1/4th Life expression for First Order Rxn

[[V]]

I must derive the 1/4th life expression for a first order rxn.
$$ln(\frac{[A]_{\circ}}{[A]_{t}})=kt$$
$$ln(\frac{[A]\circ}{\frac{1}{4}[A]_{\circ}})=kt_\frac{1}{4}$$
$$ln(4)=kt_{\frac{1}{4}}$$
do I set t=1/4 ?
$$\frac{ln(4)*4}{k}$$
$$\frac{5.545}{k}$$

What am I doing wrong here? The answer is allegedly 1.386/k

However, this answer key has been wrong before. Can someone please confirm/deny this?

Thank you

 

[Borek]

You are asked to calculate [itex]t_{\frac 1 4}[/tex] as a function of k, you can't assume t=0.25 and put it into equation. You need answer in form t=some expression, where is t in your final answer?<br /> <br /> <blockquote data-attributes="" data-quote="[V];2971106" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> [V];2971106 said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> $$ln(4)=kt_{\frac{1}{4}}$$ </div> </div> </blockquote><br /> Up to here you were right, just solve for [itex]t_{\frac 1 4}[/tex].[/itex][/itex]

 

[tiny-tim]

hi [V]!

do I set t=1/4 ?

nooo, 1/4 isn't time, it's amount (of reactant used) …

the "1/4" in t_1/4 is only a label, to remind you that it corresponds to 1/4 of the reactant being used

you're correct up to ln(4) = kt_1/4,

now just say t_1/4 = … ? ​