To solve the problem of calculating the ratio of concentrations [C]/[A] after one hour for the given reactions, we need to analyze the kinetics of both reactions. We have two reactions involving the same reactant A, converting it into products B and C with different rate constants.
Understanding the Rate Constants
We know that the rate constants for the reactions are related as follows:
- For the reaction A → B, the rate constant is k1 = x per hour.
- For the reaction A → C, the rate constant is k2 = 10x per hour (since k1:k2 = 1:10).
Setting Up the Differential Equations
Since both reactions occur simultaneously, we can express the change in concentration of A over time using the following differential equation:
Let [A] be the concentration of A at time t. The rate of change of [A] can be described by:
d[A]/dt = -k1[A] - k2[A]
Substituting the values of k1 and k2, we get:
d[A]/dt = -x[A] - 10x[A] = -11x[A]
Solving the Differential Equation
This is a first-order linear differential equation. We can separate variables and integrate:
∫(1/[A]) d[A] = -11x ∫dt
Integrating both sides gives us:
ln[A] = -11xt + C
Exponentiating both sides results in:
[A] = e^C * e^(-11xt)
At time t = 0, let’s denote the initial concentration of A as [A]₀. Thus, we can express C as:
[A] = [A]₀ * e^(-11xt)
Calculating Concentrations of B and C
Next, we need to find the concentrations of B and C after one hour (t = 1). The concentration of B can be determined using the integrated rate law for a first-order reaction:
[B] = [A]₀ - [A]
Substituting for [A]:
[B] = [A]₀ - [A]₀ * e^(-11x)
[B] = [A]₀(1 - e^(-11x))
For the concentration of C, we use the same approach:
[C] = [A]₀ - [A] - [B]
[C] = [A]₀ - [A]₀ * e^(-11x) - [A]₀(1 - e^(-11x))
[C] = [A]₀ * e^(-11x)
Finding the Ratio [C]/[A]
Now, we can calculate the ratio of [C] to [A] after one hour:
[C]/[A] = ([A]₀ * e^(-11x)) / ([A]₀ * e^(-11x)) = 1
Thus, after one hour, the ratio of the concentrations of C to A is:
[C]/[A] = 1
Final Thoughts
This result indicates that the concentration of C is equal to the concentration of A after one hour, given the specified rate constants and initial conditions. This type of analysis is fundamental in chemical kinetics, allowing us to predict the behavior of reactants and products over time.
