Chemical Kinetics: Reaction Rates and Mechanisms

Chemical Kinetics

• Chemical kinetics involves studying reaction rates, how these rates are affected by different conditions, and the mechanisms of reactions.

Reaction Rate

• For a reaction $aA + bB \rightarrow cC + dD$, the rate can be expressed as: Rate $= -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$

Rate Law

• Rate is proportional to $[A]^x [B]^y$.
• Rate $= k[A]^x [B]^y$, where k is the rate constant.
• 'x' represents the order with respect to A, 'y' is the order with respect to B, and x + y is the overall order of the reaction.
• The rate law is determined experimentally.

Integrated Rate Laws

• 0 order: Rate = k, Integrated Rate Law: $[A] = [A]_0 - kt$, Linear plot: $[A]$ vs t, Slope: -k
• 1st order: Rate = k[A], Integrated Rate Law: $ln[A] = ln[A]_0 - kt$, Linear plot: $ln[A]$ vs t, Slope: -k
• 2nd order: Rate = k$[A]^2$, Integrated Rate Law: $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$, Linear plot: $\frac{1}{[A]}$ vs t, Slope: k
• 2nd order: Rate = k[A][B], Integrated Rate Law: $ln(\frac{[B]_0[A]}{[A]_0[B]}) = ([B]_0 - [A]_0)kt$

Half-Life

• Half-life is the time it takes for a reactant concentration to decrease to half its initial value.
• For a 1st order reaction, $t_{1/2} = \frac{0.693}{k}$.

Collision Theory

• Reactions occur when molecules collide with sufficient energy and proper orientation.

Arrhenius Equation

• $k = Ae^{-E_a/RT}$
• $ln(k) = ln(A) - \frac{E_a}{RT}$
• $ln(\frac{k_2}{k_1}) = \frac{E_a}{R} (\frac{1}{T_1} - \frac{1}{T_2})$
• $E_a$ is the activation energy.
• R is the gas constant (8.314 J/mol $\cdot$ K).
• A is the frequency factor.

Reaction Mechanisms

• Reaction mechanisms are the step-by-step sequences of a chemical reaction.
• Elementary Step: A reaction that occurs in a single step.
• Reaction Intermediate: A species formed and consumed during the reaction but neither a reactant nor a product.
• Rate-Determining Step: The slowest step in the reaction mechanism.

Partial Differential Equations (PDEs)

Introduction

• Partial Differential Equations (PDEs) involve multiple independent variables and their partial derivatives.
• A PDE's general form is $F(x, y, u, u_x, u_y, u_{xx}, u_{yy}, u_{xy},...) = 0$, where x and y are independent variables, and u is the dependent variable.
• $u_x, u_y$ are the first partial derivatives of $u$ with respect to $x$ and $y$.
• $u_{xx}, u_{yy}, u_{xy}$ are the second partial derivatives of $u$.
• Wave Equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$
• Heat Equation: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
• Laplace's Equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
• The order of a PDE is determined by the highest order derivative in the equation.
• A PDE is linear if the dependent variable and its derivatives appear linearly.

Classification of Second-Order Linear PDEs

• The general form is $A\frac{\partial^2 u}{\partial x^2} + B\frac{\partial^2 u}{\partial x \partial y} + C\frac{\partial^2 u}{\partial y^2} + D\frac{\partial u}{\partial x} + E\frac{\partial u}{\partial y} + Fu = G$
• A, B, C, D, E, F, and G are functions of x and y.
• Classification depends on the discriminant $B^2 - 4AC$.
• Hyperbolic: $B^2 - 4AC > 0$ (e.g., Wave Equation)
• Parabolic: $B^2 - 4AC = 0$ (e.g., Heat Equation)
• Elliptic: $B^2 - 4AC < 0$ (e.g., Laplace's Equation)
• $u_{xx} - u_{yy} = 0$ is hyperbolic since $B^2 - 4AC = 4 > 0$.
• $u_t = u_{xx}$ is parabolic since $B^2 - 4AC = 0$.
• $u_{xx} + u_{yy} = 0$ is elliptic since $B^2 - 4AC = -4 < 0$.

Boundary and Initial Conditions

• Dirichlet Boundary Condition: Specifies the value of the function u on the boundary, such as $u(0, t) = f(t)$ and $u(L, t) = g(t)$.
• Neumann Boundary Condition: Specifies the value of the normal derivative of the function u on the boundary, such as $\frac{\partial u}{\partial x}(0, t) = f(t)$ and $\frac{\partial u}{\partial x}(L, t) = g(t)$.
• Robin Boundary Condition: A combination of Dirichlet and Neumann conditions, such as $a u(0, t) + b \frac{\partial u}{\partial x}(0, t) = h(t)$.
• Initial Condition: Specifies the value of the function u at an initial time, typically t = 0, such as $u(x, 0) = f(x)$.
• Boundary and initial conditions are vital for obtaining a unique solution to a PDE as they provide the necessary information to determine the specific solution.

Methods for Solving PDEs

• Separation of Variables: Reduces a PDE into multiple ordinary differential equations (ODEs).
• Fourier Series: Represents functions and solves PDEs, especially those with periodic boundary conditions.
• Green's Functions: Solves inhomogeneous PDEs with specific boundary conditions.

Numerical Methods

• Finite Difference Method: Approximates derivatives using difference quotients.
• Finite Element Method: Divides the domain into smaller elements.
• Finite Volume Method: Integrates PDE over control volumes.

Solving the Heat Equation using Separation of Variables

• Problem: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$, with $u(0, t) = 0, \quad u(L, t) = 0$ and $u(x, 0) = f(x)$.
• Solution: Assume $u(x, t) = X(x)T(t)$, substitute, separate variables, and solve the ODEs.
• Applying boundary conditions leads to $\lambda_n = (\frac{n\pi}{L})^2, \quad n = 1, 2, 3,...$
• $X_n(x) = \sin(\frac{n\pi x}{L})$
• General Solution: $u(x, t) = \sum_{n=1}^{\infty} B_n e^{-\alpha (\frac{n\pi}{L})^2 t} \sin(\frac{n\pi x}{L})$.

Final Solution

• Results from applying the initial condition to get the Fourier sine series coefficients for $f(x)$.

Vector Spaces

Definition 4.1.1

• A vector space over a field $F$ is a nonempty set $V$ with two operations:

  • Addition: $V \times V \longrightarrow V$ defined by $(u, v) \longmapsto u+v$.
  • Scalar Multiplication: $F \times V \longrightarrow V$ defined by $(\alpha, v) \longmapsto \alpha v$. Which satisfy the following properties:
  • Associativity: $u+(v+w)=(u+v)+w, \forall u, v, w \in V$.
  • Commutativity: $u+v=v+u, \forall u, v \in V$.
  • Existence of a Neutral Element: There exists an element $0 \in V$ such that $u+0=u, \forall u \in V$.
  • Existence of an Opposite Element: For each $u \in V$, there exists an element $-u \in V$ such that $u+(-u)=0$.
  • Distributivity with Respect to Scalar Addition: $\alpha(u+v)=\alpha u + \alpha v, \forall \alpha \in F, \forall u, v \in V$
  • Distributivity with Respect to Vector Addition: $(\alpha+\beta)u=\alpha u + \beta u, \forall \alpha, \beta \in F, \forall u \in V$.
  • Mixed Associativity: $\alpha(\beta u)=(\alpha \beta)u, \forall \alpha, \beta \in F, \forall u \in V$.
  • Existence of a Unit Element: $1u=u, \forall u \in V$.

• Elements of $V$ are called vectors; elements of $F$ are called scalars. The vector $0$ is called the zero vector.

Vector Space Examples

• The set of $m \times n$ matrices with elements in a field $F$, denoted $M_{m \times n}(F)$, is a vector space over $F$ with the usual matrix addition and scalar multiplication operations.
• $F^{m \times n}$ specifically forms a vector space over $F$.
• The set of polynomials with coefficients in a field $F$, denoted $P(F)$, is a vector space over $F$ with the usual polynomial addition and scalar multiplication. The set of real-valued functions of a real variable, denoted $F(\mathbb{R})$, is a vector space over $\mathbb{R}$ with the usual function addition and scalar multiplication operations. The set of real number sequences, denoted $S(\mathbb{R})$, is a vector space over $\mathbb{R}$ under ordinary sequence addition and scalar multiplication.
• $\mathbb{R}^n$ forms a vector space over $\mathbb{R}$ with the usual vector addition and scalar multiplication.
• $\mathbb{C}$ is a vector space over $\mathbb{R}$ with the standard addition of complex numbers and multiplication by real scalars.