For a first order reaction, [tex] t_{99.99\%} = y \times t_{90\%} [/tex] Find the value of [tex] y [/tex].
Answer (1)
To solve this problem, we need to understand the concept of a first-order reaction and how to determine the time taken for such reactions to reach a certain percentage completion.
A first-order reaction is one where the rate of reaction is directly proportional to the concentration of one reactant. The integrated rate law for a first-order reaction is given by:
[ A ] = [ A ] 0 × e − k t
where
[ A ] is the concentration of the reactant at time t ,
[ A ] 0 is the initial concentration,
k is the rate constant, and
t is the time.
The time taken for a reaction to reach a certain percentage completion can be found by rearranging the integrated rate law equation. Specifically, the formula for the time t needed to reach x % completion is:
t x = k 2.303 lo g ( 100 − x 100 )
Let's solve for t 99.99% and t 90% using this formula:
Calculate t 99.99% :
(x = 99.99% t 99.99% = k 2.303 lo g ( 0.01 100 ) = k 2.303 lo g ( 10000 )
Calculate t 90% :
(x = 90% t 90% = k 2.303 lo g ( 10 100 ) = k 2.303 lo g ( 10 )
Solve for y such that t 99.99% = y × t 90% :
k 2.303 lo g ( 10000 ) = y × k 2.303 lo g ( 10 )
Simplifying,
lo g ( 10000 ) = y × lo g ( 10 )
Since lo g ( 10000 ) = 4 and lo g ( 10 ) = 1 ,
4 = y × 1
Thus, y = 4 .
Therefore, the value of y is 4.
