QJEEM23Ch8af1: JEE Main 2023 Chemistry - Class 12 JEEM23Ch8af1 Solution: is the time required for the reaction to undergo completion and …

For a first-order reaction, the relation between the reaction rate constant (k) and time (t) for a given percentage of completion (p) is:

$\ln\left(\frac{1}{1-p}\right) = kt$

For t₅₀ (50% completion), p = 0.5:

$\ln\left(\frac{1}{1-0.5}\right) = k\cdot t_{50}$

$\ln\left(\frac{1}{0.5}\right) = k\cdot t_{50}$

$\ln(2) = k\cdot t_{50}$

For t₈₇.₅ (87.5% completion), p = 0.875:

$\ln\left(\frac{1}{1-0.875}\right) = k\cdot t_{87.5}$

$\ln\left(\frac{1}{0.125}\right) = k\cdot t_{87.5}$ $\ln(8) = k\cdot t_{87.5}$

Now, we need to find the relationship between t₈₇.₅ and t₅₀:

$\frac{k\cdot t_{87.5}}{k\cdot t_{50}} = \frac{\ln(8)}{\ln(2)}$

Since the k's cancel out, we have:

$\frac{t_{87.5}}{t_{50}} = \frac{\ln(8)}{\ln(2)}$

Using the property of logarithms, we get:

$\frac{t_{87.5}}{t_{50}} = \frac{\ln(2^3)}{\ln(2)}$

$\frac{t_{87.5}}{t_{50}} = 3$

So, the value of x is 3.