The rate constant of a first-order reaction depends on:

The correct answer is Option (1) → Temperature

• The rate constant k is the proportionality factor in the rate law. For a first order reaction:

      \( \displaystyle \text{Rate} = k\,[\ce{R}] \)

• The rate changes as [R] changes, but k does not. For a given reaction, k changes only with temperature (and with a catalyst, which lowers E_a), as the Arrhenius equation shows:

      \( \displaystyle k = A\,e^{-E_a/RT} \)

  A rise in T makes the exponential term larger, so k increases with temperature.

Why the other options are incorrect

Option (2) Initial concentration of the reactant — Incorrect. A larger [R]₀ gives a larger initial rate, but k stays the same. The unit of a first order k, \( \mathrm{s^{-1}} \), has no concentration term in it.

Option (3) Time for which the reaction has run — Incorrect. In \( k=\frac{2.303}{t}\log\frac{[\ce{R}]_0}{[\ce{R}]} \), [R] falls as t grows, and the two changes cancel: k has the same value at every instant of the reaction.

Option (4) Volume of the reaction vessel — Incorrect. A change of volume changes the concentrations, and so the rate, but not k at a fixed temperature.

NCERT (2026-27 edition):
3.4 Temperature Dependence of the Rate of a Reaction – “It has been found that for a chemical reaction with rise in temperature by 10°, the rate constant is nearly doubled. The temperature dependence of the rate of a chemical reaction can be accurately explained by Arrhenius equation (3.18). …

    \( \displaystyle k = A\,e^{-E_a/RT} \qquad (3.18) \)”