Glossary

9.1 Chemical potential and Gibbs-Duhem equation

3 min read•july 23, 2024

and are key concepts in thermodynamics. They help us understand how energy changes in chemical systems, driving reactions and phase transitions. These ideas are crucial for predicting stability and equilibrium in various processes.

The shows how chemical potentials in a system are connected. It's a powerful tool for analyzing mixtures and solutions, helping us grasp how changes in one component affect others. This equation has wide-ranging applications in thermodynamics and chemistry.

Chemical Potential and Gibbs Free Energy

Chemical potential and Gibbs free energy

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  • Chemical potential (μi\mu_i) partial molar Gibbs free energy represents change in Gibbs free energy when one mole of component ii added to system at , pressure, and composition of other components
  • Mathematically defined as μi=(∂G∂ni)T,P,nj≠i\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j \neq i}} where GG Gibbs free energy, nin_i number of moles of component ii
  • In multi-component system, total Gibbs free energy sum of chemical potentials of each component multiplied by their respective mole numbers G=∑iμiniG = \sum_{i} \mu_i n_i
  • Useful for understanding driving forces behind chemical reactions and phase transitions (melting, boiling)
  • Helps predict stability and equilibrium of chemical systems (solubility, vapor pressure)

Gibbs-Duhem Equation and Its Applications

Derivation of Gibbs-Duhem equation

  • Derived from total differential of Gibbs free energy dG=−SdT+VdP+∑iμidnidG = -SdT + VdP + \sum_{i} \mu_i dn_i
  • At constant temperature and pressure, dG=∑iμidnidG = \sum_{i} \mu_i dn_i
  • Dividing by total number of moles (ntn_t) yields dGnt=∑iμidxi\frac{dG}{n_t} = \sum_{i} \mu_i dx_i where xix_i mole fraction of component ii
  • Since ∑ixi=1\sum_{i} x_i = 1, differentiating gives ∑idxi=0\sum_{i} dx_i = 0
  • Combining equations results in Gibbs-Duhem equation ∑ixidμi=0\sum_{i} x_i d\mu_i = 0
  • Shows chemical potentials in system are interdependent change in chemical potential of one component must be balanced by changes in chemical potentials of other components

Applications of Gibbs-Duhem equation

  • For binary system (components 1 and 2) at constant temperature and pressure, Gibbs-Duhem equation simplifies to x1dμ1+x2dμ2=0x_1 d\mu_1 + x_2 d\mu_2 = 0 rearranged to dμ2=−x1x2dμ1d\mu_2 = -\frac{x_1}{x_2} d\mu_1
  • By integrating, change in chemical potential of component 2 calculated from change in chemical potential of component 1 and composition of system
  • Can relate changes in chemical potentials to changes in activity coefficients or partial pressures in non-ideal systems
  • Used to derive other important thermodynamic relations (, )
  • Helps understand behavior of mixtures and solutions (azeotropes, )

Factors affecting chemical potential

  • Effect of temperature given by (∂μi∂T)P,nj=−Si\left(\frac{\partial \mu_i}{\partial T}\right)_{P,n_j} = -S_i where SiS_i partial molar entropy of component ii
    • Increase in temperature generally decreases chemical potential
  • Effect of pressure given by (∂μi∂P)T,nj=Vi\left(\frac{\partial \mu_i}{\partial P}\right)_{T,n_j} = V_i where ViV_i partial molar volume of component ii
    • Increase in pressure generally increases chemical potential
  • Effect of composition depends on specific system and interactions between components
    • In ideal systems, chemical potential of component related to its mole fraction by μi=μi0+RTln⁡xi\mu_i = \mu_i^0 + RT \ln x_i where μi0\mu_i^0 standard chemical potential and RR gas constant
    • In non-ideal systems, activity coefficients or fugacities used to account for deviations from ideal behavior (intermolecular forces, hydrogen bonding)

Key Terms to Review (19)

Chemical Potential: Chemical potential is a measure of the potential energy of a substance in a system, reflecting how its concentration influences the system's thermodynamic behavior. It plays a crucial role in determining the direction of chemical reactions and phase changes, linking closely with concepts such as free energies and equilibrium conditions.
Clausius-Clapeyron equation: The Clausius-Clapeyron equation is a fundamental relationship in thermodynamics that describes the phase transition between two phases of a substance, particularly relating the change in vapor pressure with temperature. This equation helps to understand how the pressure of a substance changes as it transitions from liquid to gas or solid to liquid, providing insight into phase equilibria and the behavior of materials under varying conditions.
Colligative properties: Colligative properties are characteristics of solutions that depend on the number of solute particles present in a given quantity of solvent, rather than the identity of those solute particles. These properties arise from the interactions between solute and solvent molecules, influencing behaviors such as vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure. Understanding colligative properties is crucial for analyzing how solutions behave under different conditions.
Constant pressure: Constant pressure refers to a thermodynamic condition where the pressure of a system remains unchanged throughout a process. This concept is crucial in understanding various phenomena, particularly in the context of phase transitions and chemical reactions, as it helps in analyzing how energy is exchanged and how substances behave under specific conditions.
Constant temperature: Constant temperature refers to a state in which the temperature of a system remains unchanged over time, despite any changes in other properties such as pressure or volume. This condition is crucial in thermodynamics as it allows for the analysis of various processes, especially those involving ideal gases and phase transitions. When dealing with constant temperature scenarios, certain thermodynamic relationships, such as those involving chemical potential or statistical mechanics principles, become significantly simplified.
Duhem-Margules Equation: The Duhem-Margules equation is a fundamental relationship in thermodynamics that connects the chemical potential of components in a solution with their concentrations. It is particularly important for understanding how changes in the concentration of one component affect the chemical potential of others in multicomponent systems, thus providing insight into phase equilibria and the behavior of mixtures.
Gibbs Free Energy: Gibbs free energy is a thermodynamic potential that measures the maximum reversible work obtainable from a closed system at constant temperature and pressure. It is a crucial concept because it helps predict the direction of chemical reactions and phase transitions, determining whether a process will occur spontaneously based on changes in enthalpy and entropy.
Gibbs-Duhem equation: The Gibbs-Duhem equation is a fundamental relation in thermodynamics that describes how the chemical potential of a system changes with its composition at constant temperature and pressure. This equation connects changes in chemical potential to variations in the amount of components in a system, highlighting the interdependence of different phases and components in thermodynamic systems.
Helmholtz Free Energy: Helmholtz free energy is a thermodynamic potential that measures the work obtainable from a closed system at constant temperature and volume. It is defined as the difference between the internal energy of the system and the product of its temperature and entropy, expressed mathematically as $$A = U - TS$$. This concept is crucial for understanding how systems exchange energy and work in various thermodynamic processes, particularly in relation to other potentials like Gibbs free energy.
Ideal gas behavior: Ideal gas behavior refers to the theoretical conditions under which a gas behaves in a predictable manner, following the ideal gas law, which states that the pressure, volume, and temperature of a gas are related in a specific way. This concept is foundational for understanding how gases interact at a molecular level and allows for simplifications in calculations involving chemical potential and thermodynamic properties. In ideal gas behavior, it is assumed that gas molecules have negligible volume and do not experience intermolecular forces, making it easier to predict how they will respond to changes in temperature and pressure.
Internal Energy: Internal energy is the total energy contained within a thermodynamic system, encompassing kinetic and potential energies of all the molecules in the system. This energy is crucial in determining the state of the system and plays a key role in thermodynamic processes, including heat transfer and work done on or by the system.
Isothermal process: An isothermal process is a thermodynamic process in which the temperature of the system remains constant while heat is exchanged with the surroundings. This constant temperature implies that any internal energy changes in the system are fully compensated by heat transfer, making it an essential concept in understanding how systems behave under thermal equilibrium and the laws governing energy conservation.
Josiah Willard Gibbs: Josiah Willard Gibbs was an American physicist, chemist, and mathematician known for his contributions to thermodynamics and statistical mechanics. His work laid the foundation for understanding free energies, chemical potentials, and the behavior of particles in different states, making significant impacts across various fields of science.
Ludwig Boltzmann: Ludwig Boltzmann was an Austrian physicist who made significant contributions to the field of statistical mechanics and thermodynamics, particularly known for his formulation of the statistical interpretation of entropy. His work established a crucial link between microscopic particle behavior and macroscopic physical properties, providing a deeper understanding of the second law of thermodynamics and the nature of entropy.
Mixing entropy: Mixing entropy is a measure of the increase in disorder or randomness when two or more substances are mixed together. It quantifies the amount of energy that becomes unavailable for doing work as a result of mixing, and it plays a crucial role in determining the spontaneity of a process. The concept is essential for understanding chemical potential and the Gibbs-Duhem equation, as it relates to how the thermodynamic properties of a system change during mixing.
Non-ideal solutions: Non-ideal solutions are mixtures where the interactions between different components cause deviations from Raoult's law, meaning that the properties of the solution cannot be accurately predicted based solely on the properties of the pure components. This behavior arises due to molecular interactions that are not purely ideal, such as hydrogen bonding, dipole-dipole interactions, or significant size differences among solute and solvent molecules. Understanding non-ideal solutions is crucial for applications involving chemical potential and the Gibbs-Duhem equation, as these concepts rely on the accurate modeling of how components interact within a mixture.
Partial Molar Property: A partial molar property is a thermodynamic property of a component in a mixture that represents the change in the overall property of the mixture when an infinitesimal amount of that component is added, while keeping the temperature and pressure constant. This concept is crucial for understanding how different components interact in solutions and helps explain how the properties of mixtures differ from those of their pure components.
Phase Equilibrium: Phase equilibrium refers to a condition in which distinct phases of a substance coexist in a stable manner, with no net change in their respective quantities over time. This balance occurs when the rates of transition between phases, such as solid, liquid, and gas, are equal, leading to an overall stability in the system. Understanding phase equilibrium is essential for analyzing latent heat during phase transitions, chemical potential in thermodynamic systems, the construction of phase diagrams, and the behavior of gases under varying conditions.
Reaction quotient: The reaction quotient, denoted as Q, is a measure of the relative concentrations of products and reactants at any given point in a chemical reaction. It provides insight into the direction in which a reaction will proceed to reach equilibrium by comparing its value to the equilibrium constant, K. If Q is less than K, the reaction favors products, while if Q is greater than K, it favors reactants. This concept links closely to the understanding of free energy changes and chemical potential.
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