Discrete Probability Distribution

• A random variable's value is a numerical outcome of a random phenomenon.
• A discrete random variable's value is a numerical outcome, taking either a finite or countably infinite set of values.
• A probability distribution for a discrete random variable lists probabilities for each possible value; also known as a probability mass function (PMF).

Requirements for Discrete Probability Distribution

• Each probability, P(x), falls between 0 and 1, inclusive; 0 ≤ P(x) ≤ 1.
• The sum of all probabilities equals 1; Σ P(x) = 1.

Mean of a Discrete Probability Distribution

• Calculated by μ = Σ [x * P(x)].

Variance of a Discrete Probability Distribution

• Calculated by σ² = Σ [(x - μ)² * P(x)].
• Alternative Calculation: σ² = Σ [x² * P(x)] - μ².

Standard Deviation of a Discrete Probability Distribution

• Calculated by σ = √(σ²).

Range Rule of Thumb

• Minimum usual value = μ - 2σ.
• Maximum usual value = μ + 2σ.

Rare Event Rule

• If the probability of an observed event is extremely small under a given assumption, the assumption is likely incorrect.

Identifying Unusual Results with Probabilities

• Unusually High: x successes in n trials, P(x or more) ≤ 0.05.
• Unusually Low: x successes in n trials, P(x or fewer) ≤ 0.05.

Binomial Experiment

• Defined by a fixed number of trials.
• Trials being independent.
• Each trial having two outcomes: success and failure.
• The probability of success being constant across trials.

Notation for Binomial Distributions

• P(S) = p (probability of success).
• P(F) = 1 - p = q (probability of failure).
• n = number of trials.
• x = number of successes in n trials.

Binomial Probability Formula

• P(x) = (n choose x) * p^x * q^(n-x) = nCx * p^x * q^(n-x).
• For x = 0, 1, 2,..., n

Mean for the Binomial Distribution

• μ = n * p.

Variance for the Binomial Distribution

• σ² = n * p * q.

Standard Deviation for the Binomial Distribution

• σ = √(n * p * q).

Range Rule of Thumb for Binomial Distribution

• Minimum usual value = μ - 2σ.
• Maximum usual value = μ + 2σ.

Poisson Distribution

• Discrete probability distribution for the number of occurrences of an event over a specified interval.
• x represents the number of event occurrences in the interval.

Requirements for the Poisson Distribution

• x is the number of occurrences over an interval.
• Occurrences are random and independent.
• Occurrences are uniformly distributed.

Notation for the Poisson Distribution

• P(x): Probability of x occurrences in an interval.
• μ: Mean number of occurrences in an interval.
• e: Euler's constant ≈ 2.71828.

Formula for the Poisson Distribution

• P(x) = (μ^x * e^(-μ)) / x!.

Mean for the Poisson Distribution

• μ.

Variance for the Poisson Distribution

• σ² = μ.

Standard Deviation for the Poisson Distribution

• σ = √μ.

Basic Concepts

• Dimensions such as mass, length, time, and temperature.
• Units measure dimensions (e.g., kg, m, s, °C).

Systems of Units

• SI: kg, m, s, K
• CGS: g, cm, s, °C
• American Engineering: lb, ft, s, °R

Mole and Molecular Weight

• Mole: Substance amount with the same number of entities as atoms in 12 g of carbon-12.
• Molecular weight: Mass of one mole of a substance.

Composition of Substances

• Mass fraction: Component mass / total mass.
• Mole fraction: Component moles / total moles.

Density and Specific Gravity

• Density: Mass per unit volume.
• Specific gravity: Substance's density relative to a reference substance (usually water).

Flow Rate

• Mass flow rate: Mass of substance flowing per unit time.
• Volumetric flow rate: Volume of substance flowing per unit time.

Chemical Reactions and Stoichiometry

• Stoichiometry: Quantitative relationships between reactants and products.
• Balancing chemical equations is essential.
• Limiting and excess reactants must be identified.
• Conversion, yield, and selectivity are key performance metrics.

Ideal Gas Law

• Key Equation: PV = nRT
• P = pressure, V = volume, n = moles, R = ideal gas constant, T = temperature.

Real Gases

• Deviate from ideal behavior at high pressures/low temperatures.
• Compressibility factor (Z): Z = PV/nRT

Vapor Pressure

• Pressure exerted by a vapor in equilibrium with its condensed phases.
• Clausius-Clapeyron equation: ln(P2/P1) = (ΔHvap/R) * (1/T1 - 1/T2)

Humidity

• Amount of water vapor in the air.
• Absolute humidity: Mass of water vapor / volume of air.
• Relative humidity: (Partial pressure of water vapor / vapor pressure of water) * 100%

Energy Balance

• First law of thermodynamics: Energy conservation.
• Enthalpy: H = U + PV
• Heat capacity: Cp = (∂H / ∂T)p
• Heat of reaction: Enthalpy change during a chemical reaction at constant pressure.

Phase Equilibrium

• Gibbs phase rule: F = C - P + 2
• F = degrees of freedom, C = components, P = phases.

Fluid Mechanics

• Viscosity is a resistance to flow.
• Newtonian vs non-Newtonian fluids differentiate flow behavior.
• Bernoulli equation: P + 1/2 * ρ * V^2 + ρ * g * h = constant.
• Describes relationships between pressure, velocity, and height in a fluid

Heat Transfer

• Conduction occurs through solids.
• Convection uses fluid motion.
• Radiation uses electromagnetic waves.

Mass Transfer

• Diffusion moves substances from high to low concentration areas.
• Mass transfer coefficient quantifies efficiency.
• Distillation, absorption, and extraction are important processes.

Kinetics

• Rate law relates reaction rate to reactant concentrations.
• Reaction order and activation energy impact reaction speed.
• Arrhenius equation: k = A * exp(-Ea/RT).

Reactor Design

• Types include batch, CSTR, and PFR.
• Design variables include conversion and space time.

Thermodynamics

• First law: Energy is conserved.
• Second law: Entropy increases.
• Third law: Entropy of a perfect crystal at absolute zero is zero

Thermodynamic Properties

• Enthalpy, internal energy and entropy.
• Gibbs free energy.

Phase Equilibrium

• Vapor-liquid equilibrium (VLE)

Distillation

• Vapor-liquid separation
• The performance is measured relative to volatility.
• Refux ratio and number of trays

Absorption

• Gas-liquid separation

Extraction

• Extratcion is a liquid seperation
• Selectivity and distribution coefficient

Control Systems

• Feedback control
• Feedforward control
• PID control

Instrumentation

• Control relies on instrumentation
• Instruments like sensors, transmitters, control valves

Hazard Identification

• Identified through HAZOP Study and What-If Analysis

Risk Assessment

• Requires Frequency and Consequence assessment

Safety Measures

• Measured with Prevention, Protection and Mitigation