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Answer 1

Hint: A first-order reaction is a type of reaction which proceeds at a rate which depends linearly on only one reactant concentration. The half-life is a timescale in which the initial concentration is decreased by half of its original value and rate constant is the coefficient of proportionality to the product of concentrations of the reactants. By applying this information, we can derive the relation between them and formulate a mathematical equation.

Complete step by step solution:
-From the integrated rate law, we can write the rate constant as follows
\[k=\dfrac{2.303}{t}{{\log }_{10}}\dfrac{{{\left[ A \right]}_{0}}}{{{\left[ A \right]}_{t}}}\]
Where, $k$ is the rate constant
$t$ is the time
${{\left[ A \right]}_{0}}$. is the initial concentration at t=0.
${{\left[ A \right]}_{t}}$ is the final concentration at time t.
As we mentioned half-life is the time taken for the concentration of reactant to reach half of its initial value. Hence for a half-life reaction $t={{t}_{{1}/{2}\;}}$
At half-life the ${{\left[ A \right]}_{t}}$will be half of the initial concentration ${{\left[ A \right]}_{0}}$.Thus ${{\left[ A \right]}_{t}}$ can be written as
\[{{\left[ A \right]}_{t}}=\dfrac{{{\left[ A \right]}_{0}}}{2}\]
Substituting this value of ${{\left[ A \right]}_{t}}$in the equation of integrated rate law for first order reaction,
\[k=\dfrac{2.303}{{{t}_{{1}/{2}\;}}}{{\log }_{10}}\dfrac{{{\left[ A \right]}_{0}}}{\dfrac{{{\left[ A \right]}_{0}}}{2}}\]
On rearranging the terms, we get,
\[k=\dfrac{2.303}{{{t}_{{1}/{2}\;}}}{{\log }_{10}}\dfrac{{{\left[ A \right]}_{0}}}{1}\times \dfrac{2}{{{\left[ A \right]}_{0}}}\]
On simplification of terms we get,
\[k=\dfrac{2.303}{{{t}_{{1}/{2}\;}}}{{\log }_{10}}(2)\]
As we know the logarithmic value,${{\log }_{10}}(2)=0.301$ and substituting this value in the above equation we get,
\[k=\dfrac{2.303}{{{t}_{{1}/{2}\;}}}\times 0.301\]
By doing the calculations we get,
\[k=\dfrac{0.693}{{{t}_{{1}/{2}\;}}}\]
Since we are asked to find the half-life by rearranging the above equation, we get
\[{{t}_{{1}/{2}\;}}=\dfrac{0.693}{k}\]
Thus, the relation between rate constant and half-life for a first order reaction is derived.

Note: It should be noted that from the above relation between half-life and rate constant, the half-life of a first-order reaction is independent of the initial concentration of reactants. This is a unique factor for first-order reactions. Also, for a first-order reaction, the rate constant and half-life are inversely proportional.