12: Free Energy
- Page ID
- 7290
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 12.1: Review of Internal Energy and Enthalpy
- This page covers thermodynamic equations for internal energy and enthalpy, focusing on different types of work in a system. It clarifies how the First Law of Thermodynamics addresses irreversible work and its effect on entropy. The text stresses the importance of recognizing various configuration work forms beyond PdV work, leading to modified expressions for dU and dH. It also explains the conditions under which heat added corresponds to changes in internal energy or enthalpy.
- 12.2: Free Energy
- This page clarifies the distinction between Helmholtz free energy (A or F) and Gibbs free energy (G), addressing the confusion from using the term "free energy" interchangeably. The author advocates for differentiating these forms by using A for Helmholtz and G for Gibbs, suggesting that the ambiguous symbol F should be avoided.
- 12.3: Helmholtz Free Energy
- This page defines Helmholtz free energy (A) as A = U - TS and discusses its significance in isothermal processes. It indicates that during such processes, the change in A corresponds to the reversible work done on the system. Moreover, it notes that when a machine performs reversible work at constant temperature, A decreases equivalently. This emphasizes Helmholtz free energy as the energy available for useful work, underscoring its identity as "free energy."
- 12.4: Gibbs Free Energy
- This page explains the concept of Gibbs free energy (G) and its significance in thermodynamics. It presents the equations G=H-T S and G=A+P V. The differential form of G highlights how changes in free energy (dG) at constant temperature and pressure relate to reversible work in a system. An increase in G indicates useful work done on the system, whereas a decrease indicates work done by the system, emphasizing the interplay of energy transformations in thermodynamic processes.
- 12.5: Summary, the Maxwell Relations, and the Gibbs-Helmholtz Relations
- This page covers essential thermodynamic functions—internal energy, enthalpy, Helmholtz free energy, and Gibbs free energy—emphasizing their differential forms under reversible conditions. It highlights the dependence on state variables and introduces Maxwell Relations to relate partial derivatives of these variables. Furthermore, it explains the Gibbs-Helmholtz Relations that link internal energy and enthalpy to free energy and entropy.
- 12.6: The Joule and Joule-Thomson Coefficients
- This page discusses Chapter 10, which covers the derivation of the Joule and Joule-Thomson coefficients using Helmholtz and Gibbs functions alongside Maxwell relations. It explains how the Joule coefficient is derived by analyzing temperature changes with volume at constant internal energy, and how the Joule-Thomson coefficient is derived from temperature changes with pressure at constant enthalpy. The chapter emphasizes the use of state functions and equations of state in these derivations.
- 12.7: The Thermodynamic Functions for an Ideal Gas
- This page explains thermodynamic changes for an ideal gas during transitions between states, focusing on isothermal and adiabatic processes. It details the work done, internal energy changes, heat absorption, entropy, and free energy variations. It emphasizes that in isothermal processes, heat absorbed matches the work done, whereas in adiabatic processes, energy changes differ.
- 12.8: The Thermodynamic Functions for Other Substances
- This page discusses the calculation of changes in thermodynamic functions during reversible transformations between defined states by pressure, volume, and temperature. It highlights entropy as a state function reliant on these variables, derives expressions for entropy change using heat capacity, and outlines methods to calculate changes, especially for ideal gases.
- 12.9: Absolute Entropy
- This page explains how to calculate the molar entropy of hydrogen gas at 25°C, outlining a five-step process that includes heating, liquefying, vaporizing, and adjusting temperature and pressure, resulting in a total entropy of 124000 J K−1 kmole−1. It emphasizes the challenge of determining true "absolute" entropy without 0 K values, relating this to the Third Law of Thermodynamics. The page also foreshadows a discussion on non-PdV work scenarios, such as battery charging.
- 12.10: Charging a Battery
- This page explains "non-PdV work" using battery charging as an example, detailing how the Gibbs function increases (qE) when a charge is introduced to an electric cell at constant temperature and pressure. It derives the increase in enthalpy through the Gibbs-Helmholtz relation, demonstrating the relationship between enthalpy change and the temperature dependence of the cell's electromotive force (EMF). This effectively links the concepts of Gibbs function and enthalpy in thermodynamics.
- 12.11: Surface Energy
- This page explores surface tension as a non-PdV work in liquids, emphasizing its role in minimizing surface area and adopting a spherical shape. It describes the thermal implications of increasing surface area through a Carnot cycle, detailing relationships among work, heat, internal energy, and Helmholtz free energy.
- 12.12: Fugacity
- This page explains the derivation of the change in Gibbs free energy for an ideal gas during isothermal pressure increases, represented by the equation \( G-G_{0}=R T \ln \left(P / P_{0}\right) \). It highlights the need for different equations of state for non-ideal gases, using fugacity in calculations, and discusses the connection between fugacity and Gibbs free energy.