To find the half-life of a compound A that decomposes in a first-order reaction at [tex]$25^{\circ} \text{C}$[/tex] with a given rate constant, we can use the formula for the half-life of a first-order reaction.
The half-life ([tex]\( t_{1/2} \)[/tex]) for a first-order reaction is given by the equation:
[tex]\[ t_{1/2} = \frac{0.693}{k} \][/tex]
Where:
- [tex]\( t_{1/2} \)[/tex] is the half-life,
- [tex]\( k \)[/tex] is the rate constant of the reaction.
In this problem:
- The rate constant [tex]\( k \)[/tex] is [tex]\( 0.450 \, S^{-1} \)[/tex].
Let's plug this value into the formula:
[tex]\[ t_{1/2} = \frac{0.693}{0.450} \][/tex]
Now, divide [tex]\( 0.693 \)[/tex] by [tex]\( 0.450 \)[/tex]:
[tex]\[ t_{1/2} = 1.540 \, \text{seconds} \][/tex]
Therefore, the half-life of compound A at [tex]\( 25^{\circ} \text{C} \)[/tex] is approximately [tex]\( 1.54 \)[/tex] seconds.
