Question

Point out the wrong statement: for a first order reaction:
A. Time for half-change $\left ({{{\text{t}}_{1/2}}} \right)$ is independent of initial concentration
B. Change in the concentration unit does not change the rate constant (K)
C. Time for half-change $ \times $ rate constant =$0.693$
D. The unit of K is ${\text{mol}}{{\text{e}}^{ - 1}}{\min ^{ - 1}}$

Answer 1

Hint:The first-order reaction is the reaction in which the rate of reaction is directly proportional to the concentration of the reactant. The half-life of the first-order reaction is inversely proportional to the rate constant.

Complete step by step answer:
When zinc is burned in the presence of air, woolly tufts form which is known as ‘Lana philosophica’ in As we known the first-order rate constant formula is,
${\text{k}}\,\,{\text{ = }}\,\dfrac{{{\text{2}}{\text{.303}}}}{{\text{t}}}{\text{log}}\dfrac{{{{\text{A}}_{\text{o}}}}}{{{{\text{A}}_{\text{x}}}}}$
We can multiply the expression from $2.303$ to convert ln into log.
${\text{k}}\,\,{\text{ = }}\,\dfrac{{{\text{2}}{\text{.303}}}}{{\text{t}}}{\text{log}}\dfrac{{{{\text{A}}_{\text{o}}}}}{{{{\text{A}}_{\text{x}}}}}$
Where,
${\text{k}}$ is the first-order rate constant. Unit of first order rate constant is ${\text{tim}}{{\text{e}}^{ - 1}}$.
${\text{t}}$ is the time.
${{\text{A}}_{\text{o}}}$ is the initial concentration of the reactant.
${{\text{A}}_{\text{x}}}$ is the concentration of the reactant left at time ${\text{t}}$.
Half-life is the time at which the concentration of the reactant becomes half of the initial
 concentration. So, if the initial concentration is $1$ at half-life the concentration will be $1/2$.
The first-order half-life formula is as follows:
\[\,\,{\text{k = }}\,\dfrac{{{\text{2}}{\text{.303}}}}{{{{\text{t}}_{{\text{1/2}}}}}}{\text{log}}\,\frac{{\text{1}}}{{{\text{1/2}}}}\]
Where,
\[{{\text{t}}_{{\text{1/2}}}}\] is the half-life.
${\text{k}}\,\,{\text{ = }}\,\dfrac{{{\text{0}}{\text{.693}}}}{{{{\text{t}}_{{\text{1/2}}}}\,\,}}$
So, according to the formula which relates rate constant and half-life relation of first-order we can say the half-life does not depend upon initial concentration of reactant. So, statement A. is true.
Rate constant of first-order reaction does not depend on concentration, so change in the concentration unit does not change the rate constant (K), so statement B is true.
We can rearrange the half-life formula for as follows:
${{\text{t}}_{1/2}}\,{{ \times }}\,{\text{k}}\,\, = \,0.693$
So, we can say the product of half-life and rate constant is equal to $0.693$, so statement C is true.
The relation between rate constant, rate and concentration is as follows:
${\text{rate = K}}{\left[ {{\text{concentration}}} \right]^1}$
Concentration is taken in terms of molarity. Unit of molarity is ${\text{mol}}\,\,{{\text{L}}^{ - 1}}$.
Unit of rate is ${\text{mol}}\,\,{{\text{L}}^{ - 1}}\,{\text{tim}}{{\text{e}}^{ - 1}}$.
So, ${\text{K}}\, = \,\dfrac{{{\text{rate}}}}{{{{\left[ {{\text{concentration}}} \right]}^1}}}$
${\text{K}}\, = \,\dfrac{{{\text{mol}}\,\,{{\text{L}}^{ - 1}}\,{\text{tim}}{{\text{e}}^{ - 1}}}}{{{{\left[ {{\text{mol}}\,\,{{\text{L}}^{ - 1}}} \right]}^1}}}$
${\text{K}}\, = \,{\text{tim}}{{\text{e}}^{ - 1}}$
So, the rate constant of first-order reaction is ${\text{tim}}{{\text{e}}^{ - 1}}$. So, statement D is false.
Therefore, option (D) is correct.

Note: The plot of half-life v/s reactant concentration will give a straight constant line. According to the half-life formula, half-life and rate constant is inversely proportional. More the half-life less will be the rate constant. The unit of half-life is time and the unit of the rate constant is ${\text{tim}}{{\text{e}}^{ - 1}}$ and the time can be taken in a second, minute, hour, or year.