Chemical Kinetics: Rates and Influences

Chemical Kinetics

• Chemical kinetics studies the rates of chemical reactions.
• Reaction rate defines how fast a chemical reaction occurs.
• All reactions proceed at a certain speed.

Factors Affecting Reaction Rates

• Reaction rates are influenced by:
  • Concentration of reactants.
  • Temperature.
  • Catalysts.

Reactant Concentration

• Increasing reactant concentration leads to more molecular collisions.
• Increased concentration = increased reaction rate

Temperature

• Higher temperature = increased kinetic energy and collision frequency.
• Increased temperature = increased reaction rate.

Catalysts

• Catalysts accelerate reactions without being consumed.

Rate Laws

• Rate laws (or rate equations) relate reaction rate to the rate constant and reactant concentrations raised to certain powers.
• For a reaction $aA + bB \longrightarrow cC + dD$, the rate law is Rate = $k[A]^m[B]^n$.
  • k: rate constant
  • m & n: reaction orders

Reaction Order

• Reaction order is the sum of the exponents of reactant concentrations in the rate law.
• m: reaction order with respect to A
• n: reaction order with respect to B
• m + n: overall reaction order

Types of Rate Laws

• Differential Rate Law: Describes rate dependence on concentration.
• Integrated Rate Law: Describes concentration dependence on time.

First-Order Reactions

• Reaction rate depends on the concentration of one reactant raised to the first power.
• For $A \longrightarrow \text{Product}$, Rate = $-\frac{\Delta[A]}{\Delta t} = k[A]$.
  • k: first-order rate constant

Integrated Rate Law

$$ \ln[A]_t - \ln[A]_0 = -kt $$

$$ \ln\frac{[A]_t}{[A]_0} = -kt $$

$$ \ln[A]_t = -kt + \ln[A]_0 $$

• $[A]_t$: concentration of A at time t
• $[A]_0$: initial concentration of A at t=0

Half-Life

• Half-life is the time needed for a reactant's concentration to halve its initial value.
• At $t = t_{1/2}$, $[A]_t = \frac{1}{2}[A]_0$.

$$ \ln\frac{\frac{1}{2}[A]_0}{[A]0} = -kt{1/2} $$

$$ \ln\frac{1}{2} = -kt_{1/2} $$

$$ \ln0.5 = -kt_{1/2} $$

$$ t_{1/2} = \frac{\ln2}{k} = \frac{0.693}{k} $$

• The half-life of a first-order reaction is constant.

Second-Order Reactions

• The reaction rate depends on the concentration of one reactant raised to the second power or on the concentrations of two different reactants, each raised to the first power.
• For $A \longrightarrow \text{Product}$, Rate = $-\frac{\Delta[A]}{\Delta t} = k[A]^2$.
  • k is the second-order rate constant

Integrated Rate Law

$$ \frac{1}{[A]_t} = kt + \frac{1}{[A]_0} $$

• $[A]_t$: concentration of A at time t
• $[A]_0$: initial concentration of A at t=0

Half-Life

• At $t = t_{1/2}$, $[A]_t = \frac{1}{2}[A]_0$.

$$ \frac{1}{\frac{1}{2}[A]0} = kt{1/2} + \frac{1}{[A]_0} $$

$$ \frac{2}{[A]_0} - \frac{1}{[A]0} = kt{1/2} $$

$$ \frac{1}{[A]0} = kt{1/2} $$

$$ t_{1/2} = \frac{1}{k[A]_0} $$

• The half-life of a second-order reaction is not constant.

Zero-Order Reactions

• The reaction rate is independent of reactant concentrations.
• For $A \longrightarrow \text{Product}$, Rate = $-\frac{\Delta[A]}{\Delta t} = k$.
  • k is the zero-order rate constant

Integrated Rate Law

$$ [A]_t = -kt + [A]_0 $$

• $[A]_t$: concentration of A at time t
• $[A]_0$: initial concentration of A at t=0

Half-Life

• At $t = t_{1/2}$, $[A]_t = \frac{1}{2}[A]_0$.

$$ \frac{1}{2}[A]0 = -kt{1/2} + [A]_0 $$

$$ kt_{1/2} = [A]_0 - \frac{1}{2}[A]_0 $$

$$ kt_{1/2} = \frac{1}{2}[A]_0 $$

$$ t_{1/2} = \frac{[A]_0}{2k} $$

• The half-life of a zero-order reaction is not constant.

Collision Theory

Collision Theory

• Molecules must collide to enable a reaction.
• Higher concentration = more collisions
• Higher temperature = more collisions with greater energy.
• Molecules must collide with sufficient energy.

Activation Energy

• Activation energy is the minimum energy for a chemical reaction to start.
• An energy barrier needs overcoming.

Transition State

• Species at the peak of the energy barrier during an elementary reaction.
• Also known as the activated complex.
• Neither reactant nor product.

Arrhenius Equation

• It is a mathematical connection between rate constant, activation energy, and temperature.

$$ k = Ae^{-E_a/RT} $$

• k: rate constant
• $E_a$: activation energy (J/mol)
• R: gas constant (8.314 J/K·mol)
• T: absolute temperature (K)
• A: frequency factor

Frequency Factor

• Rate of collision occurrence.
• Consists of an orientation factor.

How Ea Affects Rate Constants

• Small Ea: larger k (faster reaction).
• Large Ea: smaller k (slower reaction).

Two-Point Form

• Used when two rate constants are known at two temperatures.

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

Reaction Mechanisms

Reaction Mechanism

• The order of elementary steps producing product creation.

Elementary Step

• A reaction occurring in a single step.

Reaction Intermediates

• Species present in a reaction mechanism but not the final balanced equation.

Molecularity

• Count of molecules reacting in an elementary step.
  • Unimolecular: one molecule
  • Bimolecular: two molecules
  • Termolecular: three molecules

Rate-Determining Step

• The slowest step in a reaction mechanism.
• Controls the rate of the overall reaction.

Catalysis

• A substance that increases the rate of a chemical reaction without being consumed.
• Provides a mechanism with lower activation energy.
• Catalysts: homogenous or heterogeneous.

Homogenous Catalysis

• The catalyst exists in the same phase as the reactants.

Heterogeneous Catalysis

• The catalyst exists in a different phase as the reactants.
• Usually involves gaseous reactants adhering to a solid catalyst's surface.

Null Hypothesis Significance Testing

• Focuses on determining the likelihood of obtaining observed results if the null hypothesis is true.

P-values

• The probability of getting test results as extreme as observed, assuming the null hypothesis is correct.
• A small P-value indicates deviations from the null hypothesis.
• Example:
  • Coin flipping experiment: 10 flips, 9 heads.
    • $H_0$: the coin is fair, i.e. $p = 0.5$
    • $H_A$: the coin is not fair, i.e. $p \neq 0.5$
    • $$ P = P(X \geq 9 \text{ or } X \leq 1 | p = 0.5) = \binom{10}{9}(0.5)^9(0.5)^1 + \binom{10}{1}(0.5)^1(0.5)^9 \approx 0.02 $$

Significance Level

• Denoted as $\alpha$.
• The probability of wrongly rejecting the null hypothesis if it's true.
• A significance level of 0.05 means a 5% risk of detecting a difference where none exists.

Type I and Type II Errors

• Type I error is a false positive.
• Type II error is a false negative.

Statistical Power

• It is the probability to reject the null hypothesis when it's wrong.
• Power can be improved by increasing sample size (n), significance level ($\alpha$), or reducing standard deviation ($\sigma$).
• Power = $1 - \beta$, where $\beta$ represents Type II error probability.
• Higher power lowers the chances of committing a Type II error.

Multiple Testing

• Running more tests raises the probability of Type I errors.
• Bonferroni correction: Multiple comparison technique to adjust for conducting many tests and reduces false positives.
• For n independent tests, evaluate each hypothesis at a level of $\alpha/n$ instead of $\alpha$

t-test

One sample t-test

• Used to test the mean of one population versus known or hypothesized value.
• Assumptions:
  • Data is independent.
  • Data is approximately normally distributed.
• Test statistic: $$ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} $$ where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $s$ is the sample standard deviation, and $n$ is the sample size.
• Degrees of freedom: $df = n - 1$.

Two sample t-test

• Used to compare the means of two populations.

  • Assumptions:
    • Data independence.
    • Approximate normal distribution of both datasets.
    • Equal variances within both populations.

• Test statistic: $$ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_p^2(1/n_1 + 1/n_2)}} $$ where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $n_1$ and $n_2$ are the sample sizes, and $s_p^2$ is the pooled variance.

• Pooled variance: $$ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} $$ where $s_1^2$ and $s_2^2$ are the sample variances.

• Degrees of freedom: $df = n_1 + n_2 - 2$.

Paired t-test

• Compare means from two related/paired groups.

  • Assumptions:
    • Data independence.
    • Approximate normality of the dataset.

• Test statistic: $$ t = \frac{\bar{d}}{s_d/\sqrt{n}} $$ where $\bar{d}$ is the sample mean of the differences, $s_d$ is the sample standard deviation of the differences, and $n$ is the sample size.

• Degrees of freedom: $df = n - 1$.

$\chi^2$ Test

Goodness of Fit

• Test for how well observed data matches expected distributions.

  • $H_0$: the observed distribution matches the expected distribution.
  • $H_A$: the observed distribution does not match the expected distribution.

• Test statistic: $$ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $$ where $O_i$ is the observed frequency and $E_i$ is the expected frequency.

• Degrees of freedom: $df = k - 1$, where $k$ is the number of categories.

Independence

• Assesses whether two categorical variables are independent.

  • $H_0$: the two categorical variables are independent.
  • $H_A$: the two categorical variables are not independent.

• Test statistic: $$ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} $$ where $O_{ij}$ is the observed frequency and $E_{ij}$ is the expected frequency.

• Expected frequency: $$ E_{ij} = \frac{(\text{row total})(\text{column total})}{\text{grand total}} $$

• Degrees of freedom: $df = (r - 1)(c - 1)$, where $r$ is the number of rows and $c$ is the number of columns.

Algorithmic Game Theory

Definition

• Uses game theory and algorithm design to create and study systems wherein strategic agents interact.
• Game Theory: The study of strategic decision making, outcome influenced by multiple player choices.
• Algorithm Design: Effective and efficient data algorithms for computing problems.

Examples

• Sponsored search: selling ad slots to advertisers using auctions.
• Network routing: users optimizing travel paths by minimizing time taken.
• Online Markets: sellers setting up prices while buyers make decisions.

Selfish Routing

• Model:
  • A network is described as a directed graph $G = (V, E)$.
  • Each edge $e \in E$ has a cost function $c_e(x)$, the cost per unit of traffic and edge $e$ gets heavier when this traffic rises to $x$.
  • There are $k$ items with a supply station, a place of end and a rate the stuff goes from station to destination.

Flows

• A flow $f_i$ for good $i$ distributes the stuff from station to destination.
• A flow transports stuff between stations and destinations just between stations.
• The total stream on an edge $e$ can be denoted as $f_e = \sum_{i=1}^{k} f_i(e)$, with the flow of that edge between stations.

Social Cost

• Total spent by users

Wardrop Equilibrium

• $$ \sum_{e \in P} c_e(f_e) \le \sum_{e \in P'} c_e(f_e) $$
• A path from A to B cannot cost less than an alternate route