Read the passage carefully and answer the questions.
The chemical reactions proceed at different rates which depend on various factors like concentration, temperature, etc. Rate law is an expression which relates the rate of reaction with concentration of various reacting species in the reaction. For a first order reaction, the concentration of the reactant after time \( t \) is related with its initial concentration by the relation, \( C = C_0e^{-kt} \). The rate constants for different orders have different expressions. For a first order reaction, \( k =\frac{2.303}{t}\log\frac{a}{a-x} \), where \( a \) is the initial concentration and \( x \) is the extent of reaction.
Which plot is linear with negative slope for a first order reaction
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → ln[C] versus time
For a first order reaction the concentration falls exponentially with time, \( C = C_0\,e^{-kt} \). Taking natural logarithms of both sides turns this into a straight-line equation:
\( \displaystyle \ln C = \ln C_0 - kt \)
\( \displaystyle \underbrace{\ln C}_{y} = \underbrace{(-k)}_{m}\,\underbrace{t}_{x} + \underbrace{\ln C_0}_{c} \)
Comparing with \( y = mx + c \), a plot of ln[C] against time is a straight line with slope = −k (negative) and intercept \( \ln C_0 \). In the base-10 form, \( \log C \) against t also gives a straight line, with slope \( -k/2.303 \).
Why the other options are incorrect
Option (1) k vs ln[C] — Incorrect. The rate constant does not depend on concentration, so k stays the same whatever ln[C] is: a horizontal line with zero slope.
Option (2) ln k versus ln C — Incorrect. For the same reason ln k does not change with ln C; the plot is again a horizontal line.
Option (3) ln k versus time — Incorrect. At a fixed temperature k does not change as the reaction proceeds, so this plot also has zero slope. A straight line with negative slope involving ln k is obtained only against 1/T (Arrhenius plot, slope \( -E_a/R \)).
NCERT (2026-27 edition):
First Order Reactions – …
