Chemical Kinetics: Reaction Rates and Rate Laws

Chemical Kinetics

• Chemical kinetics studies the rates of chemical reactions.
• Reaction rate represents the speed at which a chemical reaction occurs.

Reaction Rate Definition

• Reaction rate refers to the change in concentration of a reactant or product over time.
• For a reaction A → B, the rate can be expressed as Rate = -Δ[A]/Δt or Rate = Δ[B]/Δt.
  • Δ[A] is the change in reactant A concentration.
  • Δ[B] is the change in product B concentration.
  • Δt is the change in time.
• Reaction rate is always a positive value.
• The rate of disappearance of a reactant equals the rate of appearance of its product.
• For 2HI(g) → H2(g) + I2(g), Rate = -1/2 Δ[HI]/Δt = Δ[H2]/Δt = Δ[I2]/Δt.

The General Form

• For a general reaction aA + bB → cC + dD, the rate is: Rate = -1/a Δ[A]/Δt = -1/b Δ[B]/Δt = 1/c Δ[C]/Δt = 1/d Δ[D]/Δt.

Rate Laws

• Rate laws are mathematical expressions linking reaction rate with reactant concentrations.
• Rate laws are determined experimentally.
• For a reaction aA + bB → cC + dD, the rate law is Rate = k[A]^m[B]^n.
  • k is the rate constant.
  • [A] and [B] are the concentrations of reactants A and B.
  • m and n are the reaction orders with respect to A and B.
• Overall reaction order equals the sum of the individual orders (m + n).
• Reaction orders (m, n) are found experimentally and do not necessarily correspond to stoichiometric coefficients.

Rate Law Determination

• Determining rate law involves measuring initial reaction rates at varying initial reactant concentrations.
• The method of initial rates serves as a common experimental technique.

Example 1

• Reaction: NH4+(aq) + NO2-(aq) → N2(g) + 2H2O(l)
• At 25°C, the data indicates first order reactions with respect to both NH4+ and NO2-.
• Rate Law: Rate = k[NH4+][NO2-]
• Overall reaction order: second
• The rate constant is calculated as k = 2.7 × 10-4 M^-1s^-1.

Example 2

• Reaction: 2N2O5(g) → 4NO2(g) + O2(g)
• At 25°C, experiments reveal the reaction is first order.
• Rate Law: Rate = k[N2O5]
• Overall reaction order: first
• The rate constant is calculated as k = 7.0 × 10-5 s^-1.

First-Order Reactions

• Rate depends on the concentration of one reactant raised to the first power.
• For A → products, Rate = -Δ[A]/Δt = k[A].
• The integrated rate law is ln[A]t - ln[A]0 = -kt
  • [A]t is the concentration at time t.
  • [A]0 is the initial concentration.
  • k is the rate constant.
• The law can be expressed as ln([A]t/[A]0) = -kt or [A]t = [A]0e^-kt.

Half-Life

• It measures the time for reactant concentration to halve.
• For first-order reactions, it's independent of initial concentration: t1/2 = ln2/k = 0.693/k.

Example

• Insecticide decomposition follows a first-order rate law with k = 1.45/yr.
• Initial concentration: 5.0 × 10^-7 g/mL.
• After one year, the concentration drops to 1.18 × 10^-7 g/mL.
• Reducing insecticide concentration to 3.0 × 10^-7 g/mL will take approximately 0.35 years.
• The insecticide's half-life is 0.48 years.

Cylinder Centroid Example

• For a cylinder, defined by where x^2 + y^2 ≤ R^2 and 0 ≤ z ≤ h, and with constant density δ.
• Coordinates of Centroid: (x,y,z)
• x = y = 0 due to symmetrical nature
• z = h/2

Moment of Inertia

• Measures an object's resistance to angular acceleration.
• Requires specifying an axis of measurement.
• For a point mass m at distance r, I = mr^2.
• For multiple masses, I = Σ(mi * ri^2)
• For a solid W with density δ(x, y, z), the moments of inertia are defined as follows:
  • Along z axis I_z = ∫∫∫_W (x^2 + y^2) δ dV.
  • Along x axis I_x = ∫∫∫_W (y^2 + z^2) δ dV.
  • Along y axis I_y = ∫∫∫_W (x^2 + z^2) δ dV.

Example

• For solid cylinder with constant density moment of inertia about z-axis is: I_z = 1/2 * M * R^2
  • M is the total mass.

Game Theory

• Explores strategic interactions between decision-makers.
• Decision makers can include individuals, companies, & algorithms
• It offers a framework for understanding interaction outcomes.

Algorithmic Game Theory (AGT)

• AGT considers players' computational limits.
• Considers algorithms' game outcome impact.
• AGT designs algorithms for good outcomes in strategic settings.

Selfish Routing Model

• Consists of a network with n nodes and m edges.
• Each edge has a traffic cost function: ce(x).
• There are k commodities with a source(si) and destination(ti), with traffic amount di.

Definition (Nash Equilibrium)

• State where no player improves cost by changing strategy alone. For each commodity i, flow from si to ti follows cheapest routes.

Flow Cost

• Defined as the sum of traffic costs across all edges.
• Equation C(f)= Σ fece(fe) calculates flow cost f.

Social Optimum

• The flow that yields lowest total cost.
• Defined as: f^* = argmin_f C(f).

Price of Anarchy (PoA)

• Is the cost ratio between worst Nash equilibrium & social optimum.
• Defined as: PoA = C(f) / C(f*).

Complex Eigenvalues Example

• x' = x + y
• y' = -x + y
• Can be written as x' = Ax with A = [[1, 1], [-1, 1]].
• Eigenvalues are λ = 1 ± i.
• For λ = 1 + i, the eigenvector is v = [[1], [i]].
• For λ = 1 - i, the eigenvector is v = [[1], [-i]].
• Solutions x = c1e^((1+i)t)[[1], [i]] + c2e^((1-i)t)[[1], [-i]].
• Transforming Complex Solutions into Real Solutions
• • Eulers formula is: e^(it) = cos t + i sin t
• Equation: e^(λt)v = u(t) + iv(t). where
• u(t) = e^(αt) (cos(βt)a - sin(βt)b)
• v(t) = e^(αt) (sin(βt)a + cos(βt)b)

Example continued

• Real-valued solutions:
• u(t) = e^(t)[[cos t], [-sin t]]
• v(t) = e^(t)[[sin t], [cos t]]
• General solution x = c1e^(t)[[cos t], [-sin t]] + c2e^(t)[[sin t], [cos t]].

Chemical Properties

• The study of matter, its properties, and changes it undergoes
• Matter is anything that possesses mass and volume.
• Properties provide each matter with a unique identity

Physical Properties

• These properties are shown without interaction to another substance
• Melting point, electrical conductivity, and density,

Chemical Properties

• Interactions with another substance
• Flammability, corrosivity, and reactivity

States of Matter

• Solids posses a fixed shape and volume
• Liquids have fixed volume that assumes the shape of it's container
• Gas will conform to the shape of any container

Changes

– Physical Change- Changing the physical make up of matter without changing it’s compostition – Chemical reactions- Converting one substance into another

Chemical Rxn Properties

• Reactants are those present before the change, while products are those present after.

Intensive Vs Extensive

– Intensive are independent of amount ex density, temperature

• Extensive are dependent of the amount ex mass or volume

Separation Techniques

– Filtration- separates a solid from a liquid – Distillation- separation based on BP – Chromatography- separation on solubility

Mixture Composition

• Substance: fixed composition
• Element: cannot be broken down by chemical means.

Chemical Mixtures

• Compound: two or more elements chemically combined.
  • -Mixture: physically combined only.
• Homogenous: uniform
• Heterogenous: not uniform.

Chemical Elements

• Metals: lustrous, conductors, and typically solids (except mercury).
• Nonmetals: typically gases or dull, brittle solids.

Element Properties

• Metalloids: Properties intermediate between metals and nonmetals.
• A visual table showing all the elements ( Periodic Table ).

Chemical Nomenclature

– Nomenclature: systematic naming – compound types ionic,molecular, acids

Key Definition- Ions

• These are charged particles via gain or loss of an electron to the atomic structure.
• Cations: positively charged ions.
• Anions: negatively charged ions.

Naming Binary

1. Cation name (transition metal charge in Roman numerals)
2. Anion base name + "-ide" suffix

Compound Examples

• NaCl: Sodium chloride
• MgBr2: Magnesium bromide
• FeCl3 : Iron(III) chloride

Key Definition- Polyatomic Ions

Ions composed of multiple atoms.

• Examples NH_4, NO_3, SO_4, PO_4

Ionic w/Polyatomic

1. Cation name (metal charge) 2.Then the anion name

• ex NaOH- Sodium Hydroxide, & $FeNO_3$ Iron III Nitrate

Naming Molecular

1. First.Element.Name
2. 2nd.Element.Name + ‘ide’

Naming Acids

– Binary Acids, hydro,anion base ic-Acid – Examples: Hydrogen Chloride, ,HBr Hydro-bromic-Acid

Oxyacids Name Rule

– ‘ate anion become ic acid otherwise ‘ite anion becomes ous acid

Matrices - Definition and Types

• Matrix: Rectangular numerical symbol arrangement.
• Square: Row count equals column count.
• Identity: Main diagonals display number one the rest are zero
• Zero: All elements have a value of zero

Basic Operations - Matrices

• Addition: Add same arrangement elements of same matrix sizes
• Subtraction: Matrix same size and arrangement subtract

Operations and Definitions

• *Multiplication by a Scalar: Multiply each element Matrix
• Matrix Multiplication: Row multiply by row

Operations Continued Matrix’s

Transpose: Interchanging rows and columns

Key Matrix Definitions

• *Determinant (2x2): ad – bc – Inverse: a matrix that AB = BA = 1 – Rank: The maximum linearly independent rows and columns – Eigenvalues Eigen Vectors that satisfy specific equation.

General Matrix Applications

Linear Transformations, computer , quantum mechanics

• Systems of Linear Equations
• Network Analysis

Bernoulli's Principle

• States that fluid speed increases with a reduction in pressure or potential energy.

Equation

• P + 1/2 ρV^2 + ρgh = constant
• Where:
• P = pressure of the fluid
• ρ = density
• v = velocity
• h = height of the container

Applications

• Airplanes: Air flows faster over the top of the wing, creating lower pressure and lift the plane higher.
• Race cars: create the opposite lower pressure from the bottom wing- which creates pressure downward to keep the car on the track
• Chimneys: Wind blowing across creates low pressure, sucking the smoke up

Limitations

• Applicable to inviscid(no viscosity ) fluids
• Applicable to incompressible fluids (density unchanging) Applicable to steady flow(nothing changes with time)

Stress Types

Axial Stress:

• Equation- σ=P/ A- P standds for load -A are is the area-Sigma stands for stress Shear Stress: Equation τ=V/A

Strain

• Axial: Equation Epsiolon, d=delta/ Length
• Deformation over the original length
• Shear- Shear, strain, -delta sub s /length
• Side deformation divided by Length

Law of Matter

Elastic constants E- Epsiolon, Stress and strain, =Hook’s Constant

• Shear Constants- G gamoa Shear

Ratio Definitions Constants

Ratio: V Constant equation- Minus sign and ratios/ratio

Temperature Rules

• Thermal Constant Epsiolon T=Alpha constant and Delta T (change Temp

Equation Application of Constants

• Constant G = E/2(1+Constant v)

This lesson explores chemical kinetics, focusing on the rates of chemical reactions. It defines reaction rate as the change in reactant or product concentration over time. The lesson also covers the general form of rate expressions and introduces rate laws, which are mathematical expressions linking reaction rate with reactant concentrations.