Question
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Which of the following reactions is correct for a first order reaction? (k = rate constant; r = rate of reaction; c = concentration of reaction)
(a)- $k=r\text{ x }{{\text{c}}^{2}}$
(b)- $k=\text{ r x c}$
(c)- $k=\dfrac{c}{r}$
(d)- $k=\dfrac{r}{c}$

Answer
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Hint: The chemical reactions are classified on the basis of the dependency on the number of reactants in the chemical reaction. If the rate of reaction depends only on the concentration of one reactant, then it is a first-order reaction.

Complete answer:
The chemical reactions are classified on the basis of the dependency on the number of reactants in the chemical reaction. This is measured by the order of the reaction and the rate of the reaction tells the change in concentration of the reactant or product. These are classified as:
If the order of the reaction is zero then the rate of reaction doesn’t depend on any concentration of the reactant.
If the order of the reaction is first then the rate of reaction depends only on the concentration of one reactant.
If the rate of reaction is second then the rate of reaction depends on the concentration of two reactants. And so on.
Suppose a reaction:
$A\to Products$
So in this reaction, the rate of reaction for first order reaction will be:
$\text{Rate of reaction = k }\!\![\!\!\text{ A }\!\!]\!\!\text{ }$
So given in the question, k = rate constant; r = rate of reaction; c = concentration of reaction, so the above equation will be:
$\text{r = k x c}$
$k=\dfrac{r}{c}$
So the rate constant is the ratio of the rate of reaction to the concentration of the reactant.

Therefore, the correct answer is an option (d)- $k=\dfrac{r}{c}$.

Note:
We can calculate the rate constant for the first-order reaction by the formula $k=\dfrac{2.303}{t}\log \dfrac{{{[A]}_{o}}}{[A]}$ where t is the time taken, ${{[A]}_{o}}$ is the initial concentration of the reactant, and $[A]$ is the final concentration of the reactant.