Chemical Kinetics: Reaction Rates and Rate Laws

Introduction to Chemical Kinetics

• Chemical kinetics is defined as the study of the factors that affect the rates of chemical reactions. One primary factor involves how temperature influences these rates.

• The speed at which a chemical reaction occurs is termed the reaction rate.

• Reaction rate can be specifically defined in two ways:

  • With respect to the rate of product(s) formation.

  • With respect to the rate of reactant(s) consumption.

• Conceptually, a rate represents how much a particular quantity changes within a specific period of time. This is analogous to driving a car, where speed is measured as distance traveled (miles) over time (hours), resulting in the unit $mi/hr$.

• Kinetics is a critical field for two main reasons:

  • Understanding Reaction Mechanisms: It provides insight into the molecular-level occurrences during a reaction, which is fundamental to organic chemistry.

  • Temperature Dependence: It allows for the calculation of how reaction rates change with temperature ($T$), essential for chemical engineering applications.

Differential vs. Integrated Rate Laws

• Differential Rate Law: This law illustrates how the reaction rate depends specifically on concentration. It answers questions such as:

  • How fast is the reaction proceeding at the exact moment $y$ moles per liter of $A$ are mixed with $z$ moles per liter of $B$?

• Integrated Rate Law: This law utilizes calculus to show how the concentrations of species in a reaction depend on time. It addresses questions such as:

  • What is the concentration of $B$ when the concentration of $A$ is $x$ moles per liter?

  • How long will it take to consume $x$ moles per liter of $A$?

  • What is the concentration of $A$ after $y$ minutes of the reaction proceeding?

Defining and Measuring Reaction Rate

• The rate of a chemical reaction is generally measured by the decrease in reactant concentration or the increase in product concentration over a defined time interval.

• By convention, the reaction rate is always a positive value.

• For a general reaction of the form $aA + bB \rightarrow cC + dD$, the rate expression incorporates the stoichiometric coefficients ($a, b, c, d$) to ensure the rate is consistent regardless of which species is being measured:

  • $\text{Rate} = -\frac{1}{a} \frac{\Delta[A]}{\Delta t} = -\frac{1}{b} \frac{\Delta[B]}{\Delta t} = +\frac{1}{c} \frac{\Delta[C]}{\Delta t} = +\frac{1}{d} \frac{\Delta[D]}{\Delta t}$

• The negative sign is applied to reactants to account for their decreasing concentration, ensuring the resulting rate is positive.

• Typical units for reaction rate are molarity per second ($M/s$ or $mol\,L^{-1}\,s^{-1}$), though these can vary based on the timescale of the reaction.

Timescale Examples of Chemical Reactions

• Rhodopsin Isomerization: LIGHT absorption causes rhodopsin to isomerize in approximately $200\,fs$ ($2 \times 10^{-13}\,s$).

• Acetylene Burning: The oxidation of $C_2H_2$ in a flame occurs in roughly $10^{-6}\,s$.

• Diamond Oxidation: The oxidation of diamond to $CO_2$ is a process so slow that it is impossible to measure at room temperature ($RT$).

Reaction Rate and Stoichiometry Requirements

• If the rate of change for one specific reactant or product is known, the rate of change for any other participant in the reaction, as well as the overall reaction rate, can be calculated using the balanced equation.

• It is critical to note that the reaction rate cannot be determined from the balanced equation alone; it must be determined experimentally.

• Reaction rates are not necessarily constant; they often change as the reaction progresses over time.

Stoichiometric Calculation Example

• For the reaction $3NO_2(g) + H_2O(g) \rightarrow 2HNO_3(g) + NO(g)$, if the initial instantaneous rate of change of $NO_2$ concentration is $-5.05 \times 10^{-3}\,M/s$, the initial instantaneous rate of the overall reaction is determined by accounting for the coefficient of $3$:

  • $\text{Rate} = -\frac{1}{3} \frac{\Delta[NO_2]}{\Delta t}$

Average vs. Instantaneous Rate

• Average Rate: This is calculated using $\Delta$ to represent the change over a specific range of concentration and time ($Rate = -\frac{\Delta[A]}{\Delta t}$).

• Instantaneous Rate: This represents the rate at a specific single point in time. It is defined using the differential $d$ rather than $\Delta$.

  • The instantaneous rate is equal to the slope of the tangent to the concentration-vs-time curve at that specific time.

  • For a reaction involving $H_2$ and $HI$, the instantaneous rate at $50\,s$ is calculated as the negative slope of the $H_2$ tangent or half the slope of the $HI$ tangent.

• Example calculation for an instantaneous rate at $50\,s$:

  • $\text{Rate} = -\frac{\text{slope}}{2} = -\frac{-0.28}{40} = 0.0070\,M/s$

  • $\text{Rate} = \frac{1}{2} \frac{\Delta[HI]}{\Delta t} = \frac{1}{2} \frac{0.56}{40} = 0.0070\,M/s$

The Rate Law

• Reaction rates often change over time because the rate is dependent on the concentration of the reactants, which decrease as the reaction proceeds.

• If the rate depends only on reactant concentration, a mathematical expression called the Rate Law can be written.

• This assumes the reaction proceeds in one direction and the reverse reaction does not occur to any significant extent.

• For a generic reaction $A \rightarrow \text{products}$, the rate law is:

  • $\text{Rate} = k[A]^n$

• In this expression, $k$ is the rate constant. This constant does not change during a reaction, even if the rate itself changes.

• $n$ is the reaction order, which determines how the rate is affected by the concentration of $A$.

General Rate Law for Multiple Reactants

• For the reaction $aA + bB \rightarrow cC + dD$, the rate law is written as:

  • $\text{Rate} = k[A]^m[B]^n$

• $m$ is the reaction order with respect to reactant $A$.

• $n$ is the reaction order with respect to reactant $B$.

• The overall reaction order is the sum of the individual orders: $m + n$.

• Crucially, the overall reaction order and individual orders are not necessarily related to the stoichiometric coefficients in the balanced equation.

Disconnect Between Stoichiometry and Reality

• Example 1: $F_2(g) + 2ClO_2(g) \rightarrow 2FClO_2(g)$

  • Stoichiometry suggests: $\text{Rate} = k[F_2][ClO_2]^2$ (Order of 3).

  • Experimental reality: $\text{Rate} = k[F_2][ClO_2]$ (Order of 2).

• Example 2: $2NO(g) + 2H_2(g) \rightarrow 2N_2(g) + 2H_2O(g)$

  • Stoichiometry suggests: $\text{Rate} = k[NO]^2[H_2]^2$ (Order of 4).

  • Experimental reality: $\text{Rate} = k[NO]^2[H_2]$ (Order of 3).

Units of the Rate Constant ($k$)

• The units for $k$ vary depending on the overall reaction order to ensure that the units for Rate are always $M\,s^{-1}$.

• Zero Order ($n=0$): Units are $M\,s^{-1}$.

• First Order ($n=1$): Units are $s^{-1}$.

• Second Order ($n=2$): Units are $M^{-1}\,s^{-1}$ (or $L\,mol^{-1}\,s^{-1}$).

• Third Order ($n=3$): Units are $M^{-2}\,s^{-1}$ (or $L^2\,mol^{-2}\,s^{-1}$).

Analysis of Reaction Orders

Zero Order Reactions

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

• The reaction rate (slope) has no dependence on the concentration of $A$.

• Example: Evaporation. The rate depends on the surface area, not the total amount of water in the container ($H_2O(l) \rightarrow H_2O(g)$).

• Example: Decomposition of ammonia ($NH_3$) on a Platinum ($Pt$) catalyst. The rate is governed by temperature or the catalyst surface area rather than $[NH_3]$.

  • $2NH_3(g) \xrightarrow{\Delta, Pt} N_2(g) + 3H_2(g)$

  • $\text{Rate} = k[NH_3]^0 = k$

First Order Reactions

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

• The reaction rate is directly (linearly) proportional to the concentration of $A$.

• Proportionality Rules: If $[A]$ doubles, the rate doubles. If $[A]$ is halved, the rate is halved.

• Example: Decomposition of hydrogen peroxide ($2H_2O_2(l) \rightarrow 2H_2O(l) + O_2(g)$).

  • $\text{Rate} = k[H_2O_2]^1$

Second Order Reactions

• Rate Law: $\text{Rate} = k[A]^2$

• The reaction rate is proportional to the square of the concentration of $A$.

• Proportionality Rules: If $[A]$ doubles, the rate quadruples ($2^2 = 4$). If $[A]$ is halved, the rate decreases to $\frac{1}{4}$ of the original value.

• Example: Bimolecular reactions, such as the formation of a hydrogen molecule ($H(g) + H(g) \rightarrow H_2(g)$).

• Example: Dimerization reactions common in polymer chemistry.

Determining Reaction Order: Method of Initial Rates

• Reaction orders and the rate constant are determined experimentally by varying initial concentrations and measuring initial rates.

• Qualitative Inspection Rules:

  • If the rate does not change when concentration varies, it is zeroth order.

  • If the rate increases by the same factor as the concentration (e.g., doubles when concentration doubles), it is first order.

  • If the rate increases by the square of the factor (e.g., quadruples when concentration doubles), it is second order.

• Algebraic Method: Use the ratio of two trials to solve for the order $n$:

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

• Once the reaction order is found, $k$ can be calculated for each trial using the formula $k = \frac{\text{Rate}}{[A]^n}$. Experimental uncertainty may cause small deviations in $k$ values between trials, but they should be relatively close.

Complex Example Data Analysis

• For a reaction such as $CHCl_3(g) + Cl_2(g) \rightarrow CCl_4(g) + HCl(g)$, multiple trials are required where only one reactant concentration changes at a time to isolate the effect of each component on the rate law ($\text{Rate} = k [CHCl_3]^n [Cl_2]^m$).