Chemical Kinetics: Reaction Rates and Rate Laws

Chemical Kinetics

• Chemical kinetics is the study of reaction rates.

Reaction Rate

• For a reaction $aA + bB \rightarrow cC + dD$, the reaction rate is expressed as:

$rate = -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$

where a, b, c, and d are stoichiometric coefficients, and $\frac{d[ ]}{dt}$ represents the change in concentration.

• A minus sign is included for reactants because they are consumed in the reaction

Rate Law

• The rate law relates the rate of a reaction to the concentrations of the reactants:

$rate = k[A]^x [B]^y$

where k is the rate constant, x is the order with respect to reactant A, and y is the order with respect to reactant B.

• The overall order of the reaction is x + y.
• The values of x, y, and k must be determined experimentally.

Integrated Rate Laws

• The integrated rate laws relate the concentration of reactants or products to time

• Zero Order:

  • Rate Law: $rate = k$
  • Integrated Rate Law: $[A]_t = -kt + [A]_0$
  • Plot for Straight Line: $[A]_t$ vs t
  • Slope: -k
  • Half-life: $t_{1/2} = \frac{[A]_0}{2k}$

• First Order:

  • Rate Law: $rate = k[A]$
  • Integrated Rate Law: $ln[A]_t = -kt + ln[A]_0$
  • Plot for Straight Line: $ln[A]_t$ vs t
  • Slope: -k
  • Half-life: $t_{1/2} = \frac{0.693}{k}$

• Second Order:

  • Rate Law: $rate = k[A]^2$
  • Integrated Rate Law: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
  • Plot for Straight Line: $\frac{1}{[A]_t}$ vs t
  • Slope: k
  • Half-life: $t_{1/2} = \frac{1}{k[A]_0}$

• Second Order:

  • Rate Law: $rate = k[A][B]$
  • Integrated Rate Law: $\frac{1}{[A]_0 - [B]_0}ln\frac{[B]_0[A]_t}{[A]_0[B]_t} = kt$

• nth Order ($n \neq 1$):

  • Rate Law: $rate = k[A]^n$
  • Integrated Rate Law: $\frac{1}{(n-1)[A]_t^{n-1}} = \frac{(n-1)kt}{[A]_0^{n-1}}$
  • Plot for Straight Line: $\frac{1}{[A]_t^{n-1}}$ vs t
  • Slope: $(n-1)k$
  • Half-life: $t_{1/2} = \frac{2^{n-1}-1}{k(n-1)[A]_0^{n-1}}$

Arrhenius Equation

• The Arrhenius equation describes the temperature dependence of the rate constant:

$k = Ae^{-\frac{E_a}{RT}}$

where k is the rate constant, A is the frequency factor, $E_a$ is the activation energy, R is the gas constant (8.314 J/mol·K), and T is the temperature in Kelvin.

• Can also be expressed as:

$ln \frac{k_2}{k_1} = \frac{E_a}{R} (\frac{1}{T_1} - \frac{1}{T_2})$

Algorithmic Game Theory

• Focuses on the computational aspects of games, mechanism design, social networks, and the internet.

What is Game Theory?

• Game theory involves mathematical models of agents' strategic interactions.
• Its origins lie in economics and the study of markets, firms, and consumers.
• Applied in social science, logic, systems science, and computer science.

Selfish Routing

• Selfish routing studies how traffic flows in a network when agents act selfishly.
• A network that is represented by a graph $G = (V, E)$., with 'e' being an edge.
• Each edge e has a cost function $l_e(x)$ for each unit of traffic.
• There are k commodities, each with a source $s_i$, destination $t_i$, and traffic rate $r_i > 0$.
• Total traffic represented as $r = \sum_{i=1}^k r_i$.

Definition

• A flow f is a specification of a rate $f_p \ge 0$ for each path P in the network.
• For each commodity i, the flow f satisfies $\sum_{P \in P_{s_i, t_i}} f_p = r_i$, where $P_{s_i, t_i}$ is the set of paths from $s_i$ to $t_i$
• The flow on edge e is $f_e = \sum_{P: e \in P} f_P$
• The cost on edge e is $l_e(f_e)$

Cost

• The cost of path P is $C_p(f) = \sum_{e \in P} l_e(f_e)$
• The social cost of flow f is $C(f) = \sum_{P \in P} f_p C_p(f) = \sum_{e \in E} f_e \cdot l_e(f_e)$

Wardrop Equilibrium

• A flow f is in Wardrop equilibrium if for any $s_i - t_i$ pair and any paths $P, P'$ in $P_{s_i, t_i}$ with $f_P > 0$, we have $C_P(f) \le C_{P'}(f)$

Price of Anarchy

• Price of Anarchy (PoA) quantifies network performance degradation due to selfish routing.

$$ PoA = \frac{\text{Social cost in the worst Nash equilibrium}}{\text{Optimal social cost}} $$

Algorithmic Complexity

• Centers on how resource needs (time or space) grow as input size increases
• A tool that predicts program performance and find design trade-offs

Key Aspect

• Ignores constant factors and lower-order terms.

Big O Notation

• $O(f(n))$ denotes the set of algorithms with resource needs growing no faster than $f(n)$, where n is the input size.
• $g(n) \in O(f(n))$ if there exist positive constants c and $n_0$ such that $g(n) \le cf(n)$ for all $n \ge n_0$

Common Complexities

• Constant: $O(1)$
• Logarithmic: $O(\log n)$
• Linear: $O(n)$
• N-Log-N: $O(n \log n)$
• Quadratic: $O(n^2)$
• Cubic: $O(n^3)$
• Polynomial: $O(n^k)$
• Exponential: $O(2^n)$
• Factorial/EXP: $O(n!)$ or $O(n^n)$

The Laplace Transform

• It is a transform technique that converts differnetial equations in the time domain to algebraic equations in the frequency domain
• Important for analyzing linear time-invariant systems

Definition

• For $t \geq 0$, the Laplace transform of $f(t)$, is defined by

$$ F(s) = \mathcal{L}{f(t)} = \int_{0}^{\infty} e^{-st} f(t) dt $$

provided that the integral converges

• Where s is a complex number

Examples

• $f(t) = 1$

$$ \mathcal{L}{1} = \int_{0}^{\infty} e^{-st} dt = \lim_{b \to \infty} \int_{0}^{b} e^{-st} dt = \lim_{b \to \infty} \left[ -\frac{1}{s} e^{-st} \right]{t=0}^{t=b} = \lim{b \to \infty} \left( -\frac{1}{s} e^{-sb} + \frac{1}{s} \right) = \frac{1}{s}, \quad s>0 $$

• $f(t) = e^{at}$:

$$ \mathcal{L}{e^{at}} = \int_{0}^{\infty} e^{-st} e^{at} dt = \int_{0}^{\infty} e^{-(s-a)t} dt = \lim_{b \to \infty} \int_{0}^{b} e^{-(s-a)t} dt = \lim_{b \to \infty} \left[ -\frac{1}{s-a} e^{-(s-a)t} \right]_{t=0}^{t=b} = \frac{1}{s-a}, \quad s>a $$

• $f(t) = t$:

$$ \mathcal{L}{t} = \int_{0}^{\infty} t e^{-st} dt $$

• Using integration by parts:

$$ \begin{array}{ll} u = t & dv = e^{-st} dt \ du = dt & v = -\frac{1}{s} e^{-st} \end{array} $$

$$ \mathcal{L}{t} = \left[ -\frac{t}{s} e^{-st} \right]{0}^{\infty} + \frac{1}{s} \int{0}^{\infty} e^{-st} dt = 0 + \frac{1}{s} \mathcal{L}{1} = \frac{1}{s} \cdot \frac{1}{s} = \frac{1}{s^2}, \quad s>0 $$

• $f(t) = \sin(at)$:

$$ \mathcal{L}{\sin(at)} = \int_{0}^{\infty} \sin(at) e^{-st} dt $$

• Using integration by parts twice:

$$ \begin{array}{ll} u = \sin(at) & dv = e^{-st} dt \ du = a \cos(at) dt & v = -\frac{1}{s} e^{-st} \end{array} $$

$$ \mathcal{L}{\sin(at)} = \left[ -\frac{1}{s} e^{-st} \sin(at) \right]{0}^{\infty} + \frac{a}{s} \int{0}^{\infty} \cos(at) e^{-st} dt = 0 + \frac{a}{s} \int_{0}^{\infty} \cos(at) e^{-st} dt $$

• Again, integration by parts:

$$ \begin{array}{ll} u = \cos(at) & dv = e^{-st} dt \ du = -a \sin(at) dt & v = -\frac{1}{s} e^{-st} \end{array} $$

$$ \mathcal{L}{\sin(at)} = \frac{a}{s} \left( \left[ -\frac{1}{s} e^{-st} \cos(at) \right]{0}^{\infty} - \frac{a}{s} \int{0}^{\infty} \sin(at) e^{-st} dt \right) = \frac{a}{s} \left( \frac{1}{s} - \frac{a}{s} \mathcal{L}{\sin(at)} \right) $$

$$ \mathcal{L}{\sin(at)} = \frac{a}{s^2} - \frac{a^2}{s^2} \mathcal{L}{\sin(at)} $$

$$ \left( 1 + \frac{a^2}{s^2} \right) \mathcal{L}{\sin(at)} = \frac{a}{s^2} $$

$$ \mathcal{L}{\sin(at)} = \frac{a}{s^2 + a^2}, \quad s>0 $$

Exoplanets

• Planets that orbit other stars.

Detection Methods

• Radial Velocity
• Transit
• Direct Imaging
• Astrometry
• Microlensing

Kepler Mission

• NASA mission that discovered thousands of exoplanets using transits

Radial Velocity (Doppler Spectroscopy)

• Star and planet orbit around a common center of mass
• Star's motion causes a doppler shift in it's spectrum

Diagram

• A diagram illustrating a star with a planet orbiting it
• The star's wobble is shown along with the resulting blueshift (towards) and redshift(away) in it's spectrum
• Equation: $v_{star} = (m_{planet} / m_{star}) * v_{planet}$

Advantages

• Can determine planet's mass
• Relatively easy to implement

Disadvantages

• More massive plants closer to the star easier to detect
• Sensitive ot the angle of inclination

Transit Photometry

• Planet pauses in front of the star, causing a dip in its brightness.

Diagram

• Light curve of a star with a transiting planet shows the planet's orbit from the star perspective
• A dip in brightness is visible when the planet transits.
• Equation Transit depth: $\Delta F = (R_{planet} / R_{star})^2$

Advantages

• Can determine planet's size
• Can study planet's atmosphere during transit

Disadvantages

• Smaller planets harder to detect
• High false positive rate
• Requires a specific orbit.

Direct Imagining

• Use telescopes to directly observe a planet

Diagram

• An image of a star with a faint planet nearby
• Uses a coronagraph to block light from the sun

Advantages

• Study atmosphere and surface features
• Can determine planet's orbital parameters

Disadvantages

• Easier to detect young, hot and massive planets
• Requires high-resolution telescopes and techniques to block the star's light

Laplace Distribution

• Also known as the double exponential distribution
• Defined by two exponential distributions stitched together

Probability Density Function (PDF)

• Is defined for each value of x for a distribution
• Given by

$f(x \mid \mu, b) = \frac{1}{2b} \exp \left( -\frac{|x - \mu|}{b} \right)$

- $x$ is a random variable
- $\mu$ is a location parameter
- $b > 0$ is a scale parameter

Cumulative Density Function (CDF)

• Used to describe the probability that a random variable that is less than or equal to a certain variable number
• Given by

$F(x \mid \mu, b) = \begin{cases} \frac{1}{2} \exp \left( \frac{x - \mu}{b} \right) & \text{if } x < \mu \ 1 - \frac{1}{2} \exp \left( -\frac{x - \mu}{b} \right) & \text{if } x \geq \mu \end{cases}$

Mean and Median

• The mean and median of the Laplace distribution are both equal to the location parameter $\mu$
• Expressed as:

$E[X] = \mu$ $Median(X) = \mu$

Variance

• Measures how far a random variable is from its expected variable
• Given by:

$Var(X) = 2b^2$

Standard Deviation

• Measures how spread out a distribution is
• Given by:

$\sigma = \sqrt{2b^2} = b\sqrt{2}$

Characteristic Function

• Fully defines probability for random variable
• Given by:

$\phi(t) = \frac{e^{i\mu t}}{1 + b^2t^2}$

Applications

• Widely used for numerous applications, including ML
• Data with heavier tails
• Modeling financial data
• Signal Processing applications

Relation to Other Distributions

• Exponential Distribution: The absolute difference between a Laplace-distributed random variable and its mean follows an exponential distribution.
• Normal Distribution: While both are symmetric distributions, the Laplace distribution has heavier tails than the normal distribution.

Eigenvalues and Eigenvectors

Motivation

Let $A \in \mathbb{R}^{n \times n}$. Solve:

$\qquad \dot{x}(t) = A x(t), \quad x(t) \in \mathbb{R}^n, \quad t \in \mathbb{R}$

• If A is a diagonal matrix, then:

$\qquad x(t) = \begin{bmatrix} e^{\lambda_1 t} & & \ & \ddots & \ & & e^{\lambda_n t} \end{bmatrix} x(0)$

• Is A able to be "diagonalized?"
• Suppose there exists $v \neq 0$ such that

$\qquad A v = \lambda v, \quad \lambda \in \mathbb{R}$

then v is an eigenvector of A, and $\lambda$ is an eigenvalue of A.

Definition

Let $A \in \mathbb{R}^{n \times n}$. If there exists $v \neq 0$ such that:

$\qquad A v = \lambda v, \quad \lambda \in \mathbb{R}$

then v is called an eigenvector of A, and $\lambda$ is called an eigenvalue of A.

• Finding $\lambda$ and $v$:

$\qquad A v = \lambda v \Rightarrow (A - \lambda I) v = 0$

$(A - \lambda I)$ is singular:

$\qquad \text{det}(A - \lambda I) = 0$

$\text{det}(A - \lambda I)$ is a polynomial of degree $n$ in $\lambda$, called the characteristic polynomial of A.

Examples

• A = $\begin{bmatrix} 2 & 1 \ 1 & 2 \end{bmatrix}$

  • $\qquad \text{det}(A - \lambda I) = \text{det} \begin{bmatrix} 2-\lambda & 1 \ 1 & 2-\lambda \end{bmatrix} = (2-\lambda)^2 - 1 = \lambda^2 - 4\lambda + 3 = (\lambda - 1)(\lambda - 3) = 0$
  • $\qquad \Rightarrow \lambda_1 = 1, \quad \lambda_2 = 3$
  • For $\lambda_1 = 1$:
    • $\qquad A - \lambda_1 I = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}$:

• A = $\begin{bmatrix} -i & -1 \ 1 & -i \end{bmatrix}$ - $\qquad \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \begin{bmatrix} v_1 \ v_2 \end{bmatrix} = 0 \Rightarrow v_1 + v_2 = 0 \Rightarrow v = \begin{bmatrix} 1 \ -1 \end{bmatrix}$

  • For $\lambda_2 = 3$:
    • $\qquad A - \lambda_2 I = \begin{bmatrix} -1 & 1 \ 1 & -1 \end{bmatrix}$
      • $\qquad \begin{bmatrix} -1 & 1 \ 1 & -1 \end{bmatrix} \begin{bmatrix} v_1 \ v_2 \end{bmatrix} = 0 \Rightarrow v_1 = v_2 \Rightarrow v = \begin{bmatrix} 1 \ 1 \end{bmatrix}$

• A = $\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}$

  • $\qquad \text{det}(A - \lambda I) = \text{det} \begin{bmatrix} -\lambda & -1 \ 1 & -\lambda \end{bmatrix} = \lambda^2 + 1 = 0$
  • $\qquad \Rightarrow \lambda = \pm i$
  • For $\lambda = i$:
    • $\qquad A - \lambda I = \begin{bmatrix} -i & -1 \ 1 & -i \end{bmatrix}$
      • $\qquad v_1 = i v_2 \Rightarrow v = \begin{bmatrix} i \ 1 \end{bmatrix}$
  • For $\lambda = -i$:
    • $\qquad A - \lambda I = \begin{bmatrix} -i & -1 \ 1 & -i \end{bmatrix}$
      • $\qquad \Rightarrow v_1 = -i v_2 \Rightarrow v = \begin{bmatrix} -i \ 1 \end{bmatrix}$

• A = $\begin{bmatrix} 2 & 1 \ 0 & 2 \end{bmatrix}$

  • $\qquad \text{det}(A - \lambda I) = \text{det} \begin{bmatrix} 2-\lambda & 1 \ 0 & 2-\lambda \end{bmatrix} = (2-\lambda)^2 = 0$
  • $\qquad \Rightarrow \lambda = 2$ (multiplicity 2)
  • $\qquad A - \lambda I = \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}$
  • $\qquad \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \begin{bmatrix} v_1 \ v_2 \end{bmatrix} = 0 \Rightarrow v_2 = 0 \Rightarrow v = \begin{bmatrix} 1 \ 0 \end{bmatrix}$

Properties

Let $A \in \mathbb{R}^{n \times n}$.

1. The sum of the eigenvalues of $A$ is equal to the trace of $A$, i.e $\sum_{i=1}^n \lambda_i = \text{tr}(A) = \sum_{i=1}^n A_{ii}$.
2. The product of the eigenvalues of $A$ is equal to the determinant of $A$, i.e. $\prod_{i=1}^n \lambda_i = \text{det}(A)$.
3. If $A$ is symmetric, then all eigenvalues are real.
4. If $A$ is symmetric and $\lambda_1 \neq \lambda_2$, then the corresponding eigenvectors $v_1$ and $v_2$ are orthogonal, i.e. $v_1^T v_2 = 0$.

Explore chemical kinetics: the study of reaction rates, rate laws, and integrated rate laws. Understand how reactant concentrations change over time. Learn about zero order reactions.