Chemical Kinetics Formula Sheet

Last-minute revision sheet for the entire Chemical Kinetics chapter: rate of reaction, rate law, order and molecularity, integrated rate equations, half-life, the Arrhenius equation, and catalysis. Every formula, relation, and shortcut below is pulled straight from the chapter pages.

Chapter Map

graph TD
    A[Chemical Kinetics] --> B[Rate of Reaction]
    A --> C[Rate Law & Order]
    A --> D[Integrated Rate Laws]
    A --> E[Half-Life]
    A --> F[Arrhenius / Temperature]
    A --> G[Catalysis]
    C --> C1[Order vs Molecularity]
    C --> C2[Units of k]
    D --> D1[Zero / First / Second]
    F --> F1[Activation Energy]
    F --> F2[Collision Theory]

Rate of Reaction

For a general reaction $aA + bB \rightarrow cC + dD$, the unique rate (divide by stoichiometric coefficients):

$$\boxed{r = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = +\frac{1}{c}\frac{d[C]}{dt} = +\frac{1}{d}\frac{d[D]}{dt}}$$

Quantity	Formula	Notes
Average rate	$r_{avg} = -\dfrac{\Delta[R]}{\Delta t} = +\dfrac{\Delta[P]}{\Delta t}$	Over finite interval (slope of chord)
Instantaneous rate	$r = -\dfrac{d[R]}{dt} = +\dfrac{d[P]}{dt}$	At one instant (slope of tangent)
Units of rate	$\text{mol L}^{-1}\text{s}^{-1}$ or $\text{M s}^{-1}$	Always positive when called “rate of reaction”

Rate Law and Rate Constant

For $aA + bB \rightarrow \text{Products}$:

$$\boxed{r = k[A]^m[B]^n}$$

• $k$ = rate constant (specific rate constant)
• $m$ = order w.r.t. A, $n$ = order w.r.t. B
• Overall order $= m + n$ ( determined experimentally, not from stoichiometry)

Units of Rate Constant

$$\boxed{\text{Units of } k = (\text{concentration})^{1-n} \times (\text{time})^{-1}}$$

Order $n$	Rate law	Units of $k$
0	$r = k$	mol L⁻¹ s⁻¹ (M s⁻¹)
1	$r = k[A]$	s⁻¹
2	$r = k[A]^2$ or $k[A][B]$	L mol⁻¹ s⁻¹ (M⁻¹ s⁻¹)
3	$r = k[A]^2[B]$	L² mol⁻² s⁻¹ (M⁻² s⁻¹)
$n$	—	L⁽ⁿ⁻¹⁾ mol⁽¹⁻ⁿ⁾ s⁻¹

Initial Rate Method

Comparing two experiments where only one concentration changes:

$$\frac{r_2}{r_1} = \left(\frac{[A]_2}{[A]_1}\right)^{m} \implies 2^m = \frac{r_2}{r_1}\ \text{(if conc. doubled)}$$

Order vs Molecularity

Property	Order	Molecularity
Definition	Sum of powers in rate law	No. of molecules colliding in elementary step
Determination	Experimental	From mechanism (theoretical)
Value	0, fractional, integer, negative	Only positive integer (1, 2, 3)
Can be zero / fraction?	Yes	No
Max value	No limit	3 (termolecular, rare)
Applies to	Overall or elementary	Elementary steps only

Pseudo-First-Order Reactions

When one reactant is in large excess its concentration stays nearly constant:

$$r = k[A][B] \approx k'[A], \quad \text{where } k' = k[B]$$

Example — acid hydrolysis of ester (water in excess):

$$CH_3COOC_2H_5 + H_2O \xrightarrow{H^+} CH_3COOH + C_2H_5OH$$

Integrated Rate Equations

Zero Order

$$-\frac{d[A]}{dt} = k \qquad \boxed{[A]_t = [A]_0 - kt}$$

First Order

$$-\frac{d[A]}{dt} = k[A]$$$$\boxed{\ln\frac{[A]_0}{[A]_t} = kt} \qquad k = \frac{2.303}{t}\log\frac{[A]_0}{[A]_t}$$$$[A]_t = [A]_0\, e^{-kt}$$

Second Order (Type I: $r = k[A]^2$)

$$-\frac{d[A]}{dt} = k[A]^2 \qquad \boxed{\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt}$$

Type II ($r = k[A][B]$, with $[A]_0 \neq [B]_0$):

$$\frac{1}{[A]_0 - [B]_0}\ln\frac{[B]_0[A]_t}{[A]_0[B]_t} = kt$$

Master Summary Table

Order	Differential	Integrated	Linear plot	Slope	Half-life	$k$ units
0	$-\frac{d[A]}{dt}=k$	$[A]_t=[A]_0-kt$	$[A]$ vs $t$	$-k$	$\frac{[A]_0}{2k}$	M s⁻¹
1	$-\frac{d[A]}{dt}=k[A]$	$\ln[A]_t=\ln[A]_0-kt$	$\ln[A]$ vs $t$	$-k$	$\frac{0.693}{k}$	s⁻¹
2	$-\frac{d[A]}{dt}=k[A]^2$	$\frac{1}{[A]_t}=\frac{1}{[A]_0}+kt$	$\frac{1}{[A]}$ vs $t$	$+k$	$\frac{1}{k[A]_0}$	M⁻¹ s⁻¹

First-Order Completion Shortcuts

$$t_{90\%} = \frac{2.303}{k}, \qquad t_{99\%} = \frac{2\times 2.303}{k} = 2\,t_{90\%}$$

So $t_{99\%} = 2\,t_{90\%}$ for any first-order reaction.

Half-Life

$$\text{Half-life } t_{1/2}: \text{ time for } [A]_t = \frac{[A]_0}{2}$$

After $n$ half-lives:

$$\boxed{\frac{[A]_t}{[A]_0} = \left(\frac{1}{2}\right)^{n}}$$

Order	$t_{1/2}$	Dependence on $[A]_0$	Successive half-lives
0	$\dfrac{[A]_0}{2k}$	$\propto [A]_0$	Decrease (1, ½, ¼, …)
1	$\dfrac{0.693}{k} = \dfrac{\ln 2}{k}$	Independent	Constant (1, 1, 1, …)
2	$\dfrac{1}{k[A]_0}$	$\propto \dfrac{1}{[A]_0}$	Increase (1, 2, 4, …)

Mean (Average) Life — First Order

$$\boxed{\tau = \frac{1}{k} = 1.443\, t_{1/2}}, \qquad t_{1/2} = 0.693\,\tau$$

Order from Half-Life

$$\frac{t_{1/2}^{(1)}}{t_{1/2}^{(2)}} = \left(\frac{[A]_0^{(1)}}{[A]_0^{(2)}}\right)^{1-n}$$

Arrhenius Equation and Activation Energy

$$\boxed{k = A\,e^{-E_a/RT}}$$

• $A$ = pre-exponential / frequency factor
• $E_a$ = activation energy, $R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$, $T$ in Kelvin

Logarithmic Forms ($\ln k$ vs $1/T$ is a straight line)

$$\ln k = \ln A - \frac{E_a}{RT} \qquad \log k = \log A - \frac{E_a}{2.303RT}$$

Slope of $\ln k$ vs $1/T$ plot $= -\dfrac{E_a}{R}$, intercept $= \ln A$, so $E_a = -\text{slope}\times R$.

Two-Temperature Form (most used in JEE)

$$\boxed{\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)}$$$$\log\frac{k_2}{k_1} = \frac{E_a}{2.303R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$$

Energy Profile Relations

$$\boxed{\Delta H = E_{a,\text{forward}} - E_{a,\text{reverse}}}$$

• Exothermic ($\Delta H < 0$): $E_{a,f} < E_{a,r}$
• Endothermic ($\Delta H > 0$): $E_{a,f} > E_{a,r}$

Temperature Coefficient

$$Q_{10} = \frac{k_{T+10}}{k_T} \approx 2\ \text{to}\ 3 \quad (\text{rate roughly doubles per } 10^\circ\text{C rise})$$

Catalyst Rate Enhancement

$$\boxed{\frac{k_{cat}}{k_{uncat}} = e^{-(E_{a,cat} - E_{a,uncat})/RT}}$$

graph LR
    A[Reactants] -->|"Ea forward"| B[Activated Complex]
    B -->|"Ea reverse"| C[Products]

Collision Theory

$$r = Z \times f \times p, \qquad f = e^{-E_a/RT}$$$$k = Z\,p\,e^{-E_a/RT} \implies A = Z \times p$$

• $Z$ = collision frequency
• $f$ = fraction of molecules with energy $\geq E_a$
• $p$ = steric (orientation) factor

A reaction occurs only when molecules collide, with energy $\geq E_a$, and proper orientation.

Catalysis

A catalyst provides an alternative pathway with lower $E_a$, so $k_{cat} \gg k_{uncat}$.

Type	Phase relation	Key examples
Homogeneous	Catalyst same phase as reactants	$H^+$ in ester hydrolysis; NO in lead chamber process
Heterogeneous	Catalyst different phase (usually solid)	Fe in Haber; V₂O₅ in Contact; Ni in oil hydrogenation; Pt/Pd/Rh converters
Enzyme	Biological (active site)	Carbonic anhydrase, catalase, urease
Autocatalysis	Product catalyzes the reaction	$Mn^{2+}$ in $KMnO_4$–oxalic acid

Heterogeneous mechanism (3 steps): Adsorption → Surface reaction → Desorption. Activity $\propto$ surface area.

Enzyme Kinetics — Michaelis–Menten

$$\boxed{v = \frac{V_{max}[S]}{K_M + [S]}}$$

When $[S] = K_M$, $v = \tfrac{1}{2}V_{max}$.

Key Equations at a Glance

Topic	Headline formula
Unique rate	$r = -\frac{1}{a}\frac{d[A]}{dt} = +\frac{1}{c}\frac{d[C]}{dt}$
Rate law	$r = k[A]^m[B]^n$
Zero order	$[A]_t = [A]_0 - kt$, $\ t_{1/2}=\frac{[A]_0}{2k}$
First order	$k=\frac{2.303}{t}\log\frac{[A]_0}{[A]_t}$, $\ t_{1/2}=\frac{0.693}{k}$
Second order	$\frac{1}{[A]_t}=\frac{1}{[A]_0}+kt$, $\ t_{1/2}=\frac{1}{k[A]_0}$
Mean life	$\tau = \frac{1}{k} = 1.443\,t_{1/2}$
Arrhenius	$k = A e^{-E_a/RT}$
Two-temperature	$\ln\frac{k_2}{k_1}=\frac{E_a}{R}\left(\frac{1}{T_1}-\frac{1}{T_2}\right)$
Energy profile	$\Delta H = E_{a,f} - E_{a,r}$
Catalyst boost	$\frac{k_{cat}}{k_{uncat}} = e^{-(E_{a,cat}-E_{a,uncat})/RT}$
Michaelis–Menten	$v = \frac{V_{max}[S]}{K_M+[S]}$

Further Reading

• Rate of Reaction
• Rate Law and Rate Constant
• Order and Molecularity
• Integrated Rate Equations
• Half-Life
• Arrhenius Equation
• Catalysis