Read the passage carefully and answer the Questions.
The concentration dependence of the rate of a reaction is called the differential rate equation. It is not always convenient to determine the instantaneous rate, as it is measured by calculating the slope of the tangent at point 't' in the concentration vs time plot. This makes it difficult to determine the rate and hence the order of the reaction. This difficulty is overcome by integrating the differential rate equation. This integrated rate equation gives a direct relation between concentrations at different times and the rate constant. The integrated rate equations are different for the reactions having different reaction orders.
For an integrated first order reaction, a straight line graph is plotted. The incorrect co-ordinates and resulting slope of the graph are
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(ln[R]\) vs \(t\); Slope = \(k\)
For a first order reaction the integrated rate equation is
\( \displaystyle \ln[\mathrm{R}] = -kt + \ln[\mathrm{R}]_0 \)
which has the form \( y = mx + c \). Every straight-line plot of first order kinetics comes from rearranging this one equation, and the slope is read off as the coefficient of t:
\( \displaystyle \log[\mathrm{R}] = -\frac{k}{2.303}\,t + \log[\mathrm{R}]_0 \)
\( \displaystyle \ln\frac{[\mathrm{R}]_0}{[\mathrm{R}]} = kt \)
\( \displaystyle \log\frac{[\mathrm{R}]_0}{[\mathrm{R}]} = \frac{k}{2.303}\,t \)
Option (1) \(\log[R]\) vs \(t\); Slope = \(-k/2.303\) — Correct co-ordinates and slope. Dividing the ln form by 2.303 gives log[R] with slope \( -k/2.303 \).
Option (2) \(ln\{[R]_0/[R]\}\) vs \(t\), slope = \(k\) — Correct co-ordinates and slope. \( \ln([\mathrm{R}]_0/[\mathrm{R}]) = kt \) is a straight line through the origin with slope k.
Option (3) \(ln[R]\) vs \(t\); Slope = \(k\) — The incorrect co-ordinates and slope. A plot of ln[R] against t is a straight line, but its slope is −k, not k. The concentration of the reactant falls with time, so the line slopes downwards; its intercept is ln[R]0.
Option (4) \(\log \{[R]_0 /[R]\}\) vs \(t\), slope = \(k/2.303\) — Correct co-ordinates and slope. The same line in log form has slope \( k/2.303 \); this plot is the usual way to find k from experimental data.
NCERT (2026-27 edition):
First Order Reactions – …
