The ratio of the rate constants at 298 K and 260 K is 3.45.

Arrhenius Equation

The Arrhenius equation models the temperature dependence of the rate constant k as k = A e-Ea/RT, where A is the frequency factor, Ea is the activation energy, R is the gas constant, and T is the absolute temperature. Since A remains constant, the ratio k298/k260 = e-Ea/R (1/298 - 1/260).

Step-by-Step Calculation

Convert Ea = 21 kJ mol-1 to 21,000 J mol-1 for unit consistency with R = 8.314 J K-1 mol-1. Compute 1/260 - 1/298 = 0.003846 - 0.003356 = 0.000490 K-1. Then, (Ea/R) × 0.000490 = 2524.3 × 0.000490 = 1.237, so e1.237 = 3.45 (rounded to two decimal places).

The activation energy 21 kJ/mol rate constants ratio at 298K and 260K is a key Arrhenius equation problem for CSIR NET aspirants studying chemical kinetics. This calculation demonstrates how temperature influences reaction rates when the frequency factor stays constant.

Core Concept

In chemical kinetics, the Arrhenius equation k = A e-Ea/RT links rate constant k to activation energy Ea = 21 kJ mol-1. For two temperatures, the ratio simplifies to kT2/kT1 = exp[ Ea/R (1/T1 - 1/T2) ], with T2 = 298 K (higher, faster rate) and T1 = 260 K.

Detailed Solution Walkthrough

  • Ea = 21000 J/mol, R = 8.314 J K-1 mol-1.
  • 1/260 = 0.003846 K-1, 1/298 = 0.003356 K-1.
  • Difference: 0.000490 K-1.
  • Exponent: 21000 / 8.314 × 0.000490 = 1.237.
  • Ratio: e1.237 = 3.45. Thus, k298/k260 = 3.45.

Why No Options?

This numerical fill-in question (common in CSIR NET) has no multiple-choice options; the answer is directly computed and rounded to two decimals. Common errors include mismatched units (kJ vs J) or inverting temperatures, yielding ~0.29 instead.

CSIR NET Relevance

Mastering Arrhenius equation rate constant ratio optimizes exam prep for biochemistry enzyme kinetics and reaction mechanisms.