CUET UG Chemistry: Physical Chemistry - Chemical Kinetics, Rate Laws, Arrhenius Equation, Half-Life

Must-Know

• The rate law expresses the rate of a reaction as proportional to the product of the concentrations of reactants raised to some powers, e.g., for a reaction ( aA + bB \rightarrow products ), rate = ( k[A]^m[B]^n ), where ( m ) and ( n ) are determined experimentally.
• The Arrhenius equation is ( k = A e^{-E_a/RT} ), where ( k ) = rate constant, ( A ) = pre-exponential factor, ( E_a ) = activation energy, ( R ) = 8.314 J/mol·K, and ( T ) = temperature in Kelvin.
• Taking natural log, the Arrhenius equation becomes ( \ln k = \ln A - \frac{E_a}{RT} ), which is linear in form ( y = mx + c ), used to determine ( E_a ) from a plot of ( \ln k ) vs ( 1/T ).
• For a first-order reaction, half-life ( t_{1/2} = \frac{0.693}{k} ); it is independent of initial concentration, e.g., radioactive decay follows this.
• For a zero-order reaction, ( t_{1/2} = \frac{[R]_0}{2k} ); half-life is directly proportional to initial concentration.
• For a second-order reaction with equal initial concentrations, ( t_{1/2} = \frac{1}{k[R]_0} ); half-life is inversely proportional to initial concentration.
• The unit of rate constant ( k ) for zero-order is mol L?¹ s?¹; for first-order, it is s?¹; for second-order, it is L mol?¹ s?¹.
• Activation energy ( E_a ) is the minimum energy required for a reaction to occur; it is always positive.
• A catalyst lowers the activation energy of both forward and reverse reactions equally, increasing the rate but not affecting equilibrium.
• The temperature coefficient of a reaction is the ratio of rate constants at temperatures differing by 10°C, typically 2–3 for many reactions.
• For every 10°C rise in temperature, the rate of reaction approximately doubles due to increased fraction of molecules with energy? ( E_a ).
• In the Arrhenius plot (( \ln k ) vs ( 1/T )), the slope = ( -E_a/R ), so ( E_a = -slope \times R ).
• If the rate of reaction increases from ( k_1 ) to ( k_2 ) when temperature rises from ( T_1 ) to ( T_2 ), then ( \log \frac{k_2}{k_1} = \frac{E_a}{2.303R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) ).
• A reaction with high ( E_a ) is more temperature-sensitive than one with low ( E_a ).
• The order of a reaction is the sum of the exponents in the rate law and can be fractional, zero, or negative, but molecularity is always a whole number-1.
• Molecularity refers to the number of reacting species in an elementary reaction; it cannot be zero or fractional.
• Pseudo-first-order reactions appear first-order when one reactant is in large excess, e.g., hydrolysis of ethyl acetate in excess water: rate = ( k'[CH_3COOC_2H_5] ).
• The half-life of a first-order reaction is constant and does not depend on initial concentration.
• For a first-order reaction, the time required for 99% completion is approximately 6.6 times the half-life (( t_{99\%} = \frac{4.606}{k}, t_{1/2} = \frac{0.693}{k} )).
• The rate of reaction is defined as the change in concentration of a reactant or product per unit time, e.g., rate = ( -\frac{d[R]}{dt} = \frac{d[P]}{dt} ).

Difficulty Level

Intermediate — requires understanding of mathematical forms, graph interpretation, and conceptual clarity between order, molecularity, and temperature effects.

Common CUET Traps

• Trap: Assuming half-life is always independent of initial concentration.
  Avoid: Remember only first-order reactions have concentration-independent half-life; zero and second order do not.
• Trap: Confusing order with molecularity; e.g., saying a reaction is trimolecular because three molecules appear in the balanced equation.
  Avoid: Molecularity applies only to elementary reactions; order is experimental and may not match stoichiometry.
• Trap: Thinking the Arrhenius equation applies only to exothermic reactions.
  Avoid: The equation applies to all reactions regardless of enthalpy change; ( E_a ) is always positive.

Practice MCQs

1. Question: What is the unit of the rate constant for a first-order reaction?
   A) mol L?¹ s?¹
   B) L mol?¹ s?¹
   C) s?¹
   D) mol² L?² s?¹
   Answer: C
   Explanation: For first-order reactions, rate = ( k[R] ), so unit of ( k ) is s?¹.
   Why others fail: Option A is for zero-order, commonly misremembered.

2. Question: The half-life of a first-order reaction is 30 minutes. What is its rate constant?
   A) 0.0231 min?¹
   B) 0.0462 min?¹
   C) 0.0154 min?¹
   D) 0.0300 min?¹
   Answer: A
   Explanation: ( k = \frac{0.693}{t_{1/2}} = \frac{0.693}{30} = 0.0231 ) min?¹.
   Why others fail: Option B is double the correct value, a common calculation error.

3. Question: For a reaction, the rate constant increases from ( 2 \times 10^{-3} ) s?¹ at 300 K to ( 4 \times 10^{-3} ) s?¹ at 310 K. What is the activation energy? (Use ( R = 8.314 ) J/mol·K)
   A) 53.6 kJ/mol
   B) 42.1 kJ/mol
   C) 27.6 kJ/mol
   D) 38.3 kJ/mol
   Answer: A
   Explanation: Using ( \log \frac{k_2}{k_1} = \frac{E_a}{2.303R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) ), ( E_a \approx 53.6 ) kJ/mol.
   Why others fail: Option B results from using natural log instead of log, a frequent mistake.

4. Question: Which of the following is correct for a zero-order reaction?
   A) Half-life is independent of initial concentration
   B) Rate constant has units of L mol?¹ s?¹
   C) Plot of [R] vs time is linear with negative slope
   D) Rate doubles if concentration is doubled
   Answer: C
   Explanation: For zero-order, [R] = [R]? – kt, so [R] vs t is linear with slope = –k.
   Why others fail: Option D is true for first-order, often confused.

5. Question: The activation energy of a reaction is 98 kJ/mol. By what factor does the rate increase when temperature rises from 300 K to 310 K?
   A) 1.8
   B) 2.5
   C) 3.2
   D) 4.0
   Answer: B
   Explanation: Using ( \log \frac{k_2}{k_1} = \frac{98000}{2.303 \times 8.314} \left( \frac{1}{300} - \frac{1}{310} \right) \approx 0.40 ), so ( \frac{k_2}{k_1} \approx 10^{0.4} = 2.5 ).
   Why others fail: Option C comes from incorrect temperature difference or arithmetic.

Last?Minute Revision

• First-order half-life: ( t_{1/2} = \frac{0.693}{k} ) — independent of [R]?.
• Arrhenius equation: ( k = A e^{-E_a/RT} ).
• Slope of ( \ln k ) vs ( 1/T ) = ( -E_a/R ).
• Zero-order unit: mol L?¹ s?¹.
• First-order unit: s?¹.
• Second-order unit: L mol?¹ s?¹.
• For every 10°C rise, rate doubles — approximate rule.
• Pseudo-first-order: one reactant in excess, e.g., sucrose hydrolysis.
• Molecularity-order — molecularity for elementary steps only.
• Rate = ( -\frac{d[R]}{dt} = \frac{d[P]}{dt} ).
• Catalyst lowers ( E_a ) but does not change ( \Delta G ) or equilibrium constant.
• ( t_{99\%} = \frac{4.606}{k} ) for first-order — ~6.6 × ( t_{1/2} ).
• ( \log \frac{k_2}{k_1} = \frac{E_a}{2.303R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right) ).
• High ( E_a )-high temperature sensitivity.
• Order can be fractional; molecularity cannot.
• Zero-order half-life: ( t_{1/2} = \frac{[R]_0}{2k} ).
• Second-order half-life: ( t_{1/2} = \frac{1}{k[R]_0} ).
• Temperature coefficient = ( \frac{k_{(T+10)}}{k_T} ), usually 2–3.
• Mnemonic: "Only First Order Has Constant Half-Life" — OFOHOCHL.
• verify from NCERT — exact value of temperature coefficient for a specific reaction.