Partial Molar Properties in Thermodynamics | Chemical Engineering

Mole fraction of a component in a mixture is calculated as: Mole Fraction = Moles of component / Total moles in mixture. The sum of all mole fractions in a mixture equals 1. For example, if a mixture has 0.3 moles of ethanol and 0.7 moles of water, the mole fraction of ethanol is 0.3.

Total moles = moles of benzene + moles of toluene = 0.77 + 0.43 = 1.2 mol. Mole fraction of benzene = 0.77 / 1.2 ≈ 0.64, and mole fraction of toluene = 0.43 / 1.2 ≈ 0.36. Mole fractions must sum to 1.

Calculate total moles by adding moles of all components: 0.26 + 0.307 = 0.567 mol. Mole fraction of heptane = 0.26/0.567 ≈ 0.457. Mole fraction of octane = 1 - 0.457 = 0.543. These mole fractions are essential for applying Raoult's Law to find partial vapor pressures.

Mole fraction is the ratio of the number of moles of a component to the total number of moles in a mixture. It is denoted by the symbol X. For a mixture with components A, B, and C, the mole fractions are: X_A = n_A / (n_A + n_B + n_C), X_B = n_B / (n_A + n_B + n_C), X_C = n_C / (n_A + n_B + n_C). The sum of all mole fractions in a mixture always equals 1. Mole fraction is dimensionless.

Mole fraction is the ratio of moles of a component to total moles in a mixture. For a mixture with components A, B, C: X_A = n_A/(n_A+n_B+n_C), X_B = n_B/(n_A+n_B+n_C), X_C = n_C/(n_A+n_B+n_C). The sum of all mole fractions equals 1. Each mole fraction is always between 0 and 1 (exclusive), as it represents a component's proportion in the mixture.

This section establishes the foundational concepts of multivariable calculus. Second partial derivatives with respect to a single variable have clear geometric interpretations as concavity in cross-sections. Mixed second partial derivatives (FXY, FYX) fundamentally differ from single-variable partials, measuring how surfaces twist as one variable changes while tracking slopes in another direction. Partial derivatives should be conceptualized as rates of change rather than slopes. The section demonstrates that partial derivatives are essential computational tools for describing natural phenomena: Laplace's equation for electric potential, the heat equation for heat flow, the wave equation for wave propagation, and the Schrödinger equation for quantum mechanics. These physical applications illustrate why partial derivatives, despite being computationally convenient, are indispensable for modeling physical systems.

Partial derivatives extend the concept of derivative to functions of multiple variables. For f(x,y), the partial derivative with respect to x at (x₀,y₀) is found by fixing y = y₀, reducing f to a function of x, then differentiating. Notation includes f_x or ∂f/∂x. To compute partial derivatives, treat all other variables as constants and apply standard differentiation rules. For composite functions, use the chain rule by differentiating from outside to inside. Partial derivatives represent rates of change in physical contexts—for temperature distribution, the partial derivative with respect to x measures how temperature changes when moving horizontally. The value 4°C/cm means temperature increases by 4°C for every 1 cm increase in x-direction. Partial derivatives depend on both the starting point and the direction of movement. Moving horizontally from (1,2) gives one rate of change, while moving vertically from the same point gives a different rate. The units of a partial derivative are the units of the function divided by the units of the variable being differentiated. When exact function expressions are unavailable, derivatives can be estimated using the average rate of change: Δf/Δx.

Partial derivatives measure how a function changes with respect to one variable while treating all other variables as constants. For z = e^(xy) + x²y³ - ln(y), the partial derivative with respect to x is ∂z/∂x = ye^(xy) + 2xy³ (treating y as constant). The partial derivative with respect to y is ∂z/∂y = xe^(xy) + 3x²y² - 1/y (treating x as constant). The partial derivative symbol ∂ distinguishes this from regular derivatives. This concept extends calculus to three-dimensional space and is fundamental for understanding multivariable functions.

Partial derivatives are used when a function depends on multiple variables. For y = f(x, u, v), ∂y/∂x means differentiate with respect to x while treating u and v as constants. Example: For u = x²y + 2xy + 4y², ∂u/∂x = 2xy + 2y (treating y as constant), and ∂u/∂y = x² + 2x + 8y (treating x as constant). Partial derivatives are essential in physics, particularly in potential energy chapters.

Partial differentiation is used for functions of multiple variables. When differentiating with respect to one variable (say x), treat all other variables (y, z) as constants. For example, if f(x, y, z) = x² + 2xy + 4yz, then ∂f/∂x = 2x + 2y (treating y and z as constants). Similarly, ∂f/∂y = 2x + 4z and ∂f/∂z = 4y. Partial derivatives are essential in physics for work, power, energy, and electrostatics.

This section presents the fundamental laws of thermodynamics governing energy transformations. The first law states energy cannot be created or destroyed in isolated systems (ΔU = Q - W). The second law states total entropy of isolated systems always increases for spontaneous processes, explaining why heat flows from hot to cold bodies and why perpetual motion is impossible. The third law states entropy approaches minimum constant value as temperature approaches absolute zero (0 K), and absolute zero cannot be reached in finite steps. Entropy measures randomness/disorder (ΔS = q_rev/T). Gibbs free energy (ΔG = ΔH - TΔS) predicts spontaneity. State functions (U, H, G, S) depend only on state, while path functions (Q, W) depend on process.

Thermodynamics uses three important state functions to describe energy changes during transformations: Gibbs free energy (G), enthalpy (H), and entropy (S). These are related by the equation ΔG = ΔH - TΔS, where Δ represents the change between final and initial states. State functions depend only on the current state of the system, not on the path taken to reach that state.

Thermodynamic state functions are properties whose values depend only on the current state of the system, not on the path taken to reach that state. Internal energy (U) and enthalpy (H) are examples of thermodynamic state functions. Similarly, entropy (S) and Gibbs free energy (G) are also thermodynamic state functions, meaning their values are determined solely by the system's state.

Entropy (S) is a state function measuring system disorder with units J/K·mol, classified as extensive. The Second Law states spontaneous processes increase universal entropy. Thermodynamics addresses energy changes and reaction feasibility, while kinetics handles reaction rates. The relationship ΔH = ΔU + PΔV (or ΔH = ΔU + ΔnRT for gases) connects enthalpy and internal energy. Systems are classified as open (energy and matter transfer), closed (only energy transfer), or isolated (no transfer). State functions depend only on initial and final states: P, V, T, G, S, H, U, n. Path functions depend on process path: work (W), heat (q), heat capacity. For ideal gases, Cp - Cv = R and Cp/Cv = γ. Properties are extensive (mass, volume, enthalpy) or intensive (density, temperature, pressure). The relationship between standard Gibbs free energy and equilibrium constant is ΔG° = -RT ln K. Spontaneity is determined by entropy change: ΔS_total > 0 for spontaneous processes, ΔS_total = 0 at equilibrium, and ΔS_total < 0 for non-spontaneous processes.

Thermodynamics is the study of relationships between heat, work, temperature, and energy. State functions are properties whose values depend only on the current state of the system, not on the path taken. The four main state functions are Internal Energy, Entropy, Enthalpy, and Gibbs Free Energy. These functions have unique values for any given state of a substance. Understanding these foundational concepts is essential before studying the laws of thermodynamics.