# Thread: 1/4th Life expression for First Order Rxn

1. ## 1/4th Life expression for First Order Rxn

$\displaystyle ln(\frac{[A]_{\circ}}{[A]_{t}})=kt$
$\displaystyle ln(\frac{[A]\circ}{\frac{1}{4}[A]_{\circ}})=kt_\frac{1}{4}$
$\displaystyle ln(4)=kt_{\frac{1}{4}}$
do I set t=1/4 ?
$\displaystyle \frac{ln(4)*4}{k}$
$\displaystyle \frac{5.545}{k}$

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

2. Originally Posted by Vamz
$\displaystyle ln(\frac{[A]_{\circ}}{[A]_{t}})=kt$
$\displaystyle ln(\frac{[A]\circ}{\frac{1}{4}[A]_{\circ}})=kt_\frac{1}{4}$
$\displaystyle ln(4)=kt_{\frac{1}{4}}$
do I set t=1/4 ?
$\displaystyle \frac{ln(4)*4}{k}$
$\displaystyle \frac{5.545}{k}$

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
I assume that you want to calculate the value of t (?).

If so:

$\displaystyle \ln(4)=k \cdot t~\implies~t = \dfrac{\ln(4)}{k} \approx \dfrac{1.38629}{k}$

3. Originally Posted by Vamz
$\displaystyle ln(\frac{[A]_{\circ}}{[A]_{t}})=kt$
$\displaystyle ln(\frac{[A]\circ}{\frac{1}{4}[A]_{\circ}})=kt_\frac{1}{4}$
$\displaystyle ln(4)=kt_{\frac{1}{4}}$
do I set t=1/4 ?
No, you don't "set t= 1/4". It is not t that is 1/4. The value of A is 1/4 the value of $\displaystyle A_0$ which is why you replaced $\displaystyle [A]$ by $\displaystyle [A]\circ$.
What does $\displaystyle t_{1/4}$ mean? I assume the "1/4" is a subscript saying "the value of t at which A is 1/4 its original value". If so there is NO "t" in the equation. There is $\displaystyle t_{1/4}$ and that is what you want to solve for.

$\displaystyle \frac{ln(4)*4}{k}$
You have been fooled by your own notation! You have multiplied by 4 as if $\displaystyle t_{1/4}$ meant that t had been multiplied by 1/4.

$\displaystyle \frac{5.545}{k}$

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