To understand the slope of the plot of
log (reactant concentration) against time for a first order reaction, let's start by examining the fundamental relationship governing first order reactions:
A first order reaction can be described by the differential rate law:
‌=−k[A]where
[A] is the concentration of the reactant and
k is the rate constant.
To solve this differential equation, we can separate variables and integrate:
∫‌d[A]=−k‌∫dtThis gives us the integrated form of the first order rate law:
ln[A]=−kt+Cwhere
C is the integration constant that can be determined from the initial conditions. If at
t=0,
[A]=[A]0 (initial concentration of
A), then:
ln[A]0=CThus, we can rewrite the equation as:
ln[A]=ln[A]0−ktThe next step is to convert the natural logarithm to a common logarithm (base 10) using the relationship:
log[A]=‌This results in:
squarelog[A]=log[A]0−‌We can clearly see that this equation is in the form
y=mx+c where
y is
log[A],c is
log[A]0, and the slope
m is
−k/2.303.
Therefore, the slope of the plot of
log (reactant concentration) against time for a first order reaction is:
Option C:
−k/2.303