Rate laws Knowledge-Based Study

Study notes

1. Defining the Rate Law

   In chemical kinetics, the rate of a reaction often depends directly on the concentrations of the reactants. A rate law (or differential rate law) is an experimentally determined mathematical expression that quantifies this relationship.

   For a general reaction of the form:

   aA + bB → Products

   The rate law is expressed as:

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

   • Rate represents the reaction rate, typically in units of $\text{M}\cdot\text{s}^{-1}$ or $\text{mol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$.
   • k is the rate constant, which is specific to a given reaction at a constant temperature.
   • [A] and [B] represent the molar concentrations of reactants A and B.
   • m and n are the reaction orders with respect to reactants A and B.

   Key Concept: Reaction Orders

   The exponents $m$ and $n$ are typically small integers (such as 0, 1, or 2), though they can occasionally be fractional or negative. Crucially, reaction orders cannot be deduced from stoichiometric coefficients (a and b) of a balanced chemical equation; they must be determined experimentally.

   The overall reaction order is the sum of the individual exponents:

   $\text{Overall order} = m + n$

   Understanding the Rate Constant (k)

   The rate constant, $k$, is a proportionality constant that links reaction rate to reactant concentrations. While $k$ is independent of concentration, it depends strongly on temperature and the presence of a catalyst.

   Determining the Units of k

   Because the unit of rate is always concentration divided by time (typically $\text{M}\cdot\text{s}^{-1}$), the units of $k$ vary depending on the overall reaction order.

   Using dimensional analysis for an overall order $x$:

   $\text{Units of } k = \text{M}^{1 - x} \cdot \text{time}^{-1}$

   • Zero-order overall ($x = 0$): $k$ has units of $\text{M}\cdot\text{s}^{-1}$.
   • First-order overall ($x = 1$): $k$ has units of $\text{s}^{-1}$.
   • Second-order overall ($x = 2$): $k$ has units of $\text{M}^{-1}\cdot\text{s}^{-1}$ (or $\text{L}\cdot\text{mol}^{-1}\cdot\text{s}^{-1}$).
   • Third-order overall ($x = 3$): $k$ has units of $\text{M}^{-2}\cdot\text{s}^{-1}$ (or $\text{L}^2\cdot\text{mol}^{-2}\cdot\text{s}^{-1}$).

   Examining the units of $k$ provides a reliable method to identify the overall order of an unfamiliar reaction.

   The Method of Initial Rates

   The most common experimental approach to establish a rate law is the method of initial rates. In this method, a chemist measures the instantaneous rate at the very start of the reaction ($t = 0$) across multiple trials with varying initial concentrations.

   Systematic Isolation of Exponents

   To determine the order with respect to a specific reactant, identify two trials where that reactant's concentration changes while all other reactant concentrations remain constant.

   • Zero Order ($m = 0$): Doubling $[\text{A}]$ causes no change in the initial rate ($2^0 = 1$). The rate is independent of $[\text{A}]$.
   • First Order ($m = 1$): Doubling $[\text{A}]$ doubles the initial rate ($2^1 = 2$). The rate is directly proportional to $[\text{A}]$.
   • Second Order ($m = 2$): Doubling $[\text{A}]$ quadruples the initial rate ($2^2 = 4$). The rate scales with the square of $[\text{A}]$.

   Calculating the Value of k

   Once the reaction orders $m$ and $n$ are established, substitute the experimental rate and concentrations from any single trial into the rate law to solve algebraically for $k$, ensuring the correct units are included.

   Concentration as a Function of Time

   While differential rate laws express rate as a function of concentration, integrated rate laws express reactant concentration directly as a function of elapsed time ($t$). Each reaction order displays a characteristic linear graph.

   First-Order Processes

   For a reaction that is first order in reactant A ($\text{Rate} = k[\text{A}]$):

   $\ln[\text{A}]_t = -kt + \ln[\text{A}]_0$

   • A plot of $\ln[\text{A}]$ versus time yields a straight line with a slope of $-k$.
   • The half-life ($t_{1/2}$) is constant and concentration-independent: $t_{1/2} = \frac{\ln 2}{k} \approx \frac{0.693}{k}$.

   Second-Order Processes

   For a reaction that is second order in reactant A ($\text{Rate} = k[\text{A}]^2$):

   $\frac{1}{[\text{A}]_t} = kt + \frac{1}{[\text{A}]_0}$

   • A plot of $\frac{1}{[\text{A}]}$ versus time yields a straight line with a positive slope equal to $k$.
   • The half-life increases as concentration decreases: $t_{1/2} = \frac{1}{k[\text{A}]_0}$.

   Zero-Order Processes

   For a zero-order reaction ($\text{Rate} = k$):

   $[\text{A}]_t = -kt + [\text{A}]_0$

   • A direct plot of $[\text{A}]$ versus time is linear with a slope of $-k$.
   • The half-life decreases as the initial concentration decreases: $t_{1/2} = \frac{[\text{A}]_0}{2k}$.