Chemical Kinetics: Reaction Rate and Rate Law

Chemical Kinetics - Reaction Rate

• Reaction rate is the speed at which reactants are used or products are created.
• Rate is expressed as the change in concentration of a substance per unit time: $Rate = \frac{\Delta[A]}{\Delta t}$.
• Reaction rate is affected by reactant concentration, temperature, surface area, catalysts, and pressure.
• Higher reactant concentrations yield faster rates due to more molecules being available for reaction.
• Increased temperature typically increases the reaction rate by providing more energy to molecules.
• Increased surface area (for solids) increases the reaction rate.
• Catalysts increase reaction rate without being consumed, offering a lower activation energy pathway.
• Pressure affects gas reactions; increased pressure means higher concentration & a faster reaction rate.

Chemical Kinetics - Rate Law

• Rate law expresses the relationship between reaction rate and reactant concentrations: $Rate = k[A]^m[B]^n$.
• $k$ is the rate constant, reflecting the reaction's intrinsic speed and is temperature-dependent.
• $[A]$ and $[B]$ represent the concentrations of reactants A and B.
• $m$ and $n$ are reaction orders, determined experimentally and indicating how rate changes with concentration.
• Rate laws are determined by measuring initial rates at different reactant concentrations.
• The method of initial rates compares reaction rates by varying the concentration of one reactant at a time in order to determine reaction orders

Chemical Kinetics - Reaction Order

• Reaction order is the exponent to which a reactant's concentration is raised in the rate law.
• It indicates how reactant concentration affects reaction rate.
• Reaction orders are typically integers, may also be fractions or zero.
• Zero Order: Rate is independent of reactant concentration.
• First Order: Rate is directly proportional to reactant concentration.
• Second Order: Rate is proportional to the square of reactant concentration.
• Overall reaction order is the sum of all reaction orders for each reactant in the rate law.

Chemical Kinetics - Arrhenius Equation

• Arrhenius equation describes the temperature dependence of the rate constant, $k = A \cdot e^{-\frac{E_a}{RT}}$.
• $A$ is the pre-exponential factor, representing collision frequency.
• $E_a$ is the activation energy, which is the minimum energy required for a reaction to occur.
• $R$ is the gas constant, $8.314 , J/(mol \cdot K)$.
• $T$ is the absolute temperature in Kelvin.
• Activation energy ($E_a$) is the energy barrier for a reaction. Lower $E_a$ gives faster reactions.
• $E_a$ can be determined by measuring $k$ at different temperatures and plotting $\ln(k)$ versus $1/T$.

Extended Particle Systems - Introduction

• A particle system developed in Chapter 3, is enhanced with behaviors like forces, constraints, collision response, and custom attributes.
• Adding forces such as gravity, wind, and drag creates more dynamic particle motion.
• Constraints are used by keeping particles within a certain area.
• Collision response involves making particles bounce off surfaces.
• Custom attributes include color, size, and lifetime.
• Particle creation and destruction techniques, enables effects like explosions and smoke.

Extended Particle Systems - Forces

• Forces are vectors that represent a push or pull on a particle.
• The net force is the vector sum of all individual forces affecting a particle.
• Newton's Second Law states $F = ma$, where $F$ is net force, $m$ is mass, and $a$ is acceleration.
• To apply a force: calculate net force, calculate acceleration ($a = F / m$), update velocity ($v = v + a * dt$), and update position ($x = x + v * dt$).

Extended Particle Systems - Common Forces

• Gravity: A constant downward force, $F_{gravity} = mg$, where $g$ is gravitational acceleration.
• Wind: Force in a particular direction, constant or variable.
• Drag: $F_{drag} = -kv$, where $k$ is a drag coefficient and $v$ is velocity.
• Spring: $F_{spring} = -k(x - x_{rest})$, where $k$ is the spring constant, $x$ is current position, and $x_{rest}$ is rest position.

Extended Particle Systems - Example: Gravity and Drag

• To implement gravity and drag, use the following snippet of code:

//Apply gravity
Vec2 gravity(0.0f, -9.8f);
acceleration += gravity;

//Apply drag
float dragCoefficient = 0.1f;
Vec2 dragForce = velocity * -dragCoefficient;
acceleration += dragForce;

Extended Particle Systems - Constraints

• Constraints restrict particle motion.
• Constraints can be a function returning true if a particle is within and false otherwise.
• If violated, particle's position or velocity must be altered to comply.

Extended Particle Systems - Common Constraints

• Bounding Box: Keeps particle within a rectangular area.
• Sphere: Restricts particle to within a spherical volume.
• Plane: Keeps particle on one side of a defined plane.

Extended Particle Systems - Example: Bounding Box

• Use the following snippet of code to implement a Bounding Box:

//Bounding Box Constraint
if (position.x  maxX) {
    position.x = maxX;
    velocity.x *= -bounceFactor;
}
//Repeat for Y axis

Extended Particle Systems - Collision Response

• Collision response defines particle reaction upon surface impact.
• To implement: detect collision, find normal vector at collision point, reflect velocity vector across normal, and reduce reflected velocity magnitude.

Extended Particle Systems - Example: Collision with a Static Surface

• Use the following snippet of code to implement a static collision:

//Assume we have collision point 'collisionPoint' and normal 'normal'
//Reflect velocity
Vec2 reflectedVelocity = velocity - 2.0f * dot(velocity, normal) * normal;
velocity = reflectedVelocity * restitution; //restitution is energy loss

Extended Particle Systems - Custom Attributes

• Custom attributes make particles flexible.
• Store extra data, such as color, size, lifetime, and texture.

Extended Particle Systems - Example: Particle Color and Lifetime

• Use the following snippet of code to implement custom attributes:

struct Particle {
    Vec2 position;
    Vec2 velocity;
    Color color;
    float lifetime;
    float maxLifetime;
};
//...
particle.color = Color(1.0f, 0.0f, 0.0f, 1.0f); //Red
particle.lifetime = 5.0f; //5 seconds

Extended Particle Systems - Particle Creation and Destruction

• Controlling creation and destruction is key for effects.
• Particle Creation: define spawn rate, initial conditions, and emitter shape.
• Particle Destruction: destroy by lifetime, distance, or collision.

Extended Particle Systems - Example: Lifetime-Based Destruction

• Use a lifetime counter to destroy old particles:

//Inside the update loop:
particle.lifetime -= dt;
if (particle.lifetime

### What is Mach?

- Mach is a kernel that forms the foundation of macOS and iOS.
- Mach is a message-passing oriented operating system kernel.

### Mach - History

- Developed at CMU starting in 1985 as a replacement for the BSD Unix kernel.
- Influenced by the Accent operating system.
- Version 2 released in 1988 and Version 3 (microkernel) released in 1989.
- Adopted by OSF and NeXTStep operating system.

### Mach - Key Concepts

- Message Passing: The primary IPC; tasks communicate by sending messages to protected kernel objects called ports.
- Ports: Protected kernel objects representing communication channels, tasks must have proper rights to send/receive.
- Tasks and Threads: A task is an execution environment (similar to a process) and threaded within a task.
- Virtual Memory: Mach provides sophisticated virtual memory management, with copy-on-write semantics, and memory objects can be backed by files or other tasks.

### Mach - Microkernel Architecture

- A microkernel architecture with most services running outside the kernel in user space was adopted in Version 3.
- The kernel manages message passing, task and thread management, and virtual memory.
- Increased modularity, stability, and flexibility are all benefits of the microkernel architecture.

### Mach in macOS and iOS

- macOS and iOS are based on XNU kernel with hybrid architecture.
- XNU is comprised of the Mach microkernel and the BSD Unix kernel.
- Mach provides core OS services; BSD layer provides file system, networking, and POSIX compatibility.

### Mach - Advantages

- Modularity: Design allows for modifications and extensions.
- Portability: Can be ported to many different architectures.
- Scalability: Supports shared memory multiprocessors.
- Security: Provides a secure form of IPC.

### Mach - Disadvantages

- Performance: Passing messages can be slower than in shared memory.
- Complexity: More complex to develop and debug.
- Overhead: Context switching between tasks/threads can be expensive.

### Machine Learning - Definition

- Machine Learning enables machines to improve at tasks through data learning instead of explicit programming.
- A computer program learns from experience E, task T, and performance P if its performance improves with experience.

### Spam Filter Example

- Task T: Flagging spam for new emails.
- Experience E: Spam filter being flagged as spam/not spam.
- Performance Measure P: Emails correctly flagged by the spam filter.

### Types of Machine Learning

- Supervised learning: Trained on labeled data.
  - Regression: Predicts continuous values.
  - Classification: Predicts categories.
- Unsupervised learning: Trained without labeled data.
  - Clustering: Groups similar data points.
  - Dimensionality reduction: Reduces number of variables
  - Anomaly detection: Detects unusual data points
- Semi-supervised learning: Trained on partially labeled data.
- Reinforcement learning:  Trains to maximize a reward.

### Machine Learning - Supervised Learning

- Regression is used to predict continuous values, as in predicting the price of a car.
- Classification is used to predict categories, as in classifying tumors as benign or malignant.

### Machine Learning - Unsupervised Learning

- Clustering is used to group similar data points such as customer segmentation.
- Dimensionality Reduction is used to reduce variables, such as feature extraction.
- Anomaly Detection is used to detect unusual data points such as fraud detection.

### Model Selection and Training

- Training data is data used to train the model.
- Test data is used to evaluate the model.
- Model Selection is choosing the right model for the task.
- Hyperparameter tuning involves selecting appropriate hyperparameters prior to training.
- Overfitting is when the model does well on the training data but poorly on the test data.
- Underfitting is when the model performs poorly on both the training data and the test data.

### Matrices - Definition

- A matrix is a rectangular array of numbers or symbols arranged in rows and columns denoted by upper-case letters (ex. A, B, C).
- Each item in a matrix is called an element denoted by lower-case letters with subscripts to indicate its row and column position (ex. $a_{11}, a_{12}, a_{21}, a_{22}$)

### Matrices - General Form

- The general form of a matrix is:
$$
A = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix}
$$
- In the above, *m* represents the number of rows, and *n* represents the number of columns. The size of the matrix is *m x n*.

### Matrices - Example

- The following matrix is an image of a matrix with two rows, and three columns. As such, the size of the matrix is 2 x 3.
$$
A = \begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}
$$
$a_{11} = 1, a_{12} = 2, a_{13} = 3, a_{21} = 4, a_{22} = 5, a_{23} = 6$

### Matrices - Vectors

- A vector is a special case of a matrix with only one row or one column.
- A Row Vector is a 1 x n matrix:
$$
A = \begin{bmatrix}
a_{1} & a_{2} & \cdots & a_{n}
\end{bmatrix}
$$
- A Column Vector is a m x 1 matrix:
$$
A = \begin{bmatrix}
a_{1} \\
a_{2} \\
\vdots \\
a_{m}
\end{bmatrix}
$$

### Matrices - Types of Matrices

- A Square Matrix is a matrix with the same number of rows and columns. The diagonal of a square matrix consists of the elements $a_{11}, a_{22}, \cdots a_{nn}$.
- An Identity Matrix is a diagonal array as shown below, where n is the size of the matrix:
$$
I_3 = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
$$
- A Zero Matrix is a matrix in which all elements are equal to 0.

### Matrices - Transpose of a Matrix

- The transpose of a matrix A is a matrix obtained by interchanging the rows and columns of A represented by $A^T$.  If A is an m x n matrix, then $A^T$ is an n x m matrix.
- The element in the i-th row and j-th column of $A^T$ is the element in the j-th row and i-th column of A: $a_{ij}^T = a_{ji}$.

### Matrices - Transpose of a Matrix Example

- If
$$
A = \begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}
$$
then
$$
A^T = \begin{bmatrix}
1 & 4 \\
2 & 5 \\
3 & 6
\end{bmatrix}
$$