Question

A first-order reaction is 45% complete at the end of 43 minutes. What is the length of the half-life of this reaction

Answer 1

The half-life of the reaction is 50 minutes

Data;

• Time = 43 minutes
• Type of reaction = first order
• Amount of Completion = 45%

Reaction Constant

Let the initial concentration of the reaction be X$_0$

The reactant left = (1 - 0.45) X$_0$ = 0.55 X$_0$ = X

For a first order reaction

$\ln(\frac{x}{x_o}) = -kt\\ k = \frac{1}{t}\ln (\frac{x_o}{x}) \\ k = \frac{1}{43}\ln (\frac{x_o}{0.55_o})\\ k = 0.013903 min^-^1$

Half Life

The half-life of a reaction is said to be the time required for the initial amount of the reactant to reach half it's original size.

$x = \frac{x_o}{2} \\t = t_\frac{1}{2} \\t_\frac{1}{2} = \frac{1}{k}\ln(\frac{x_o}{x_o/2})$

Substitute the values

$t_\frac{1}{2} = \frac{1}{k}\ln(2)=\frac{0.6931}{0.013903}\\t_\frac{1}{2}= 49.85 min = 50 min$

The half-life of the reaction is 50 minutes

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