Calculations of Reaction Rate Laws and Kinetics

Reaction Speed and Concentration: The rate at which a chemical reaction proceeds is directly dependent upon the concentration of the reactants involved.
General Reaction Representation: For a theoretical reaction represented as $A \rightarrow B$ , the rate of the reaction is determined by the concentration of reactant $A$ .
The Brackets Convention: In chemical kinetics, square brackets placed around a chemical symbol, such as $[A]$ , denote the molarity (concentration in moles per cubic decimeter, $mol\,dm^{-3}$ ) of that specific substance.
Multi-Reactant Rate Laws: For a more complex reaction such as $A + B \rightarrow C$ , the rate law is expressed mathematically as: $\text{Rate} = k[A]^x[B]^y$

k (Rate Constant): This is a proportionality constant that is unique to every reaction at a specific temperature. Its units vary depending on the overall order of the reaction.
[A] and [B] (Molar Concentrations): These represent the molarities of the reactants $A$ and $B$ , typically measured in $M$ ( $mol\,dm^{-3}$ ).
Reaction Orders (x and y): * The superscripts " $x$ " and " $y$ " are known as the order of the reaction with respect to each reactant. * $x$ is the order specifically for reactant $A$ . * $y$ is the order specifically for reactant $B$ . * The reaction order serves as an indicator of how sensitive or dependent the reaction rate is on the concentration fluctuations of that specific substance.

The Coefficient Misconception: Ideally, it might seem that the superscripts " $x$ " and " $y$ " would match the stoichiometric coefficients from the balanced chemical equation. However, because many reactions involve complex multiple steps and the formation of intermediate substances (intermediates), these exponents are usually not the same as the coefficients.
Experimental Necessity: The only definitive method to determine the specific order (exponent) for each reactant is through laboratory experimentation; it cannot be reliably predicted from the balanced equation alone.

Reaction Chemistry: $2NO + O_2 \rightarrow 2NO_2$
Experimental Data Collected: * Trial 1: $[NO] = 0.10\,M$ , $[O_2] = 0.25\,M$ , $\text{Initial Rate} = 0.032\,M/s$ * Trial 2: $[NO] = 0.10\,M$ , $[O_2] = 0.50\,M$ , $\text{Initial Rate} = 0.032\,M/s$ * Trial 3: $[NO] = 0.20\,M$ , $[O_2] = 0.25\,M$ , $\text{Initial Rate} = 0.064\,M/s$
Determining the Order for $O_2$ : * Compare Trial 1 and Trial 2: The concentration of $NO$ is constant ( $0.10\,M$ ), while the concentration of $O_2$ doubles from $0.25\,M$ to $0.50\,M$ . * Observation: The initial rate remains unchanged at $0.032\,M/s$ . * Conclusion: The reaction is zero-order with respect to $O_2$ (order $y = 0$ ), meaning changes in $O_2$ concentration do not affect the rate.
Determining the Order for $NO$ : * Compare Trial 1 and Trial 3: The concentration of $O_2$ is constant ( $0.25\,M$ ), while the concentration of $NO$ doubles from $0.10\,M$ to $0.20\,M$ . * Observation: The initial rate doubles from $0.032\,M/s$ to $0.064\,M/s$ . * Conclusion: The reaction is first-order with respect to $NO$ (order $x = 1$ ), meaning the rate change is proportional to the concentration change.
Final Rate Law Expression: $\text{Rate} = k[NO]^1[O_2]^0 = k[NO]$

Calculating the Value of k: * Using data from Trial 1: $0.032\,M/s = k(0.10\,M)$ * $k = \frac{0.032}{0.10} = 0.32$
Determining Units for k: * $k = \frac{\text{Rate}}{[NO]} = \frac{M/s}{M} = s^{-1}$ * The value and units for $k$ are $0.32\,s^{-1}$ .
Overall Order of the Reaction: * The overall order is the sum of the individual reactant orders: $1 + 0 = 1$ . * The reaction is first-order overall.

Problem: What is the initial rate if $[NO] = 0.85\,M$ and $[O_2] = 0.63\,M$ ? * Calculation: $\text{Rate} = (0.32\,s^{-1}) \times (0.85\,M)$ * $\text{Rate} = 0.272\,M/s$
Hypothetical Complex Reaction (A, B, and C): * For a reaction involving reactants $[A]$ , $[B]$ , and $[C]$ , first find the rate law expression, $k$ value, and overall order using experimental data tables similar to the Nitric Oxide example. * If given concentrations: $[A] = 0.50\,M$ , $[B] = 1.4\,M$ , and $[C] = 0.94\,M$ , substitute these values into the established rate law equation $\text{Rate} = k[A]^x[B]^y[C]^z$ to determine the specific rate for those conditions.