To determine the value of "
y ", we need to find the relationship between the time required for
80% of a first order reaction
(t80) and the half-life period
(t1∕2) of the same reaction.
For a first-order reaction, the amount of reactant remaining at any time
t can be described by the equation:
[A]=[A]0e−kt where:
[A] is the concentration of the reactant at time
t,
[A]0 is the initial concentration of the reactant,
k is the rate constant,
t is time.
The half-life period
t1∕2 is given by:
t1∕2=‌Now, to find the time required for
80% of the reaction to complete, we need to find the time
t80 when
[A]=0.2[A]0 (since
80% of the reactant has reacted and only
20% remains). Thus, we have:
0.2[A]0=[A]0e−kt80 Simplifying this equation:
0.2=e−kt80Taking natural logarithm on both sides:
ln(0.2)=−kt80We know that:
‌ln(0.2)=ln(‌)=−ln(5)‌−kt80=−ln(5)‌kt80=ln(5) Now, substituting the value of
k from the half-life period equation
k=‌ :
‌‌t80=ln(5)‌t80=‌Using the value of
ln(5)≈1.609, we get:
‌t80=‌‌t80≈t1∕2×2.32 This shows that the time required for
80% of the reaction to complete is 2.32 times the half-life period of the same reaction. Therefore, the value of "
y " is:
Option B: 2.32