Reaction Kinetics II: Orders and Measurements

Fundamentals of Reaction Kinetics and Thermodynamics

• Thermodynamic Basis for Spontaneity: The spontaneity of a reaction is determined by the change in Gibbs free energy ($\Delta G$). The relationship is defined by the equation:

  • $\Delta G = \Delta H - T\Delta S$

  • Where $\Delta G$ is the change in Gibbs free energy.

  • $\Delta H$ represents the change in enthalpy.

  • $T$ is the absolute temperature.

  • $\Delta S$ represents the change in entropy.

• Requirement for Spontaneity: For a chemical reaction represented by $A \rightarrow P$ to occur spontaneously, the $\Delta G$ must be negative ($\Delta G < 0$).

• Distinction Between Kinetics and Thermodynamics: While thermodynamics tells us if a reaction will occur, reaction kinetics focuses on how fast the reaction proceeds (reaction rates) and the path it takes (mechanisms).

General Definitions and Rate Equations

• Rate of Reaction: Defined by the change in the concentrations of reactants ($[A]$) or products ($[P]$) over a specific period of time ($t$).

  • As the reaction $A \rightarrow P$ proceeds, the concentration of $A$ ($[A]$) decreases ($\downarrow$) and the concentration of $P$ ($[P]$) increases ($\uparrow$).

  • $\text{Rate of reaction} = \frac{\Delta[A]}{\Delta t}$

• Rate as a Function of Concentration: The rate is mathematically modeled using the rate constant ($k$) and the concentration of reactants raised to the power of the reaction order ($x$).

  • $\text{Rate} = k[A]^x$

  • $k$: Rate constant.

  • $x$: Order of reaction.

Properties of Reaction Orders

• Zero Order Kinetics:

  • The rate is independent of the concentration of reactants.

  • Rate Equation: $\text{Rate} = k$

  • Graphical Representation: Plotting Rate vs. Concentration yields a horizontal line ($y = c$).

• First Order Kinetics:

  • The rate is linearly dependent on the concentration of one reactant.

  • Rate Equation: $\text{Rate} = k[A]$

  • Graphical Representation: Plotting Rate vs. Concentration yields a straight line through the origin ($y = mx$).

• Second Order Kinetics:

  • The rate is non-linearly dependent on the concentration of a single reactant or linearly dependent on two different reactants.

  • Type 1 Rate Equation ($2A \rightarrow P$): $\text{Rate} = k[A]^2$

    • This follows a parabolic curve ($y = ax^2$) when plotting Rate vs. Concentration. rate is proportional o [A]²

  • Type 2 Rate Equation ($A + B \rightarrow P + Q$): $\text{Rate} = k[A]^1[B]^1$

    • The rate is linearly dependent on both concentration $[A]$ and $[B]$.

• Pseudo-First Order Kinetics:

  • Occurs in second-order reactions ($\text{Rate} = k[A]^1[B]^1$) when one reactant (e.g., $[B]$) is in vast excess compared to the other ($[B] \gg [A]$).

  • Example: Hydrolysis reactions where water ($H_2O$) is the solvent.

  • Because $[B]$ remains virtually constant throughout the reaction, the rate equation simplifies:

    • $\text{Rate} = k[A]^1[H_2O]^1 \approx k'[A]^1$

    • Where $k'$ is the observed pseudo-first-order rate constant.

    • apparently linearly dependent on [A]

Experimental Determination - Method 1: Initial Rates

• General Method: Measure the initial rate ($v_0$) while varying the initial concentration of reactants ($[A]_0$).

• The initial rate ($v_0$) is calculated as the gradient of the concentration-time plot at $t = 0$:

  • $v_0 = \frac{\Delta[A]}{\Delta t}$

• Make [B] »[A] → rate -k[A]^x

• Logarithmic Analysis: To determine the order ($x$), the rate equation is linearized using logarithms:

  • $\log(\text{Rate}) = \log(k) + x \log([A])$ , linear equation

• Graphical Plot: Plot $\log(v_0)$ against $\log([A]_0)$.

  • The resulting slope of the straight line equals the reaction order ($x$).

Experimental Determination - Method 2: Integrated Rate Laws

• Procedure: Measure the concentration of a reactant ($[A]$) over a period of time ($t$) to capture a time course.

• This method is specific to zero, first, and second-order reactions. The order is determined by which plot yields a straight line:

  • Zero Order: Plot $[A]$ vs. $t$ (varying). A straight line indicates zero order with slope $= -k$.

  • First Order: Plot $\ln([A])$ vs. $t$. A straight line indicates first order with slope $= -k$.

  • Second Order: Plot $\frac{1}{[A]}$ vs. $t$. A straight line indicates second order with slope $= k$.

• Handling Multiple Reactants ($\text{Rate} = k[A]^x[B]^y$):

  • To find $x$, keep $[B]$ in large excess ($[B] \gg [A]$), so $\text{Rate} = k'[A]^x$.

  • To find $y$, keep $[A]$ in large excess ($[A] \gg [B]$), so $\text{Rate} = k''[B]^y$.

Rate Equations and Mechanisms

• Stoichiometry vs. Order: Reaction order cannot be predicted from the stoichiometric coefficients of a balanced equation.

  • Example: $H_2 + Cl_2 \rightarrow 2HCl$ is second order.

  • Example: $H_2 + I_2 \rightarrow 2HI$ is complex and NOT simple second order.

• Rate-Determining Step (RDS): The reaction order reflects the number of species involved in the slowest step of the mechanism, known as the Rate-Determining Step or the formation of the transition state.

• Organic Mechanism Examples ($S_N1$ vs. $S_N2$):

  • $S_N2$ (One-step mechanism): Nucleophile ($Nu^-$) attacks the substrate ($R_3X$) as the leaving group ($X$) departs. The RDS involves both species.

    • $\text{Rate} = k[Nu^-]^1[R_3X]^1$ (Second order).

  • $S_N1$ (Two-step mechanism): The first step is the slow dissociation of the leaving group to form a carbocation intermediate. The second step is the fast attack by the nucleophile.

    • RDS is the initial dissociation.

    • $\text{Rate} = k[R_3X]^1$ (First order).

Practical Information and References

• Why Study Kinetics: It provides a "first glimpse" into reaction mechanisms, which is essential for improving chemical reactions in practical theory and analysis.

• References:

  • Crowe, 2nd edition, Chapter 14.

  • Atkins, 2nd edition, Chapter 6.

  • Averill & Eldredge, General Chemistry: Principles, Patterns, and Applications, v. 1.0 (flatworldknowledge.com).