Chemical Kinetics: Reaction Rates and Mechanisms

• Chemical kinetics studies reaction rates, influencing factors, and reaction mechanisms.

Reaction Rate

• Reaction rate quantifies reactant/product concentration changes over time.
• $Rate = -\frac{\Delta[Reactants]}{\Delta t} = \frac{\Delta[Products]}{\Delta t}$ where $\Delta[Reactants]$ and $\Delta[Products]$ are changes in concentration of reactants and products, and $\Delta t$ is the change in time.

Factors Affecting Reaction Rate

• Concentration: Higher reactant concentrations generally increase reaction rates.
• Temperature: Higher temperatures often speed up reactions by surpassing the activation energy barrier.
• Surface Area: Increased surface area boosts reaction velocity for solid-involved reactions.
• Catalysts: Catalysts accelerate reactions without being consumed; they lower activation energy.

Rate Law

• Rate law relates reaction rate to reactant concentrations.
• For the general reaction $aA + bB \rightarrow cC + dD$, the rate law is $Rate = k[A]^m[B]^n$
• $k$ is the rate constant
• $[A]$ and $[B]$ are concentrations of reactants A and B
• $m$ and $n$ are reaction orders with respect to A and B

Determining Reaction Order

• Reaction orders (m, n) are experimentally determined, indicating concentration effect on rate:
• Zero Order: Rate is independent of reactant concentration (m or n = 0).
• First Order: Rate is directly proportional to reactant concentration (m or n = 1).
• Second Order: Rate proportional to the square of reactant concentration (m or n = 2).

Examples of Rate Laws

• First-Order Reaction: $Rate = k[A]$, such as radioactive decay.
• Second-Order Reaction: $Rate = k[A]^2$ or $Rate = k[A][B]$, such as dimerization of NO2.

Activation Energy

• Activation energy ($E_a$) represents the minimum energy threshold for reactions to occur.
• It's the required energy reactant molecules need to proceed to products.

Arrhenius Equation

• The Arrhenius equation shows the relationship between rate constant $k$, activation energy $E_a$, and temperature $T$.
• $k = Ae^{-\frac{E_a}{RT}}$

Arrhenius Equation Variables

• $A$ is the pre-exponential or frequency factor.
• $R$ stands for the gas constant (8.314 J/mol·K).
• Graphing $\ln(k)$ vs $1/T$ yields $-E_a/R$ as the slope, from which $E_a$ can be derived.

Reaction Mechanisms

• Reaction mechanism describes the sequence of elementary steps in a chemical change.

Elementary Reactions

• Elementary reactions occur in a single step and cannot be broken down further.
• The rate law of an elementary reaction is determined directly by its stoichiometry.

Rate-Determining Step

• The rate-determining step is the slowest step in the reaction.
• The overall rate is ultimately determined by this rate-limiting step.

Intermediates

• Intermediates form in one step and are consumed in a subsequent step.
• Intermediates do not appear in the overall balanced equation.

Example of a Reaction Mechanism

• Consider the reaction: $2NO(g) + O_2(g) \rightarrow 2NO_2(g)$
• Possible Mechanism:
  • $NO(g) + NO(g) \rightleftharpoons N_2O_2(g)$ (fast equilibrium)
  • $N_2O_2(g) + O_2(g) \rightarrow 2NO_2(g)$ (slow, rate determining)
• Rate Law: $Rate = k[N_2O_2][O_2] = k[NO]^2[O_2]$

Catalysis

• Catalysis increases chemical reaction rates with a catalyst.
• Catalysts are not consumed during the reaction.

Types of Catalysis

• Homogeneous Catalysis: Catalyst and reactants are in the same phase.
  • Example: Acid catalysis in aqueous solution.
• Heterogeneous Catalysis: Catalyst and reactants are in different phases.
  • Example: Catalytic converters in automobiles.
• Enzyme Catalysis: Enzymes are biocatalysts with high specificity and efficiency.
  • Example: Catalysis of biochemical reactions in living organisms.

Matrizenmultiplikation (Matrix Multiplication)

• For $A = (a_{ij}) \in K^{m \times n}$ and $B = (b_{jk}) \in K^{n \times p}$, the matrix equation $C = A \cdot B = (c_{ik}) \in K^{m \times p}$ is defined by:
• $c_{ik} = \sum_{j=1}^{n} a_{ij}b_{jk}$ for $i = 1, \dots, m$ and $k = 1, \dots, p$
• Example: $$ A = \begin{pmatrix}1 & 2 & 3 \4 & 5 & 6\end{pmatrix}, \quad B = \begin{pmatrix}7 & 8 \9 & 10 \11 & 12\end{pmatrix} $$
• $$ A \cdot B = \begin{pmatrix}1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 & 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 \4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 & 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12\end{pmatrix} = \begin{pmatrix}58 & 64 \139 & 154\end{pmatrix} $$

Eigenschaften (Properties)

• Associativity: $(A \cdot B) \cdot C = A \cdot (B \cdot C)$
• Distributivity: $A \cdot (B + C) = A \cdot B + A \cdot C$ and $(A + B) \cdot C = A \cdot C + B \cdot C$
• Non-commutative: Im Allgemeinen ist $A \cdot B \neq B \cdot A$

Bemerkung (Remark)

• The # of A columns must match the # of B rows for multiplication.
• Multiplying by the identity matrix $I_n$ leaves the matrix unchanged: $A \cdot I_n = A = I_m \cdot A$
• The matrix product is a special case of the linear map.

Matriisilaskenta ja lineaarikuvausten perusteet (Matrix Calculus and Basics of Linear Mappings)

• Study of Vector spaces, Matrices, Linear transformations, Determinants and Eigenvalues/vectors.

Vektoriavaruudet (Vector Spaces)

• Vektoriavaruus $\mathbb{V}$ consists of a set with two operations:
  • Yhteenlasku (Addition): $\mathbf{u}, \mathbf{v} \in \mathbb{V} \Rightarrow \mathbf{u} + \mathbf{v} \in \mathbb{V}$
  • Skalaarikertolasku (Scalar multiplication): $a \in \mathbb{R}, \mathbf{u} \in \mathbb{V} \Rightarrow a\mathbf{u} \in \mathbb{V}$
• Conditions:
  • $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
  • $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
  • Exists $\mathbf{0} \in \mathbb{V}$ such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$ for all $\mathbf{u} \in \mathbb{V}$
  • For each $\mathbf{u} \in \mathbb{V}$, exists $-\mathbf{u} \in \mathbb{V}$ such that $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$
  • $a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}$
  • $(a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u}$
  • $a(b\mathbf{u}) = (ab)\mathbf{u}$
  • $1\mathbf{u} = \mathbf{u}$
• Examples:
  • $\mathbb{R}^n = {(x_1, x_2,..., x_n) \ | \ x_i \in \mathbb{R}}$
  • $\mathbb{P}_n$ = polynomials, with a degree high as $n$
  • $\mathbb{C}[a,b]$ = continuous functions in the interval $[a,b]$

Lineaarinen kombinaatio ja viritys (Linear Combination and Span)

• $\mathbf{v}$ is a linear combination of $\mathbf{v}_1, \mathbf{v}_2,..., \mathbf{v}_n$ if there exist scalars $c_1, c_2,..., c_n$ such that $\mathbf{v} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 +... + c_n\mathbf{v}_n$.
• The span of vectors $\mathbf{v}_1, \mathbf{v}_2,..., \mathbf{v}_n$ is the vector:$\text{span}{\mathbf{v}_1, \mathbf{v}_2,..., \mathbf{v}_n} = {c_1\mathbf{v}_1 + c_2\mathbf{v}_2 +... + c_n\mathbf{v}_n \ | \ c_i \in \mathbb{R}}$.

Lineaarinen riippumattomuus (Linear Independence)

• Vectors $\mathbf{v}_1, \mathbf{v}_2,..., \mathbf{v}_n$ are linearly independent if
  • $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 +... + c_n\mathbf{v}_n = \mathbf{0}$
• Holds only when $c_1 = c_2 =... = c_n = 0$. Otherwise they are linearly dependent.

Kanta ja dimensio (Basis and Dimension)

• The basis of vector space $\mathbb{V}$ is a set of linearly independent vectors which span $\mathbb{V}$. The dimension of the vector space is the size of its basis.
• Example: Basis for $\mathbb{R}^3$ is ${ \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix}, \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}, \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} }$, so $\text{dim}(\mathbb{R}^3) = 3$.

Aliavaruus (Subspace)

• Subspace of the vector space $\mathbb{V}$ is a set $\mathbb{H} \subseteq \mathbb{V}$, which is also a vector space.

Nolla-avaruus ja sarakeavaruus (Null Space and Column Space)

• Let $A$ be an $m \times n$ matrix.
  • The null space if matrix A is $\text{Nul}(A) = {\mathbf{x} \in \mathbb{R}^n \ | \ A\mathbf{x} = \mathbf{0}}$.
  • The column space of A is $\text{Col}(A) = \text{span}{\mathbf{a}_1, \mathbf{a}_2,..., \mathbf{a}_n}$, where $\mathbf{a}_i$ are the $A$ columns.

Advanced Higher Maths - Vectors

• Lines:

  • Vector Form: $\mathbf{r} = \mathbf{a} + t\mathbf{b}$

    • $\mathbf{a}$ is a position vector of a point on the line
    • $\mathbf{b}$ is a vector parallel to the line
    • $t$ is a parameter

  • Parametric Form: If $\mathbf{r} = \begin{pmatrix} x \ y \ z \end{pmatrix}$, $\mathbf{a} = \begin{pmatrix} a_1 \ a_2 \ a_3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} b_1 \ b_2 \ b_3 \end{pmatrix}$, then:

    • $x = a_1 + tb_1$
    • $y = a_2 + tb_2$
    • $z = a_3 + tb_3$

  • Cartesian Form: $\frac{x - a_1}{b_1} = \frac{y - a_2}{b_2} = \frac{z - a_3}{b_3}$

• Planes:

  • Vector Form: $(\mathbf{r} - \mathbf{a}) \cdot \mathbf{n} = 0$ or $\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$

    • $\mathbf{a}$ is a position vector of a point on the plane
    • $\mathbf{n}$ is a vector normal (perpendicular) to the plane

  • Cartesian Form: $ax + by + cz + d = 0$

    • $\begin{pmatrix} a \ b \ c \end{pmatrix}$ is a vector normal to the plane

Angle Between Vectors

• $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$

Angle Between Lines

• $\cos\theta = \frac{\mathbf{b_1} \cdot \mathbf{b_2}}{|\mathbf{b_1}||\mathbf{b_2}|}$
  • $\mathbf{b_1}$ and $\mathbf{b_2}$ are vectors parallel to the lines

Angle Between Planes

• $\cos\theta = \frac{\mathbf{n_1} \cdot \mathbf{n_2}}{|\mathbf{n_1}||\mathbf{n_2}|}$
  • $\mathbf{n_1}$ and $\mathbf{n_2}$ are vectors normal to the planes

Angle Between a Line and a Plane

• $\sin\theta = \frac{\mathbf{b} \cdot \mathbf{n}}{|\mathbf{b}||\mathbf{n}|}$
  • $\mathbf{b}$ is a vector parallel to the line
  • $\mathbf{n}$ is a vector normal to the plane

Vector Product (Cross Product)

• $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} = \begin{pmatrix} a_2b_3 - a_3b_2 \ a_3b_1 - a_1b_3 \ a_1b_2 - a_2b_1 \end{pmatrix}$
  • $\mathbf{a} \times \mathbf{b}$ is a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$

Scalar Product (Triple Product)

• $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix}$
  • The volume of the parallelepiped formed by the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ is $|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|$.

Exercices d'Algèbre linéaire (Linear Algebra Exercises)

Espaces vectoriels (Vector Spaces)

• Exercise 1: Let $E$ be a vector space over $\mathbb{K}$. Show that:
  • $\forall x \in E, 0_{\mathbb{K}} \cdot x = 0_{E}$
  • $\forall \lambda \in \mathbb{K}, \lambda \cdot 0_{E} = 0_{E}$
  • $\forall \lambda \in \mathbb{K}, \forall x \in E, (-\lambda) \cdot x = -(\lambda \cdot x) = \lambda \cdot (-x)$
  • $\forall \lambda \in \mathbb{K}, \forall x \in E, \lambda \cdot x = 0_{E} \Rightarrow \lambda = 0_{\mathbb{K}}$ ou $x = 0_{E}$
• Exercise 2: Let $E$ be a vector space over $\mathbb{K}$. Are the following sets subspaces of E? Justify your response.
  • $E = \mathbb{R}^{2}, F = {(x, y) \in \mathbb{R}^{2} \mid x + y = 1}$
  • $E = \mathbb{R}^{3}, F = {(x, y, z) \in \mathbb{R}^{3} \mid x = y = z}$
  • $E = \mathbb{R}^{3}, F = {(x, y, z) \in \mathbb{R}^{3} \mid x^{2} + y^{2} = z^{2}}$
  • $E = \mathbb{R}[X], F = {P \in \mathbb{R}[X] \mid P(1) = 0}$
  • $E = \mathbb{R}[X], F = {P \in \mathbb{R}[X] \mid deg(P) = n}$
  • $E = \mathbb{R}^{\mathbb{R}}, F = {f \in \mathbb{R}^{\mathbb{R}} \mid f(0) = 1}$
  • $E = \mathbb{R}^{\mathbb{R}}, F = {f \in \mathbb{R}^{\mathbb{R}} \mid f(x) = f(-x), \forall x \in \mathbb{R}}$
  • $E = \mathbb{R}^{\mathbb{R}}, F = {f \in \mathbb{R}^{\mathbb{R}} \mid f(x + 1) = f(x), \forall x \in \mathbb{R}}$
  • $E = \mathcal{M}{n}(\mathbb{R}), F = {A \in \mathcal{M}{n}(\mathbb{R}) \mid tr(A) = 0}$
  • $E = \mathcal{M}{n}(\mathbb{R}), F = {A \in \mathcal{M}{n}(\mathbb{R}) \mid A^{2} = A}$
• Exercise 3: $E$ is a vector space over $\mathbb{K}$ and $F$ and $G$ are two subspaces of vector. Show that $F \cup G$ is a subspace of E iff F belongs to G or G belongs to F.
• Exercise 4: $E$ is a vector space over $\mathbb{K}$ and $F$ and $G$ are two subspaces of vector. $F+G$ is smallest subspace of E that contains $F{\cup}G$.
• Exercise 5: $F = {(x, y, z) \in \mathbb{R}^{3} \mid x + y + z = 0}$ and $G = Vect((1, 1, 1))$. $F{\bigoplus} G = \mathbb{R}^3$.
• Exercise 6: $F = {(x, y, z, t) \in \mathbb{R}^{4} \mid x + y = 0$ and $z = 2t}$ and $G = Vect((1, 0, 1, 0), (0, 1, 1, 1))$. Determine basis of $F,G,F{\cap}G$ and $F+G$.
• Exercise 7: Show this equalitiy $dim(F + G) = dim(F) + dim(G) - dim(F \cap G)$.
• Exercise 8: $E=R_n[X]$. For $k{\in}{0,...,n}$, we note $F_k = {P{\in}E{\mid}P(K)=0}$. Calculate dim(F_k) and $dim(F_0{\cap}F_1)$.