Question

What are the units of rate constant for the first order reaction?

Answer 1

We know that rate of a reaction is explained as change in concentration of any of the reactants or product per unit time. For example:
${\text{aA + bB}} \to {\text{cC + dD}}$
Rate of reaction is equal to decrease in concentration of either A or B or increases in concentration of C or D per unit time. In the above reaction a, b, c, and d are stoichiometric coefficients. Thus rate for the above general reaction is equal to rate of removal of A or B per mole and also equal to rate appearance of C or D per mole. It can be shown is as follows:
${\text{rate}} = - \dfrac{{\text{1}}}{{\text{a}}}\dfrac{{{\text{d}}\left[ {\text{A}} \right]}}{{{\text{dt}}}} = - \dfrac{{\text{1}}}{{\text{b}}}\dfrac{{{\text{d}}\left[ {\text{B}} \right]}}{{{\text{dt}}}} \\\ \,\,\,\,\,\,\,\,\, = + \dfrac{{\text{1}}}{{\text{c}}}\dfrac{{{\text{d}}\left[ {\text{C}} \right]}}{{{\text{dt}}}}{\text{ = + }}\dfrac{{\text{1}}}{{\text{d}}}\dfrac{{{\text{d}}\left[ {\text{D}} \right]}}{{{\text{dt}}}} \\\$
Here [ ] represents concentration in mole per liter and ‘d’ represents an infinitesimally small change in concentration. Negative sign shows that concentration of reactant A and B are decreasing whereas positive sign shows concentration of product C and D are increasing.

Complete step by step answer:
As we know a first order reaction is represented as
${\text{A}} \to \,\,\,\,\,\,\,{\text{Product}}$
Initial concentration: $a$ $0$
Concentration after time: $\left( {a - x} \right)$ $x$
Now we put differential rate law:
$- \dfrac{{d\left( {a - x} \right)}}{{dt}} = + \dfrac{{dx}}{{dt}} = {k_1}\left( {a - x} \right)$
On integrating above equation we get
Integration rate law:
${k_1} = \dfrac{{2.303}}{t}\log \left( {\dfrac{a}{{a - x}}} \right)$
Here ${k_1}$ is the rate constant of a first order reaction unit that is per time or ${{\text{s}}^{ - 1}}$ since rest of the expression consists of numerical values and log does not have any units.
Thus, the unit of first order reaction is ${{\text{s}}^{ - 1}}$ .

Note:
Rate of the reaction is proportional to the product of concentration of reactants, each raised to some power. i.e.
${\text{rate}} \propto {\left[ {\text{A}} \right]^{\text{m}}}{\left[ {\text{B}} \right]^{\text{n}}} \\\ {\text{rate = k}}{\left[ {\text{A}} \right]^{\text{m}}}{\left[ {\text{B}} \right]^{\text{n}}} \\\$
Here k is rate constant. At all concentrations, rate constant k is equal to rate of a reaction.