Gibbs Free Energy

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tags: - colorclass/_synthesis - catalyst kinetics and social behavior ---Gibbs Free Energy ($G$), named after the American scientist Josiah Willard Gibbs, is a thermodynamic quantity that serves as a measure of the maximum reversible work that a system can perform at constant temperature and pressure. It is a central concept in chemical thermodynamics and physical chemistry, particularly in the study of chemical reactions and phase transitions. The Gibbs free energy is instrumental in predicting the spontaneity of processes and understanding equilibrium states.

Mathematical Definition

The Gibbs free energy $G$ of a system is defined as:

$G=H-TS$

where: - $H$ is the enthalpy of the system, - $T$ is the absolute temperature in Kelvin, - $S$ is the entropy of the system.

The change in Gibbs free energy ($\Delta G$) during a process is given by:

$\Delta G=\Delta H-T\Delta S$

where $\Delta H$ is the change in enthalpy and $\Delta S$ is the change in entropy of the system during the process.

Spontaneity and Equilibrium

The sign of $\Delta G$ determines the spontaneity of a process at constant temperature and pressure: - If $\Delta G<0$, the process is spontaneous, indicating it can proceed without the input of external energy. - If $\Delta G>0$, the process is non-spontaneous as formulated and may occur spontaneously in the reverse direction. - If $\Delta G=0$, the system is at equilibrium, and no net change can occur; the forward and reverse processes occur at the same rate.

Standard Gibbs Free Energy of Formation

The standard Gibbs free energy of formation ($\Delta G_f^{\circ }$) of a compound is the change in Gibbs free energy when one mole of the compound is formed from its elements in their standard states at $1\text{atm}$ pressure and a specified temperature, usually $298K$. These values are crucial for calculating the Gibbs free energy changes in chemical reactions under standard conditions.

Gibbs Free Energy and Chemical Potential

In the context of mixtures and solutions, the Gibbs free energy is related to the chemical potential ($\mu$), which is the change in free energy with the addition of particles at constant temperature and pressure. For a component $i$ in a mixture, the chemical potential is given by:

${\mu }_i={\left(\frac{\partial G}{\partial n_i}\right)}_{T,P,n_{j\ne i}}$

where $n_i$ is the number of moles of component $i$ and $n_{j\ne i}$ represents the moles of all other components. This relationship underscores the role of Gibbs free energy in determining the direction of mass transfer and chemical reactions in multicomponent systems.

Thermodynamic Relations Involving Gibbs Free Energy

Gibbs free energy is connected to other thermodynamic quantities through fundamental equations, such as:

- The Gibbs-Helmholtz equation, which relates the temperature dependence of $\Delta G$ to $\Delta H$ and $\Delta S$:

${\left(\frac{\partial (\Delta G/T)}{\partial T}\right)}_P=-\frac{\Delta H}{T^2}$

- The Maxwell relations and other thermodynamic identities that offer additional insights into the interdependencies between thermodynamic properties.

Applications

The concept of Gibbs free energy finds applications across various fields: - In chemistry, it predicts the direction of chemical reactions and the feasibility of processes. - In electrochemistry, it relates to the electromotive force of galvanic cells. - In materials science, it helps determine phase stability and equilibrium compositions. - In biology, it explains energy transactions in cellular processes, such as ATP hydrolysis in metabolism.

Understanding Gibbs free energy is crucial for grasping the principles that govern physical and chemical transformations, allowing scientists and engineers to predict and manipulate the behavior of natural and technological systems under a broad spectrum of conditions.