How to find Parallel first order reaction for x giving y and z with y having k1 and z having k2 ?

To analyze a parallel first-order reaction where a reactant \( x \) can transform into two products \( y \) and \( z \) with respective rate constants \( k_1 \) and \( k_2 \), we need to set up the rate equations for each pathway. This type of reaction is common in chemical kinetics and can be understood through the principles of reaction rates and concentration changes over time.

Understanding the Reaction Mechanism

In a parallel reaction, the reactant \( x \) can simultaneously convert into two different products. The reactions can be represented as follows:

• For product \( y \): \( x \xrightarrow{k_1} y \)
• For product \( z \): \( x \xrightarrow{k_2} z \)

Here, \( k_1 \) and \( k_2 \) are the rate constants for the formation of \( y \) and \( z \), respectively. The total rate of disappearance of \( x \) is the sum of the rates of the two reactions.

Setting Up the Rate Equations

The rate of change of concentration of \( x \) can be expressed as:

-\frac{d[x]}{dt} = k_1[x] + k_2[x]

This equation indicates that the rate at which \( x \) decreases is proportional to its concentration and the sum of the rate constants. We can simplify this to:

- \frac{d[x]}{dt} = (k_1 + k_2)[x]

Solving the Differential Equation

To find the concentration of \( x \) over time, we can separate variables and integrate. Rearranging gives:

\frac{d[x]}{[x]} = -(k_1 + k_2) dt

Integrating both sides yields:

\ln [x] = -(k_1 + k_2)t + C

Where \( C \) is the integration constant. Exponentiating both sides leads to:

[x] = [x]_0 e^{-(k_1 + k_2)t}

Here, \( [x]_0 \) is the initial concentration of \( x \).

Finding Concentrations of Products

Next, we can determine the concentrations of products \( y \) and \( z \). The change in concentration of \( y \) and \( z \) can be expressed as:

[y] = [x]_0 - [x] = [x]_0 - [x]_0 e^{-(k_1 + k_2)t}

Thus, simplifying gives:

[y] = [x]_0 (1 - e^{-(k_1 + k_2)t})

For product \( z \), we can use the rate constant \( k_2 \) to find its concentration:

[z] = [x]_0 e^{-k_1 t} - [x] = [x]_0 (1 - e^{-k_2 t})

Summary of Results

In summary, for a parallel first-order reaction where \( x \) converts to \( y \) and \( z \), the concentrations over time can be expressed as:

• [x] = [x]_0 e^{-(k_1 + k_2)t}
• [y] = [x]_0 (1 - e^{-(k_1 + k_2)t})
• [z] = [x]_0 (1 - e^{-k_2 t})

This framework allows you to analyze the dynamics of the reaction and predict how the concentrations of reactants and products evolve over time. Understanding these relationships is crucial in fields like chemical engineering, pharmaceuticals, and environmental science, where reaction pathways significantly impact outcomes.