Chemca Formula Sheet - Chemical Kinetics

CHEMCA

EXAM MASTER FORMULA SHEET

Chemical Kinetics

High-Yield Content for JEE Main, Advanced & NEET

1. Rate of Reaction & Rate Law

Average & Instantaneous Rate:

For a general reaction: $aA + bB \longrightarrow cC + dD$

\[ \text{Rate}_{rxn} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = +\frac{1}{c}\frac{d[C]}{dt} = +\frac{1}{d}\frac{d[D]}{dt} \]

Negative sign indicates disappearance of reactants. Positive sign indicates appearance of products.

Rate Law Expression:

\[ \text{Rate} = k[A]^x [B]^y \]

Order of Reaction ($n$): $n = x + y$

(Determined experimentally, can be zero, fraction, or negative).

Unit of Rate Constant ($k$):

\[ (\text{mol L}^{-1})^{1-n} \text{ s}^{-1} \]

• Zero Order ($n=0$): $\text{mol L}^{-1} \text{ s}^{-1}$
• First Order ($n=1$): $\text{s}^{-1}$
• Second Order ($n=2$): $\text{L mol}^{-1} \text{ s}^{-1}$

2. Integrated Rate Equations (Master Table)

Order	Integrated Rate Equation	Half-life ($t_{1/2}$)	Linear Graph (y vs x)
Zero ($n=0$)	\[ [A]_t = [A]_0 - kt \]	\[ t_{1/2} = \frac{[A]_0}{2k} \]	$[A]_t$ vs $t$ Slope = $-k$
First ($n=1$)	\[ k = \frac{2.303}{t} \log \frac{[A]_0}{[A]_t} \]	\[ t_{1/2} = \frac{0.693}{k} \]	$\log[A]_t$ vs $t$ Slope = $\frac{-k}{2.303}$
Second ($n=2$)	\[ \frac{1}{[A]_t} - \frac{1}{[A]_0} = kt \]	\[ t_{1/2} = \frac{1}{k[A]_0} \]	$\frac{1}{[A]_t}$ vs $t$ Slope = $+k$
$n$-th ($n \ge 2$)	\[ \frac{1}{n-1} \left[ \frac{1}{[A]_t^{n-1}} - \frac{1}{[A]_0^{n-1}} \right] = kt \]	\[ t_{1/2} \propto \frac{1}{[A]_0^{n-1}} \]	-

First Order Shortcuts:

$t_{75\%} = 2 \times t_{50\%}$
$t_{87.5\%} = 3 \times t_{50\%}$
$t_{99.9\%} \approx 10 \times t_{50\%}$
$[A]_t = [A]_0 \cdot e^{-kt} = \frac{[A]_0}{2^n}$ (where $n = \frac{t}{t_{1/2}}$)

Pseudo First Order Reactions:

Bimolecular reactions that follow first order kinetics because one reactant is in large excess.

Acid-Catalyzed Hydrolysis of Ester:
$\ce{CH3COOC2H5 + H2O(excess) ->[H+] CH3COOH + C2H5OH}$
Inversion of Cane Sugar (Sucrose):
$\ce{C12H22O11 + H2O(excess) ->[H+] Glucose + Fructose}$

3. Temperature Dependence (Arrhenius Equation)

The Arrhenius Equation

\[ k = A \cdot e^{-E_a/RT} \]

$A$ = Pre-exponential factor / Frequency factor

$E_a$ = Activation Energy (J/mol)

$e^{-E_a/RT}$ = Fraction of molecules having energy $\ge E_a$

Logarithmic Form (Two Temperatures)

Used to calculate $E_a$ if rate constants at two temps are known.

\[ \log \frac{k_2}{k_1} = \frac{E_a}{2.303 R} \left[ \frac{1}{T_1} - \frac{1}{T_2} \right] \]

Graph of $\log k$ vs $1/T$ has Slope = $-E_a / 2.303R$

Temperature Coefficient ($\mu$):

Ratio of rate constants at two temperatures differing by $10^\circ C$.

\[ \mu = \frac{k_{T+10}}{k_T} \approx 2 \text{ to } 3 \]

For every $10^\circ C$ rise, rate of reaction almost doubles.

Energy Profile Relation:

Enthalpy of reaction ($\Delta H$) is the difference in activation energies.

\[ \Delta H = E_{a(\text{forward})} - E_{a(\text{backward})} \]

Exothermic: $\Delta H < 0 \implies E_{a(f)} < E_{a(b)}$
Endothermic: $\Delta H > 0 \implies E_{a(f)} > E_{a(b)}$

4. Collision Theory & Catalysis

Modified Collision Theory Rate Equation:

\[ \text{Rate} = P \cdot Z_{AB} \cdot e^{-E_a/RT} \]

$Z_{AB}$ = Collision Frequency (Total collisions per second per unit volume).
$e^{-E_a/RT}$ = Energy Factor (Fraction of molecules with sufficient K.E.).
$P$ = Steric or Probability Factor (Fraction of collisions with proper orientation).

Molecularity vs Order

• Order: Experimental quantity. Can be zero, fractional, or negative. Applies to overall complex reaction.
• Molecularity: Theoretical concept. Number of reacting species participating in a simultaneous collision. Applies only to elementary (single-step) reactions. Must be a positive integer ($1, 2, 3$).

Role of a Catalyst

Provides an alternative reaction pathway with a lower Activation Energy ($E_a$).

Increases rate of both forward and backward reactions equally.
Does NOT change $\Delta G, \Delta H$, or the Equilibrium Constant ($K_{eq}$).

5. Order Determination & Parallel Reactions

Initial Rate Method

Observe the change in initial rate ($r_0$) by varying the initial concentration of one reactant while keeping others constant. If $[A]$ doubles and rate quadruples, order w.r.t $A$ is 2.

Half-life Method

Based on the relation: $t_{1/2} \propto [A]_0^{1-n}$. By measuring $t_{1/2}$ at two different initial concentrations, order $n$ can be mathematically derived.

$\frac{(t_{1/2})_1}{(t_{1/2})_2} = \left(\frac{[A_0]_2}{[A_0]_1}\right)^{n-1}$

Ostwald Isolation

Take all reactants except one in large excess. Their concentrations remain effectively constant, isolating the kinetic effect of the single limiting reactant.

Parallel (Competing) First-Order Reactions

When a reactant $A$ forms two products $B$ and $C$ via two parallel first-order paths with rate constants $k_1$ and $k_2$.

Effective Rate Constant ($k_{eff}$):

\[ k_{eff} = k_1 + k_2 \]

Percentage Yields:

$\% B = \frac{k_1}{k_1 + k_2} \times 100$
$\% C = \frac{k_2}{k_1 + k_2} \times 100$

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