0.00707 mol L^(-1) s^(-1) at a particular temperature.
To find the time it will take for the reaction to be 90% complete, we can use the integrated rate law for a first-order reaction:
ln([X]0/[X]) = kt
Where [X]0 is the initial concentration of X, [X] is the concentration of X at time t, k is the rate constant, and t is the time.
We are given that the initial concentration of X is 1.65 mol L^(-1) and we want to know the time it takes for the reaction to be 90% complete, which means [X] will be 0.1 times the initial concentration ([X] = 0.1 * [X]0).
Plugging in the given values, we have:
ln(1.65/(0.1*1.65)) = (0.00707 mol L^(-1) s^(-1)) * t
Simplifying the equation:
ln(10) = (0.00707 mol L^(-1) s^(-1)) * t
Taking the natural logarithm of both sides:
t = ln(10) / (0.00707 mol L^(-1) s^(-1))
Using a calculator, we can find that:
t ≈ 16.0 s
Therefore, it will take approximately 16.0 seconds for the decomposition reaction to be 90% complete.