Question

What is $k$ in the first order reaction?

Answer 1

The order of a reaction (different from molecularity) is the power placed on the concentration term. This number indicates the dependence of a reactant’s concentration on the rate at which the reaction takes place.

Complete answer: The reactions kinetics of reaction dynamics deals with the rates at which the reactions take place. It suggests how fast the reactants are being consumed or transformed into the products.
Let us assume a reaction in which a single reactant in being converted to a single product
$A \to B$
Where $A$ is a reactant and $B$ is a product.
Let us assume that this reaction follows the first order kinetics. This means that the concentration of the reactant at any point during the reaction is directly proportional to the rate at which the reaction takes place. The rate law can be written as follows:
$rate = k \times {[A]^1}$
Where $k$ represents the rate constant for the given first order reaction and $[A]$ represents the concentration of the reactant. It has been raised to the power one, as it is a first order reaction.
The rate of the reaction ca be expressed as the rate at which the concentration of the reactant gets consumed,
$rate = - \dfrac{{d[A]}}{{dt}}$
On equating the two rate expressions we get,
$- \dfrac{{d[A]}}{{dt}} = k \times [A]$
Which can rewritten as,
$\dfrac{{d[A]}}{{[A]}} = - kdt$
On integrating both sides, with the limits of concentration from ${[A]_0}$ which stands for initial concentration to $[A]$ which represent final concentration.
And the time goes from zero to $t$
$\int\limits_{{{[A]}_0}}^{[A]} {\dfrac{{d[A]}}{{[A]}} = } - k\int\limits_0^t {dt}$
$\ln ([A] - {[A]_0}) = - k(t - 0)$
On writing the above expression in terms of log base $10$, and rearranging the expression we get,
$k = \dfrac{{2.303}}{t}\log \dfrac{{{{[A]}_0}}}{{[A]}}$
Thus the rate constant in a first order reaction is $k = \dfrac{{2.303}}{t}\log \dfrac{{{{[A]}_0}}}{{[A]}}$

Note:
The exact value of rate constant $k$ can only be determined by knowing the initial concentration of the reactant and the concentration present at a particular time. The formula derived here only gives a relationship between the variables needed to study the kinetics of a reaction.