1/4th Life expression for First Order Rxn

**Vamz** · November 5th 2010, 06:54 PM

$ln(\frac{[A]_{\circ}}{[A]_{t}})=kt$
$ln(\frac{[A]\circ}{\frac{1}{4}[A]_{\circ}})=kt_\frac{1}{4}$
$ln(4)=kt_{\frac{1}{4}}$
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

**earboth** · November 5th 2010, 11:16 PM

Originally Posted by Vamz

$ln(\frac{[A]_{\circ}}{[A]_{t}})=kt$
$ln(\frac{[A]\circ}{\frac{1}{4}[A]_{\circ}})=kt_\frac{1}{4}$
$ln(4)=kt_{\frac{1}{4}}$
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

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

If so:

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

**HallsofIvy** · November 6th 2010, 08:36 AM

Originally Posted by Vamz

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

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 $A_0$ which is why you replaced $[A]$ by $[A]\circ$ .
What does $t_{1/4}$ 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 $t_{1/4}$ and that is what you want to solve for.

$\frac{ln(4)*4}{k}$

You have been fooled by your own notation! You have multiplied by 4 as if $t_{1/4}$ meant that t had been multiplied by 1/4.

$\frac{5.545}{k}$

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

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