Reaction order Knowledge-Based Study

Study notes

1. What is Reaction Order?

   In chemical kinetics, the rate of a reaction depends on the concentrations of the reacting species. This relationship is mathematically expressed by the differential rate law:

   $$\text{Rate} = k[A]^m[B]^n$$

   In this equation:

   • $k$ is the temperature-dependent rate constant.
   • $[A]$ and $[B]$ are the molar concentrations of reactants.
   • $m$ and $n$ are the partial reaction orders with respect to reactants $A$ and $B$, respectively.

   The overall reaction order is the sum of the individual reaction orders: $\text{Overall Order} = m + n$.

   Key Principle

   Reaction orders must be determined experimentally. They do not necessarily correspond to the stoichiometric coefficients in a balanced chemical equation unless the reaction is an elementary single-step process.

   Zero, First, and Second-Order Reactions

   Reaction orders are typically integers (0, 1, or 2), though fractional or negative orders can occasionally occur in complex reaction mechanisms.

   1. Zero-Order Reactions ($m = 0$)

   The rate is completely independent of the reactant concentration:

   $$\text{Rate} = k[A]^0 = k$$

   Doubling or tripling $[A]$ has no effect on the reaction rate. The units of the rate constant $k$ are $\text{M}\cdot\text{s}^{-1}$.

   2. First-Order Reactions ($m = 1$)

   The rate is directly proportional to the reactant concentration:

   $$\text{Rate} = k[A]^1$$

   Doubling $[A]$ doubles the rate; tripling $[A]$ triples the rate. The units of $k$ are $\text{s}^{-1}$.

   3. Second-Order Reactions ($m = 2$)

   The rate is proportional to the square of the reactant concentration:

   $$\text{Rate} = k[A]^2$$

   Doubling $[A]$ increases the rate by a factor of $2^2 = 4$. The units of $k$ are $\text{M}^{-1}\cdot\text{s}^{-1}$.

   Experimental Determination

   The method of initial rates involves conducting a series of experiments where the initial concentration of one reactant is varied while all other concentrations and temperature are kept constant.

   Mathematical Procedure

   Consider two experimental trials, 1 and 2, where $[B]$ is constant and $[A]$ changes:

   $$\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{k[A]_2^m[B]^n}{k[A]_1^m[B]^n} = \left(\frac{[A]_2}{[A]_1}\right)^m$$

   By comparing the factor by which the concentration increases to the factor by which the initial rate changes, the exponent $m$ is solved:

   • If $[A]$ doubles and the rate is unchanged, $2^m = 1 \implies m = 0$.
   • If $[A]$ doubles and the rate doubles, $2^m = 2 \implies m = 1$.
   • If $[A]$ doubles and the rate quadruples, $2^m = 4 \implies m = 2$.

   Once all partial orders are calculated, substituting the data from any single trial into the rate law yields the value and units of the rate constant $k$.

   Identifying Order Using Linear Graphs

   By measuring concentration as a function of time, integrated rate laws allow experimentalists to determine reaction order graphically.

   • Zero-Order: A plot of $[A]$ versus time $t$ gives a straight line with a slope of $-k$.
   • First-Order: A plot of $\ln[A]$ versus time $t$ gives a straight line with a slope of $-k$.
   • Second-Order: A plot of $\frac{1}{[A]}$ versus time $t$ gives a straight line with a positive slope of $+k$.

   Half-Life Signatures

   The half-life ($t_{1/2}$) behavior also distinguishes reaction orders:

   • For a first-order reaction, $t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}$. The half-life is constant and completely independent of initial concentration.
   • For a second-order reaction, $t_{1/2} = \frac{1}{k[A]_0}$. The half-life is inversely proportional to initial concentration and doubles with each successive half-life period.
   • For a zero-order reaction, $t_{1/2} = \frac{[A]_0}{2k}$. The half-life decreases as the reactant is consumed.