6: Putting the Second Law to Work
- Page ID
- 84329
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 6.1: Free Energy Functions
- The text discusses the criteria for spontaneity in chemical processes, emphasizing the Gibbs Free Energy (G) and the Helmholtz function (A). ??G, which is relevant in constant temperature and pressure conditions, must be negative for a process to be spontaneous. Similarly, ??A applies under constant temperature and volume. The text explains how entropy and enthalpy changes contribute to spontaneity and provides formulas for calculating ??G and ??A.
- 6.2: Combining the First and Second Laws - Maxwell's Relations
- This page discusses the dependence of Gibbs and Helmholtz functions on variables like temperature, pressure, and volume, highlighting the integration of the First and Second Laws into a mathematical expression. It focuses on the development of Maxwell Relations, which arise from these combined laws and offer a powerful way to substitute partial derivatives to simplify thermodynamic analyses.
- 6.3: ΔA, ΔG, and Maximum Work
- The page discusses free energy functions, A and G, which measure the maximum work available from a process. It explains how the differential of A, at constant temperature, equals the maximum reversible work. Similarly, the differential of G at constant temperature and pressure equals the maximum non p-V work. These concepts underscore the importance of these functions in optimizing systems for work, such as in engines.
- 6.4: Volume Dependence of Helmholtz Energy
- The page discusses the derivation of the Helmholtz function, A, in relation to changes in volume at constant temperature. It outlines the process starting with A = U - TS and differentiates it to relate the change in A with volume and temperature using the first and second laws of thermodynamics.
- 6.5: Pressure Dependence of Gibbs Energy
- The page discusses the pressure and temperature dependence of the Gibbs function (G). It starts with the definition of G and derives the total differential to show the natural variables are pressure and temperature. The page derives expressions for the partial derivatives related to volume (V) and entropy (S) and introduces a Maxwell relation.
- 6.6: Temperature Dependence of A and G
- The page explains the differential expressions of free energy functions (Helmholtz and Gibbs) and derives how these energies change with temperature. It introduces the Gibbs-Helmholtz equation, allowing for the determination of Gibbs free energy changes with temperature.
- 6.7: When Two Variables Change at Once
- This page discusses the complexities of analyzing thermodynamic systems where multiple variables, such as temperature and volume, change simultaneously. It emphasizes the significance of state variables like Gibbs energy (G) that allow the total differential to account for changes in multiple variables. The page provides an example of calculating entropy change for a monatomic ideal gas, demonstrating how individual changes in temperature and volume contribute to the total entropy change.
- 6.8: The Difference between Cp and Cv
- The document explains the derivation of an expression for the difference between constant pressure and constant volume heat capacities (C_p - C_V). It begins by defining these capacities in terms of enthalpy (H) and internal energy (U), then manipulates these definitions using the properties of various thermodynamic quantities (??, ?????, etc.).
- 6.E: Putting the Second Law to Work (Exercises)
- The page contains a series of thermodynamic problems primarily focused on calculating various thermodynamic properties such as Gibbs functions, Helmholtz function, entropy changes, and deriving thermodynamic equations of state. It also includes derivations for pressure and volume derivatives of thermodynamic variables and evaluates specific expressions for ideal gases and non-ideal gases such as van der Waals gas.
- 6.S: Putting the Second Law to Work (Summary)
- This section outlines the learning objectives for mastering thermodynamics concepts, focusing on the free energy functions A (Helmholtz energy) and G (Gibbs energy). It includes understanding the relationship between these functions and process spontaneity at constant volume and pressure, respectively.