Chemical Kinetics

The rate of a chemical reaction is defined as the change in the molar concentration of any one of the reactants or products per unit time. Because reactant concentrations diminish while product concentrations accumulate as time elapses, a negative sign is appended to reactant terms to ensure the reaction rate is always a positive quantity:

• Average Rate ($r_{\text{av}}$): Evaluated over a macroscopic finite time interval $\Delta t$:
  $$r_{\text{av}} = -\frac{\Delta [R]}{\Delta t} = +\frac{\Delta [P]}{\Delta t}$$
• Instantaneous Rate ($r_{\text{inst}}$): The limiting value of the average rate as the time interval approaches infinitesimal zero ($\Delta t \to 0$). Graphically, it represents the slope of the tangent drawn to the concentration-versus-time curve at that specific instant $t$:
  $$r_{\text{inst}} = -\frac{d[R]}{dt} = +\frac{d[P]}{dt}$$

When the stoichiometric coefficients of reactants and products are unequal, the individual rates of disappearance and appearance differ. For a general homogenous chemical reaction:

Example: For the synthesis of ammonia, $\text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g)$:
$$\text{Rate} = -\frac{d[\text{N}_2]}{dt} = -\frac{1}{3}\frac{d[\text{H}_2]}{dt} = +\frac{1}{2}\frac{d[\text{NH}_3]}{dt}$$ Units of rate: $\text{mol L}^{-1}\text{ s}^{-1}$ or $\text{mol dm}^{-3}\text{ s}^{-1}$; for gaseous reactions, $\text{atm s}^{-1}$ or $\text{bar s}^{-1}$.

The Rate Law is an experimentally deduced mathematical equation correlating the observed reaction rate with the molar concentrations of reactants raised to empirical powers:

• The exponents $x$ and $y$ represent the orders of reaction with respect to reactants $A$ and $B$, respectively. They cannot be deduced from the stoichiometric coefficients of a balanced overall equation; they must be determined empirically.
• Specific Reaction Rate Constant ($k$): The proportionality constant representing the reaction rate when the molar concentration of each reactant is unity ($[A] = [B] = 1\text{ mol L}^{-1}$).
• Dimensional Units of Rate Constant: For an $n^{\text{th}}$-order reaction, the unit of $k$ depends strictly on $n$:
  $$\text{Unit of } k = \frac{\text{Rate}}{(\text{Concentration})^n} = \frac{\text{mol L}^{-1}\text{ s}^{-1}}{(\text{mol L}^{-1})^n} = (\text{mol L}^{-1})^{1-n}\text{ s}^{-1} = \text{mol}^{1-n}\text{ L}^{n-1}\text{ s}^{-1}$$

A reaction is zero-order when its rate is completely independent of the concentration of reactants: $$-\frac{d[R]}{dt} = k[R]^0 = k \implies d[R] = -k\,dt$$ Integrating between boundaries $[R]_0$ at $t = 0$ to $[R]_t$ at time $t$:

• Units of $k$: $\text{mol L}^{-1}\text{ s}^{-1}$ (identical to the units of reaction rate).
• Graphical Plot: Plotting $[R]_t$ versus $t$ gives a straight line with slope $= -k$ and y-intercept $= [R]_0$.
• Half-Life ($t_{1/2}$): The time required for reactant concentration to decrease to half its initial value ($[R]_t = [R]_0 / 2$):
  $$\frac{[R]_0}{2} = [R]_0 - k t_{1/2} \implies t_{1/2} = \frac{[R]_0}{2k} \quad (t_{1/2} \propto [R]_0)$$
• Total Time for Complete Consumption ($t_{100\%}$): $t_{100\%} = \frac{[R]_0}{k} = 2 \times t_{1/2}$.
• Classical Examples: Thermal decomposition of $\text{NH}_3$ on hot platinum surface at high pressure ($2\text{NH}_3 \xrightarrow{\text{Pt}} \text{N}_2 + 3\text{H}_2$), photochemical chlorination of methane over water.

A reaction is first-order when its rate is directly proportional to the first power of reactant concentration: $$-\frac{d[R]}{dt} = k[R] \implies \frac{d[R]}{[R]} = -k\,dt$$ Integrating between limits $[R]_0$ at $t = 0$ to $[R]_t$ at time $t$:

• Units of $k$: $\text{time}^{-1}$ ($ ext{s}^{-1}, \text{min}^{-1}, \text{h}^{-1}$); completely free of concentration units.
• Graphical Plots:
  1. $\log_{10}[R]_t$ vs $t$: Straight line with slope $= -\frac{k}{2.303}$ and intercept $= \log_{10}[R]_0$.
  2. $\log_{10}([R]_0 / [R]_t)$ vs $t$: Straight line passing through the origin with slope $= +\frac{k}{2.303}$.
• Half-Life Period ($t_{1/2}$): Setting $[R]_t = [R]_0 / 2$:
  $$t_{1/2} = \frac{2.303}{k}\log_{10}(2) = \frac{2.303 \times 0.30103}{k} = \frac{0.693}{k}$$ Key Characteristic: The half-life of a first-order reaction is completely independent of the initial concentration $[R]_0$.
• Critical Board Proofs:
  $t_{99.9\%} = \frac{2.303}{k}\log_{10}\left(\frac{100}{100 - 99.9}\right) = \frac{2.303}{k}\log_{10}(10^3) = \frac{2.303 \times 3}{k} = \frac{6.909}{k} \approx 10 \times t_{1/2}$.
  $t_{75\%} = 2 \times t_{1/2}$.
• Gas-Phase First-Order Kinetics: For $A(g) \rightarrow B(g) + C(g)$ starting with initial pressure $P_i$ and reaching total pressure $P_t$ at time $t$:
  $$k = \frac{2.303}{t}\log_{10}\left(\frac{P_i}{2P_i - P_t}\right)$$

The most versatile laboratory technique for complex multi-reactant reactions. Systematic experiments are conducted where the initial concentration of one reactant is varied (e.g., doubled or tripled) while maintaining the initial concentrations of all other reactants constant:

Taking logarithms yields $x = \frac{\log(r_1 / r_2)}{\log([A]_1 / [A]_2)}$. Repeating for reactant $B$ isolates exponent $y$.

For any general $n^{\text{th}}$-order reaction, the half-life depends inversely on the initial concentration raised to power $(n - 1)$:

When a reaction involves multiple reactants (e.g., $A + B + C \rightarrow \text{products}$), all reactants except one are taken in overwhelming excess. If $[B]$ and $[C]$ are huge relative to $[A]$, their concentrations remain virtually constant, reducing the effective rate law to $\text{Rate} = k'[A]^x$, allowing the isolation of order $x$. Repeating by isolating $B$ and $C$ in turn determines the complete order $n = x + y + z$.

Molecules must collide to react, but simple collision is insufficient. Reacting species must collide with a minimum threshold kinetic energy ($E_{\text{th}}$) and an appropriate spatial orientation.

• Threshold Energy ($E_{\text{th}}$): The minimum energy that colliding molecules must possess to produce an effective chemical collision resulting in product formation.
• Activation Energy ($E_a$): The excess energy over and above the ground-state average kinetic energy of the reactants ($E_R$) that must be supplied to reach the transition state:
  $$E_a = E_{\text{th}} - E_R$$
• Activated Complex (Transition State): An unstable, high-energy intermediate configuration where old reactant bonds are partially breaking while new product bonds are partially forming simultaneously.

In 1889, Svante Arrhenius formulated the quantitative exponential relationship linking temperature and reaction rate constant:

where $A$ is the pre-exponential factor (frequency factor), $E_a$ is the activation energy in $\text{J mol}^{-1}$, $R = 8.314\text{ J K}^{-1}\text{ mol}^{-1}$ is the universal gas constant, and $T$ is absolute temperature in Kelvin.
The term $e^{-E_a / RT}$ represents the Boltzmann fraction of molecules possessing kinetic energy equal to or greater than $E_a$.

• Arrhenius Plot: A plot of $\log_{10} k$ versus $1/T$ yields a straight line with: $$\text{Slope} = -\frac{E_a}{2.303 R} \implies E_a = -2.303 R \times \text{Slope}$$ $$\text{Y-Intercept} = \log_{10} A$$
• Two-Temperature Form: If the rate constant is $k_1$ at temperature $T_1$ and $k_2$ at temperature $T_2$ ($T_2 > T_1$):
  $$\log_{10}\left(\frac{k_2}{k_1}\right) = \frac{E_a}{2.303 R}\left[\frac{T_2 - T_1}{T_1 T_2}\right] = \frac{E_a}{2.303 R}\left[\frac{1}{T_1} - \frac{1}{T_2}\right]$$

Proposed by Max Trautz and William Lewis, collision theory is based on the kinetic theory of gases. The rate of a bimolecular elementary reaction $A + B \rightarrow \text{products}$ is given by:

• Collision Frequency ($Z_{AB}$): The number of collisions per second per unit volume of the reaction mixture (typically $10^{25}\text{ to }10^{28}\text{ collisions s}^{-1}\text{ cm}^{-3}$).
• Steric / Probability Factor ($P$): Accounts for the proper spatial orientation of colliding molecules during contact. Simple diatomic molecules have $P \approx 1$, while complex organic molecules have $P \ll 1$.
• Limitations: Assumes reacting molecules are rigid hard spheres, completely ignoring vibrational-rotational modes and electronic structural distortions.

All natural and artificial radioactive decay processes adhere strictly to first-order kinetics. The rate of nuclear disintegration at any instant is directly proportional to the total number of active nuclei ($N$) present:

where $\lambda$ is the decay constant (disintegration constant), $N_0$ is the initial radioactive nuclei population, and $N_t$ is the surviving nuclei population at elapsed time $t$.