♨️Thermodynamics of Fluids Unit 11 – Solution Thermodynamics
Solution thermodynamics explores how substances mix and interact in solutions. This field is crucial for understanding chemical processes in industries like pharmaceuticals, energy, and environmental science. It covers concepts like ideal and non-ideal solutions, partial molar properties, and phase equilibria.
Key principles include Gibbs free energy, entropy of mixing, and activity coefficients. These concepts help explain solution behavior, predict equilibrium states, and design separation processes like distillation and extraction. Real-world applications range from fuel cell development to environmental remediation.
Solution a homogeneous mixture of two or more substances in which the components are uniformly distributed
Solvent the component present in the largest quantity, usually a liquid or solid
Solute the component dissolved in the solvent, typically present in a smaller quantity
Concentration a measure of the amount of solute present in a solution, expressed in various units (molarity, molality, mass fraction)
Ideal solution a solution in which the interactions between the solvent and solute molecules are similar to those between the molecules of the pure components
Non-ideal solution a solution that deviates from ideal behavior due to strong interactions between the solvent and solute molecules or between the solute molecules themselves
Partial molar property the change in an extensive property of a solution when one mole of a component is added while keeping the amounts of other components constant
Partial molar volume the change in the total volume of a solution when one mole of a component is added at constant temperature, pressure, and composition
Partial molar enthalpy the change in the total enthalpy of a solution when one mole of a component is added at constant temperature, pressure, and composition
Chemical potential a measure of the tendency of a substance to change its state or react with other substances, equal to the partial molar Gibbs free energy
Fundamental Principles of Solution Thermodynamics
Gibbs free energy the thermodynamic potential that determines the spontaneity of a process at constant temperature and pressure
For a solution, the total Gibbs free energy is the sum of the products of the chemical potential and the number of moles of each component
Entropy of mixing the increase in entropy when two or more substances are mixed to form a solution
The entropy of mixing is always positive for ideal solutions and contributes to the stability of the solution
Enthalpy of mixing the change in enthalpy when two or more substances are mixed to form a solution
The enthalpy of mixing can be positive (endothermic) or negative (exothermic) depending on the interactions between the components
Activity a measure of the effective concentration of a component in a solution, taking into account the non-ideal behavior
The activity of a component is related to its chemical potential by the equation μi=μi0+RTlnai, where μi0 is the standard chemical potential and ai is the activity
Fugacity a measure of the tendency of a component to escape from a solution, analogous to the partial pressure of a gas
The fugacity of a component is related to its activity by the equation ai=fi/fi0, where fi0 is the fugacity of the pure component at the same temperature and pressure
Raoult's law an approximation that relates the vapor pressure of a solution to the vapor pressures and mole fractions of the pure components
For an ideal solution, Raoult's law states that the partial vapor pressure of each component is equal to the product of its mole fraction and its vapor pressure in the pure state
Types of Solutions and Mixtures
Gaseous solutions mixtures of gases, such as air (nitrogen, oxygen, and other gases)
Liquid solutions mixtures of liquids or solids dissolved in liquids, such as sugar water or salt water
Solid solutions mixtures of solids, such as alloys (brass, steel) or solid solutions of gases in metals (hydrogen in palladium)
Binary solutions mixtures containing two components, such as ethanol and water
Ternary solutions mixtures containing three components, such as a mixture of water, ethanol, and glycerol
Multicomponent solutions mixtures containing more than three components, such as complex pharmaceutical formulations or industrial mixtures
Electrolyte solutions solutions containing ions, such as aqueous solutions of salts (sodium chloride in water), acids (hydrochloric acid in water), or bases (sodium hydroxide in water)
Electrolyte solutions conduct electricity due to the presence of mobile ions
Non-electrolyte solutions solutions containing neutral molecules, such as sugar water or ethanol water mixtures
Non-electrolyte solutions do not conduct electricity as they do not contain mobile ions
Partial Molar Properties
Partial molar properties are the changes in extensive properties (volume, enthalpy, entropy, Gibbs free energy) of a solution when one mole of a component is added while keeping the amounts of other components constant
Partial molar volume (Vˉi) the change in the total volume of a solution when one mole of component i is added at constant temperature, pressure, and composition
Mathematically, Vˉi=(∂V/∂ni)T,P,nj, where V is the total volume, ni is the number of moles of component i, and nj represents the number of moles of all other components
Partial molar enthalpy (Hˉi) the change in the total enthalpy of a solution when one mole of component i is added at constant temperature, pressure, and composition
Mathematically, Hˉi=(∂H/∂ni)T,P,nj, where H is the total enthalpy
Partial molar entropy (Sˉi) the change in the total entropy of a solution when one mole of component i is added at constant temperature, pressure, and composition
Mathematically, Sˉi=(∂S/∂ni)T,P,nj, where S is the total entropy
Partial molar Gibbs free energy (Gˉi) the change in the total Gibbs free energy of a solution when one mole of component i is added at constant temperature, pressure, and composition
Mathematically, Gˉi=(∂G/∂ni)T,P,nj, where G is the total Gibbs free energy
The partial molar Gibbs free energy is equal to the chemical potential of the component, Gˉi=μi
The total extensive property of a solution is the sum of the products of the partial molar properties and the number of moles of each component
For example, the total volume of a solution is given by V=∑iniVˉi, where ni is the number of moles of component i and Vˉi is its partial molar volume
Gibbs-Duhem Equation and Its Applications
The Gibbs-Duhem equation is a fundamental relationship between the changes in intensive properties (temperature, pressure, chemical potential) of a system at equilibrium
For a multicomponent system, the Gibbs-Duhem equation is given by ∑inidμi=−SdT+VdP, where ni is the number of moles of component i, μi is its chemical potential, S is the total entropy, T is the temperature, V is the total volume, and P is the pressure
The Gibbs-Duhem equation implies that the intensive properties of a system are not independent and that a change in one property must be compensated by changes in others
Applications of the Gibbs-Duhem equation include:
Derivation of the Gibbs-Helmholtz equation, which relates the temperature dependence of the Gibbs free energy to the enthalpy and entropy
Calculation of activity coefficients from experimental data on vapor-liquid equilibria or osmotic pressures
Determination of partial molar properties from experimental measurements of total properties and composition
The Gibbs-Duhem equation is also used to derive the Gibbs adsorption isotherm, which describes the relationship between the surface tension and the concentrations of components in a surface layer
In the context of solution thermodynamics, the Gibbs-Duhem equation is often used to derive relationships between the partial molar properties of the components, such as the Duhem-Margules equation
Ideal and Non-Ideal Solutions
Ideal solutions are those in which the interactions between the solvent and solute molecules are similar to those between the molecules of the pure components
In an ideal solution, the enthalpy of mixing is zero (ΔHmix=0) and the entropy of mixing is given by the Boltzmann equation, ΔSmix=−R∑ixilnxi, where xi is the mole fraction of component i
For ideal solutions, Raoult's law is obeyed, meaning that the partial vapor pressure of each component is equal to the product of its mole fraction and its vapor pressure in the pure state
Non-ideal solutions are those that deviate from ideal behavior due to strong interactions between the solvent and solute molecules or between the solute molecules themselves
In non-ideal solutions, the enthalpy of mixing is non-zero (ΔHmix=0) and can be either positive (endothermic) or negative (exothermic)
The entropy of mixing in non-ideal solutions is less than that predicted by the Boltzmann equation due to the presence of intermolecular interactions
Non-ideal solutions exhibit deviations from Raoult's law, which can be described by introducing activity coefficients (γi) that relate the activity of a component to its mole fraction, ai=γixi
The activity coefficient is a measure of the non-ideality of a solution and depends on the composition and the strength of intermolecular interactions
For ideal solutions, the activity coefficients are equal to 1, while for non-ideal solutions, they can be greater or less than 1
Activity coefficients can be determined experimentally from vapor-liquid equilibrium data or osmotic pressure measurements
Examples of non-ideal solutions include:
Ethanol-water mixtures, which exhibit a negative deviation from Raoult's law due to the formation of hydrogen bonds between ethanol and water molecules
Acetone-chloroform mixtures, which exhibit a positive deviation from Raoult's law due to the breaking of hydrogen bonds between chloroform molecules
The thermodynamic properties of non-ideal solutions can be modeled using various approaches, such as the Margules equation, the van Laar equation, or the Wilson equation, which express the activity coefficients as functions of composition and adjustable parameters that account for the intermolecular interactions
Phase Equilibria in Solutions
Phase equilibria in solutions refer to the coexistence of two or more phases (gas, liquid, or solid) at a given temperature, pressure, and composition
The condition for phase equilibrium is that the chemical potential of each component must be equal in all phases, μiα=μiβ=μiγ=..., where α, β, γ, etc. represent different phases
The most common types of phase equilibria in solutions are:
Vapor-liquid equilibrium (VLE) the coexistence of a vapor phase and a liquid phase, characterized by the equality of the partial vapor pressures and the fugacities of each component in both phases
Liquid-liquid equilibrium (LLE) the coexistence of two immiscible liquid phases, characterized by the equality of the activities or the chemical potentials of each component in both phases
Solid-liquid equilibrium (SLE) the coexistence of a solid phase and a liquid phase, characterized by the equality of the activities or the chemical potentials of each component in both phases
Phase diagrams are graphical representations of the phase equilibria in a system as a function of temperature, pressure, and composition
Binary phase diagrams show the phase behavior of two-component systems, such as the ethanol-water system or the benzene-toluene system
Ternary phase diagrams show the phase behavior of three-component systems, such as the water-ethanol-benzene system or the water-acetone-chloroform system
The lever rule is a method used to determine the relative amounts of each phase in a two-phase region of a phase diagram based on the composition of the system and the compositions of the coexisting phases
The distribution coefficient (or partition coefficient) is the ratio of the concentrations of a component in two coexisting phases at equilibrium, Ki=ciα/ciβ, where ciα and ciβ are the concentrations of component i in phases α and β, respectively
The understanding of phase equilibria is crucial for the design and operation of separation processes, such as distillation, extraction, or crystallization, which rely on the differences in the distribution of components between coexisting phases
Practical Applications and Real-World Examples
Distillation a widely used separation process that relies on the differences in the volatilities of the components in a liquid mixture
Example: the separation of ethanol from water in the production of alcoholic beverages or biofuels
The design of distillation columns requires knowledge of the vapor-liquid equilibrium (VLE) behavior of the mixture, which can be predicted using models based on Raoult's law, activity coefficients, or equations of state
Extraction a separation process that involves the transfer of a solute from one liquid phase to another immiscible liquid phase
Example: the extraction of caffeine from coffee beans using supercritical carbon dioxide as a solvent
The efficiency of extraction depends on the distribution coefficient of the solute between the two phases, which can be optimized by selecting appropriate solvents and operating conditions
Crystallization a separation process that involves the formation of solid crystals from a supersaturated solution
Example: the production of sucrose crystals from sugarcane juice or sugar beet syrup
The purity and size distribution of the crystals depend on the solubility of the solute in the solvent, which can be controlled by adjusting the temperature, pH, or the presence of additives
Gas absorption a process in which a gas mixture is contacted with a liquid solvent to selectively remove one or more components from the gas phase
Example: the removal of carbon dioxide from natural gas using aqueous solutions of amines (such as monoethanolamine or diethanolamine)
The design of gas absorption columns requires knowledge of the solubility of the gas in the liquid, which can be described by Henry's law or more complex models that account for chemical reactions or mass transfer limitations
Fuel cells electrochemical devices that convert the chemical energy of a fuel (such as hydrogen or methanol) directly into electrical energy
Example: proton exchange membrane (PEM) fuel cells, which use a polymer electrolyte membrane to separate the anode and cathode compartments and conduct protons from the anode to the cathode
The performance of fuel cells depends on the thermodynamic properties of the fuel and oxidant mixtures, as well as the kinetics of the electrochemical reactions and the transport of reactants and products across the membrane
Environmental remediation the use of physical, chemical, or biological processes to remove or degrade pollutants from contaminated soil, water, or air
Example: the use of surfactants to enhance the solubility and biodegradation of hydrophobic organic contaminants (such as polycyclic aromatic hydrocarbons) in soil
The selection of appropriate remediation strategies requires an understanding of the phase behavior and the partitioning of the contaminants between the soil, water, and air phases, as well as the interactions between the contaminants and the surfactants or other additives