COMEDK 2023 Shift 1 Paper

For a first-order reaction, the rate law can be written as:
‌

d[A]

dt

=−k[A]
where [A] is the concentration of the reactant and k is the rate constant. Integrating this equation between initial concentration [A]0 at time t=0 and concentration [A] at time t, we have:
ln[A]−ln[A]0=−kt
Rearranging gives:
ln(‌

[A]

[A]0

)=−kt
Exponentiating both sides, we find:
‌

[A]

[A]0

=e−kt
To find how much reactant remains (and by extension, how much has converted to products), first convert the time from hours to seconds (since the rate constant's unit is per second). There are 3600 seconds in an hour, so:
Using the given rate constant k=1.5×10−6 s−1, we can find:
‌

[A]

[A]0

=e−1.5×10−6×72000
Calculating this:
‌

[A]

[A]0

=e−0.108≈0.8973
This means that approximately 89.73% of the initial concentration remains unreacted after 20 hours. To find the percentage that has converted to products, subtract this from 100% :
Percentage converted to products =100%−89.73%=10.27%
The closest answer option to 10.27% is Option B, 10.23 .