1/4th Life expression for First Order Rxn

In summary, the conversation discusses the derivation of the 1/4th life expression for a first order reaction. The equation ln(\frac{[A]_{\circ}}{[A]_{t}})=kt is used to calculate [itex]t_{\frac 1 4}[/tex] as a function of k. This cannot be done by assuming t=0.25 and plugging it into the equation. The correct approach is to solve for [itex]t_{\frac 1 4}[/tex] as an expression in terms of k. The conversation also clarifies that "1/4" in t1/4 is just a label and not an actual time value.
  • #1
[V]
28
0
I must derive the 1/4th life expression for a first order rxn.
[tex]ln(\frac{[A]_{\circ}}{[A]_{t}})=kt[/tex]
[tex]ln(\frac{[A]\circ}{\frac{1}{4}[A]_{\circ}})=kt_\frac{1}{4}[/tex]
[tex]ln(4)=kt_{\frac{1}{4}}[/tex]
do I set t=1/4 ?
[tex]\frac{ln(4)*4}{k}[/tex]
[tex]\frac{5.545}{k}[/tex]

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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  • #2
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?

[V];2971106 said:
[tex]ln(4)=kt_{\frac{1}{4}}[/tex]

Up to here you were right, just solve for [itex]t_{\frac 1 4}[/tex].
 
  • #3
hi [V]! :smile:
[V];2971106 said:
do I set t=1/4 ?

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

the "1/4" in t1/4 is only a label, to remind you that it corresponds to 1/4 of the reactant being used :wink:

you're correct up to ln(4) = kt1/4,

now just say t1/4 = … ? :smile:
 

1. What is the meaning of "1/4th Life expression" in the context of a first order reaction?

The 1/4th Life expression refers to the time it takes for the concentration of a reactant to decrease by 1/4th of its initial value in a first order reaction. It is also known as the half-life of the reaction.

2. How is the 1/4th Life expression calculated for a first order reaction?

The 1/4th Life expression can be calculated by using the formula t1/4 = (ln 2)/k, where t1/4 is the 1/4th Life, ln is the natural logarithm, and k is the rate constant of the reaction.

3. What is the significance of the 1/4th Life expression in first order reactions?

The 1/4th Life expression is significant because it helps us understand the rate at which a reactant is being consumed in a first order reaction. It also provides a measure of the stability of the reaction, as a shorter 1/4th Life indicates a faster reaction.

4. Can the 1/4th Life expression be used for all types of first order reactions?

Yes, the 1/4th Life expression can be used for all types of first order reactions, whether they are chemical or biological in nature. It is a universal concept that applies to all first order reactions.

5. How does the 1/4th Life expression relate to the overall rate of a first order reaction?

The 1/4th Life expression is directly related to the overall rate of a first order reaction. As the 1/4th Life decreases, the overall rate of the reaction increases. This means that the reaction is occurring at a faster rate and will reach completion sooner.

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