Question

Derive an expression for the rate constant for a first order reaction.

Answer 1

Hint: Try to recall that first order reactions are those in which rate of reaction at any given time is directly proportional to the concentration or active mass of reactant left at that time. Now, by using this you can easily derive the expression for rate constant.

Complete step by step solution:
* It is known to you that first-order reaction is a reaction that proceeds at a rate that depends linearly on only one reactant concentration.
* Consider the general first order reaction: \[R \to products\]
* Let [R] is the concentration of reactant R and k is the rate constant for the first order reaction. For the first order reaction, the rate of this reaction is directly proportional to the concentration of the reactant R. Thus,
\[Rate = \dfrac{{ - d\left[ R \right]}}{{dt}} = k\left[ R \right]\]
This form of rate law is known as differential rate equation. Now, rearranging the above equation: \[\dfrac{{ - d\left[ R \right]}}{{\left[ R \right]}} = kdt\]
Now, integrating the above equation we get,
\[
   - \int {\dfrac{{d\left[ R \right]}}{{\left[ R \right]}}} = k\int {dt} \\
  or, - \ln \left[ R \right] = kt + I...............2 \\
\]
Where, I is the integration constant. The value of I can be calculated from the initial concentration. Let the initial concentration of R be [R₀] at t=0.
\[
   - \ln \left[ {{R_0}} \right] = k \times 0 \times I \\
  I = - \ln \left[ {{R_0}} \right] \\
\]
Substituting the value of I in equation 2, we get
\[
   - \ln \left[ R \right] = kt - \ln \left[ {{R_0}} \right] \\
  or,\ln \left[ R \right] = - kt + \ln \left[ {{R_0}} \right] \\
   \Rightarrow \ln \left[ {{R_0}} \right] - \ln \left[ R \right] = kt \\
  or\ln \dfrac{{\left[ {{R_0}} \right]}}{{\left[ R \right]}} = kt \\
  or,k = \dfrac{1}{t}\ln \dfrac{{\left[ {{R_0}} \right]}}{{\left[ R \right]}} \\
\].

Note:
* It should be remembered to you that the half life of the first order reaction of a sample is just a constant value. After every half life, the amount of the reactants gets halved.
* Also, you should remember that the rate constant and the half life are inversely proportional, none of which depends on the initial concentration of the sample.