Physical Chemistry Math Review: Foundational Topics and Skills

The instructor introduces the Calculus of One Variable course, explaining it will be taught in a friendly manner. The course covers limits, continuity, derivatives, and integrals. The instructor emphasizes that limits serve as the foundation for understanding continuity, derivatives, and integrals, making them essential for the entire course. The syllabus contains thematic contents developed throughout all sessions, with the first unit covering limits and continuity, the second unit focusing on derivatives (described as the backbone of the course), and the third unit covering integrals including improper integrals useful for statistics.

Limits are the fundamental concept that serves as the gateway to calculus. Before understanding derivatives, integrals, or other calculus concepts, one must first grasp the concept of limits. Limits define continuity, derivatives, and integrals, making them the essential starting point for all calculus study.

Calculus is built upon three core concepts: limits evaluate what happens to a function as x approaches a certain value, even when the function is undefined at that point; derivatives are functions that give the slope of the tangent line at any point, representing the instantaneous rate of change; and integration is the reverse process of differentiation, finding the anti-derivative and calculating the area under a curve or accumulated change over time. These three concepts work together to analyze how quantities change and accumulate.

Calculus is divided into three parts: Limits (සීමාව), Derivatives (අවකලනය), and Integrals (අනුකලනය). A limit describes the value a function approaches as the independent variable approaches a specific value. For f(x) = x + 4, as x approaches 3, f(x) approaches 7. Limits can be approached from left (x < 3) or right (x > 3) sides. This concept forms the foundation for understanding derivatives and integrals in calculus.

Calculus extends algebraic capabilities through three fundamental topics: limits, derivatives, and integrals. Limits enable discussion of function behavior as inputs approach values, even when undefined at those points. Derivatives allow calculation of curve slopes at specific points, unlike constant slopes of straight lines. Integrals generalize area calculation from simple geometric shapes to arbitrary curved regions between two x-values. Together, these concepts form the foundation of single-variable calculus, enabling analysis of changing quantities and accumulation processes that algebra alone cannot address.

This section covers thermodynamics and equilibrium: (1) First Law of Thermodynamics - energy is conserved, ΔU = Q + W; (2) Work at constant pressure - W = -PΔV; (3) Enthalpy - H = U + PV, with ΔH = Qp at constant pressure; (4) Heat capacity - C = Q/ΔT, with Cp - Cv = R for gases; (5) Hess's Law - total enthalpy change is path-independent; (6) Born-Haber cycle - applies Hess's Law to calculate lattice energy; (7) Entropy - measures disorder, ΔS = q_rev/T; (8) Second Law - total entropy of universe increases for spontaneous processes (ΔS_universe > 0); (9) Third Law - entropy of perfect crystal at 0 K is zero; (10) Gibbs free energy - G = H - TS, determines spontaneity (ΔG < 0 spontaneous, ΔG > 0 non-spontaneous, ΔG = 0 equilibrium); (11) ΔG° = -RT ln K connects free energy to equilibrium constant; (12) Chemical equilibrium - forward and reverse rates equal; (13) Equilibrium constant - Kc = [products]^coefficients/[reactants]^coefficients; (14) Kp = Kc(RT)^(Δn) for gases; (15) Le Chatelier's principle - systems shift to counteract disturbances (concentration, pressure, temperature).

This section covers thermodynamics and gas laws: (1) Work units include Joule, kg·m²·s⁻², and erg (1 J = 10⁷ erg); (2) One calorie equals 4.184 Joules; (3) First Law of Thermodynamics: ΔE = q + w; (4) Enthalpy is denoted by H; (5) Solution enthalpy is enthalpy change when dissolving one mole of solid; (6) Gibbs free energy is denoted by G; (7) At equilibrium, ΔG = 0; (8) Secondary forces include six types: London dispersion, dipole-dipole, dipole-induced dipole, ion-dipole, ion-induced dipole, and ion-ion forces; (9) Neutron half-life is approximately 10 minutes; (10) Gas to solid conversion is called deposition; (11) Volume SI unit is cubic meter (m³); (12) One atmosphere equals 1.013 × 10⁵ Pascals.

This extensive section covers thermodynamic principles and equilibrium chemistry: (1) Ideal gas equation PV = nRT with gas constant R = 0.0821 L·atm/mol·K; (2) Gas laws including Boyle's, Charles's, Gay-Lussac's, Avogadro's, and Dalton's partial pressures; (3) Thermodynamic systems classification (open, closed, isolated) and properties (state vs path functions, extensive vs intensive); (4) Thermodynamic processes (isothermal, isochoric, isobaric, adiabatic, cyclic) and reversibility; (5) Internal energy, first and second laws of thermodynamics, third law (S = 0 at absolute zero); (6) Gibbs free energy (G = H - TS) and bond enthalpy calculations; (7) Chemical equilibrium concepts including equilibrium constant, reaction quotient, Le Chatelier's principle, and degree of dissociation.

This comprehensive section establishes the foundational concepts of thermodynamics. The instructor defines thermodynamics as the science studying energy and its transformations, with the goal of converting thermal energy from fuel combustion into useful forms like mechanical energy. Key topics include temperature, calorimetry, enthalpy, Hess's Law, entropy, and Gibbs Free Energy. The instructor explains that thermodynamics does not consider time or reaction speed (that's kinetics). The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed. The section covers three fundamental thermodynamic terms: System, Surroundings, and Universe. Systems are classified into Open, Closed, and Isolated types based on their interaction with surroundings. The instructor distinguishes between heat (Q) as energy transfer measured in joules, and temperature (T) as a measure of heat in degrees Celsius or Kelvin. The key formula Q = m × c × ΔT is explained with detailed examples. The instructor covers specific heat capacity versus heat capacity, and distinguishes between state functions (dependent only on initial and final states) and path functions (dependent on the specific path taken). Extensive properties (enthalpy, entropy, Gibbs free energy) depend on the amount of matter, while intensive properties (specific heat, density) do not. The section covers standard enthalpy of formation and standard enthalpy of combustion, explaining when ΔH_rxn° equals ΔH_f° or ΔH_c°.

This comprehensive section covers the foundational principles of thermodynamics. It begins with thermal equilibrium and the Zeroth Law, establishing temperature as a fundamental property. The Ideal Gas Equation (PV = nRT) and Van der Waals equation for real gases are explained, including dimensional analysis of constants 'a' and 'b'. The section details work done during volume changes (W = PΔV), internal energy as a state function, and different thermodynamic processes: isothermal, isobaric, isochoric, and adiabatic. It covers the First Law of Thermodynamics (ΔU = Q - W), sign conventions for heat and work, and the relationship between P-V diagram area and work. The Joule-Thomson effect and enthalpy (H = U + PV) are also explained.

The video demonstrates dimensional analysis: length, area, volume, and time can be combined: length²=area, length³=volume, length/time=velocity, length²/time=acceleration. It shows that dimensional analysis can be used to check equations: [L][T]⁻¹=[L][T]⁻¹. The video demonstrates coordinate systems: (x,y), (x,y,z), (x,y,z,t). It shows that coordinates can be transformed: (x,y)→(x+1,y+1), (x,y)→(x-1,y-1), (x,y)→(x+1,y-1), (x,y)→(x-1,y+1).

The Cartesian coordinate system consists of two perpendicular axes: the X-axis (horizontal) and Y-axis (vertical). The X-axis has positive values to the right and negative values to the left. The Y-axis has positive values upward and negative values downward. The origin (0,0) is where the axes intersect. To plot a point (X, Y): start at the origin, move horizontally by the X value (right for positive, left for negative), then move vertically by the Y value (up for positive, down for negative). To analyze a function: (1) Check surjectivity by comparing the range with the codomain; (2) Check injectivity by verifying no repeated outputs; (3) If both conditions are met, the function is bijective. Functions with repeated outputs (like f(x) = x²) are not injective. Functions where the range doesn't match the codomain are not surjective.

Coordinate systems use x, y, and z axes with positive/negative directions to locate points. The x-axis is positive right (east) and negative left (west); y-axis is positive up (north) and negative down (south); z-axis is positive toward viewer (out of page) and negative away (into page). The four quadrants are defined by combinations of positive/negative x and y values. Dimensional analysis uses symbols L (length), M (mass), and T (time) to represent physical quantities. Speed has dimensions L/T, acceleration has L/T². This method verifies equation correctness, simplifies factors, derives units for constants, and determines dimensions of unknown quantities. For Newton's Second Law (F = ma), force has dimensions ML/T². For pressure (P = F/A), pressure has dimensions M/(LT²).

The coordinate system (rectangular coordinate system) consists of two perpendicular lines: the horizontal x-axis and vertical y-axis intersecting at the origin. Each point has coordinates (x, y) written in parentheses. To plot a point, move horizontally by the x-value (right for positive, left for negative), then vertically by the y-value (up for positive, down for negative). The axes divide the plane into four quadrants: Quadrant I (top-right, x>0, y>0), Quadrant II (top-left, x<0, y>0), Quadrant III (bottom-left, x<0, y<0), and Quadrant IV (bottom-right, x>0, y<0). A function assigns exactly one output to each input and can be represented by ordered pairs, tables, graphs, or formulas. To evaluate a function, substitute the x-value into the formula. For f(x) = 2x² - 1, f(2) = 7, f(0) = -1, f(-1) = 1. For rational functions like g(x) = 2x/(x² - 1), substitute and simplify: g(-2) = -4/3, g(0) = 0, g(3) = 3/4.

A coordinate system (Koordinatensystem) is a visual tool used to represent Zuordnungen and functions. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), intersecting at the origin. Functions are plotted by connecting elements from the source set to elements in the target set. For example, plotting cola bottle quantities against prices creates points like (1, 90), (2, 180), (3, 270), (4, 360), and (5, 450). A valid function requires each source element connects to exactly one target element, with no element in the source set connecting to multiple elements in the target set.