Which of the following represents the expression for 3/4th life of 1st order reaction

The general formula for the time taken in a first-order reaction is:

t = (2.303 / k) * log(a / (a - x))

• t: Time elapsed.
• k: Rate constant.
• a: Initial concentration of the reactant.
• x: Amount of reactant consumed at time t.

Step-by-Step Solution:

1. Understand the Situation:

   At 3/4th life (t_3/4), 75% of the reactant has reacted, leaving only 1/4th of the initial concentration. Thus:

   x = (3a / 4)
    

   Remaining concentration = a - x = a - (3a / 4) = a / 4.

2. Substitute in the Formula:

   Replace x = 3a / 4 and a - x = a / 4 into the first-order reaction formula:

   t3/4 = (2.303 / k) * log(a / (a - 3a / 4))
    

3. Simplify the Expression:

   t3/4 = (2.303 / k) * log(a / (a / 4))
   t3/4 = (2.303 / k) * log(4)
    

Final Answer:

The expression for the 3/4th life of a first-order reaction is:

t3/4 = (2.303 / k) * log(4)
 

Key Point:

To find which of the following represents 3:4, always interpret it as a ratio indicating the consumption or remaining fraction of the reactant, which helps in solving questions like this.

Conclusion:

The correct representation of the expression for the 3/4th life of a first-order reaction is:

Option (C): t_3/4 = (2.303 / k) * log(4).