Rates of Reaction & Potential Energy Diagrams

Rates of Reaction: Collision Theory

• Reactant particles must collide for a reaction to occur.
• Particles have to possess sufficient kinetic energy to reach activation energy.
• Correct collision geometry also needs to be achieved

Potential Energy Diagrams

• These diagrams describe the energy fluctuations that occur during a chemical reaction.
• Activation Energy ($E_a$): This is the smallest amount of energy required for a reaction to start.
• Activated Complex: An unstable arrangement of atoms at maximized energy barrier.
• Enthalpy Change ($\Delta H$): Depicts the discrepancy in potential energy between reactants and products.
• $\Delta H = H_{products} - H_{reactants}$
• Catalyst: It lowers the activation energy ($E_a$) by introducing an alternate pathway.

Sample Potential Energy Diagram

• Illustrates the energy profile for catalyzed and uncatalyzed reactions
• Potential energy of reactants, products, and the activated complex are detailed.
• Activation energy and enthalpy change are specified too.

Maxwell-Boltzmann Distribution Curves

• Indicates the range of kinetic energies that reactant molecules have.
• Reaction only occurs if the molecules possess sufficient kinetic energy ($E_a$) during the collision.
• Temperature is proportional to the average kinetic energy of molecules.
• Catalysts do not impact the distribution of kinetic energies.

Sample Graph

• Plots the number of molecules against kinetic energy, showing it at temperatures $T_1$ and $T_2$ ($T_2 > T_1$).
• Illustrates that elevated temperature means there is more molecules exceeding the activation energy $E_a$.

Factors Affecting Reaction Rates

• Nature of Reactants: Reactions involving ions in aqueous solutions proceed more rapidly because no bonds need to be broken.
• Reactions with bond breaking/forming proceed slowly
• Concentration: Frequency of collisions is increased with more concentration. Applies to gases and solutions.
• Surface Area: Increased surface area increases the frequency of collisions, applies to heterogeneous reactions only (different phases).
• Temperature: Elevated temperature increases the frequency of collisions; additionally, the fraction of reacting particles can overcome $E_a$ to achieve effective collisions.
• Catalysts: Introduces a reaction mechanism with a lower $E_a$.
  • Homogenous Catalyst: It is present in the same phase as the reactants.
  • Heterogeneous Catalyst: It is present in a different phase than the reactants.

Catalyzed vs Uncatalyzed Reactions

• Catalyzed reactions proceed rapidly, while uncatalyzed reactions proceed slowly.

How Planes Fly

• Lift is the force that holds a plane up in the air.
• Lift results from uneven pressure on its wings
• Air moves faster over the curved upper surface of a wing and slower across the flat lower surface.
• Pressure of a fluid lessens when its speed increases.
• Higher pressure below pushes the wing up toward lower pressure above.
• Lift maximizes when the difference in air pressure of the surfaces is greatest.

Applications of the Bernoulli Effect

The Carburetor

• Air drawn through a narrow nozzle (Venturi tube) increases its speed in a carburetor.
• Vaporization occurs when air pressure drops, mixing gasoline with air.

The Atomizer

• Air is forced through a narrow opening, which increases its speed in an atomizer.
• As air pressure drops, the liquid rises via the tube and mixes with the air.

The Chimney

• The faster wind at the top of the chimney creates a lower pressure.
• The higher pressure at the bottom draws the smoke up.

Spoilers on Race Cars

• Designed to create downward force, which helps to keep the car on the ground.
• By deflecting air upwards, a low-pressure region above the spoiler is created.
• Pulling it downwards and pushing the car down

Lab 2: Modeling the Motion of Falling Objects

Objective

• Determine how drag force relies on the object's speed experimentally.
• Construct a computational model predicting the motion of an object subject to both gravity and drag force.
• Predict motion based on computational model and compare them with experimental measurements.

Activities

• Motion of a falling object is observed and its position is collected vs time data.
• Graph the objects speed in relation to time.
• Find connection between the drag force and the objects speed.
• The experimental data estimates the parameters of the drag force.
• Predict motion of an object given initial conditions and drag force using a computational model.
• Predictions of the computational model vs measurements of the experimental model is compared for validation.

The Physics

• Objects fall because of Earths gravitational force
• Gravitational force exerted is represented as $\vec{F}_{grav} = mg$ , where $m$ is the mass of the object and $g = 9.8 N/kg$.
• Air applies a "drag" force on the object, which is defined as: The drag force increases with object speed.
• Drag force acts in the opposite direction of direction of velocity of the object.
• Magnitude of the drag force is approximately proportional to the object’s speed.
• $|\vec{F}_{air}| = b|\vec{v}|$, where $b$ is a constant that depends on the size/shape of the object and density of air.
• For larger objects moving faster, the magnitude of the drag force is about proportional to the square of its speed.
• $|\vec{F}_{air}| = C|\vec{v}|^2$ where $C$ is a constant that depends on the size/shape of the object and the density of air.
• For smaller objects $\vec{F}_{air} = -k|\vec{v}|^n \hat{v}$ should be used where $k$ is a constant, $n$ is an integer, and $\hat{v}$ is a unit vector in the direction of the object’s velocity.

Preliminary Questions

• A coffee filter has a mass of 1 gram. The gravitational force exerted on the coffee filter by Earth?
• It is falling at a constant speed; the net force on the coffee filter and magnitude of the drag force exerted?
• When $|\vec{F}_{air}| = C|\vec{v}|^2$ exhibits magnitude of drag, then what is the constant C?
• An expression for the terminal speed $v_{term}$ of the coffee filter in terms of $m$, $g$, and $C$.

Instructions

• The procedure requires the observation of a falling coffee filter.
• Its vertical position needs to be recorded as a function of time.
• A graph of the vertical position is needed.
• A speed vs time graph must be made and be analyzed to observe the terminal speed.
• Use experimental to determine whether drag force relates to speed or speed squared.
• The value of the constant k equation $\vec{F}_{air} = -k|\vec{v}|^n \hat{v}$ is also required to write a program.
• The program must include both gravitational force and drag force.
• The constant k can be adjusted in the program with experimental data if necessary.
• Increase the amount of coffee filters and experiment with it.

Exercice 1

• Let f be the function defined on $f(x) = x^2 - 4x + 3$

Question 1

• Show that $f(x) = (x-2)^2 - 1$
• Show that $f(x) = (x-3)(x-1)$

Question 2

• Using the most adapted form of $f(x)$:
  • Compute f(0)
  • Compute f(1)
  • Solve f(x)=0
  • Solve f(x)=3
  • Determine the minimum of $f$ on $\mathbb{R}$.

Question 3

• Draw the table of variations of $f$.

Question 4

• Graph the curve of $f$ in an orthonormal.

Question 5

• Resolve graphically the equation $f(x) = -0,5$.

Thermodynamics

Introduction

• Thermodynamics studies energy, its transformations, and relation to matter. Governed by laws of conservation.

Key Concepts

• System: Region of space or amount of substance being observed.
• Surroundings: Everything outside the system.
• Boundary: Surface that separates the system from surroundings.
• Types of Systems:
  • Isolated System: No exchange of mass or energy with the surroundings.
  • Closed System: Exchange of energy but not mass with the surroundings.
  • Open System: Exchange of both mass and energy with the surroundings.
• Properties of a System:
  • Intensive Properties: Independent of the amount of substance (e.g., temperature, pressure, density).
  • Extensive Properties: Dependent on the amount of substance (e.g., mass, volume, energy).
• State of a System: Defined by the values of its properties.
• Process: Change in the state of a system.
• Equilibrium: No changes in the system's properties with time.

Laws of Thermodynamics

Zeroth Law

• If two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This defines temperature measurement.

First Law

• Energy cannot be created or destroyed. $\Delta U = Q - W$
  • $\Delta U$ change in internal energy.
  • $Q$ is the heat added to the system.
  • $W$ is the work done by the system.

Second Law

• The total entropy of an isolated system may only increase over time. $\Delta S \geq 0$
  • $\Delta S$ is the change in entropy of the system

Third Law

• Entropy in a system approaches a minimum or zero. For a perfect crystal at absolute zero (0 K), the entropy is zero. $S \rightarrow 0 \text{ as } T \rightarrow 0K$

Thermodynamic Processes

• Isothermal Process: Happens at constant temperature.
• Isobaric Process: Happens at constant pressure.
• Isochoric (or Isometric) Process: Happens at constant volume.
• Adiabatic Process: No heat exchange.
• Reversible Process: A process that can be reversed without leaving any trace on the surroundings.
• Irreversible Process: Cannot be reversed without trace.

Applications

• Applied in various fields, including:
  • Engineering
  • Chemistry
  • Materials Science
  • Environmental Science

Conclusion

• Thermodynamics describes interactions of energy, matter and conservation.