Which of the following relation about first order and zero order reaction is correct?

To determine the correct relationship between first-order and zero-order reactions, let us analyze the kinetics and timescales associated with each type of reaction.

First, let us look at a first-order reaction. The rate of a first-order reaction depends on the concentration of only one reactant. The integrated rate equation for a first-order reaction is given by:
$t=\frac{2.303}{k}\log \left(\frac{{\left[A\right]}_0}{\left[A\right]}\right)$
where ${\left[A\right]}_0$ is the initial concentration of the reactant, $\left[A\right]$ is the concentration of the reactant at time $t$, and $k$ is the rate constant.

The half-life ($t_{1/2}$) is the time required for the reactant concentration to decrease to half of its initial value, meaning $\left[A\right]=\frac{{\left[A\right]}_0}{2}$. Substituting this into the first-order rate equation gives:
$t_{1/2}=\frac{\ln \left(2\right)}{k}\approx \frac{0.693}{k}$
This shows that the half-life of a first-order reaction is a constant, independent of the initial concentration.

The time required for 100% completion of the reaction ($t_{100\%}$) corresponds to the concentration of reactant reaching zero ($\left[A\right]=0$). If we substitute $\left[A\right]=0$ into the integrated rate equation:
$t_{100\%}=\frac{1}{k}\ln \left(\frac{{\left[A\right]}_0}{0}\right)=\infty$
Because the term $\ln (\infty )$ approaches infinity, a first-order reaction theoretically takes an infinite amount of time to reach 100% completion. Therefore, we can state that:
$\left(t_{100\%}\right)\text{first order}=\text{Infinite time of}{t}_{1/2}$

For comparison, in a zero-order reaction, the rate is independent of the reactant concentration. The integrated rate equation is:
$\left[A\right]={\left[A\right]}_0-kt$
The time for 100% completion ($\left[A\right]=0$) is a finite value:
$t_{100\%}=\frac{{\left[A\right]}_0}{k}$
While the half-life is:
$t_{1/2}=\frac{{\left[A\right]}_0}{2k}$
Thus, for a zero-order reaction, the relationship is $t_{100\%}=2\left(t_{1/2}\right)$, which is not represented correctly in the incorrect options.

Hence, the correct statement is:
$\left(t_{100\%}\right)\text{first order}=\text{Infinite time of}{t}_{1/2}$