22: Helmholtz and Gibbs Energies
- Page ID
- 62256
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 22.1: Helmholtz Energy
- This page explores entropy and spontaneity in thermodynamics with a focus on constant volume and temperature conditions. It details how changes in internal energy, via Helmholtz energy (A), signal spontaneity, noting that A decreases in spontaneous processes. The page includes an example of gas mixing under isothermal conditions, highlighting that Helmholtz free energy indicates the maximum work achievable when volume is fixed.
- 22.2: Gibbs Energy Determines the Direction of Spontaneity at Constant Pressure and Temperature
- This page discusses the Helmholtz energy and its relation to Gibbs energy, which is better analyzed under constant pressure conditions. It outlines how Gibbs energy (G) predicts spontaneity in reactions, utilizing the formula ΔG = ΔH - TΔS, where positive ΔG denotes non-spontaneous processes and ΔG = 0 indicates equilibrium. Temperature influences spontaneity, illustrated through an example involving NH3 and HCl.
- 22.3: The Maxwell Relations
- This page explores the mathematical connections between Gibbs and Helmholtz functions, emphasizing their dependence on temperature, pressure, and volume, and utilizing the First and Second Laws of thermodynamics to derive Maxwell Relations for simplifying calculations.
- 22.4: The Enthalpy of an Ideal Gas is Independent of Pressure
- This page discusses the relationship between pressure and enthalpy (H) through key thermodynamic equations related to Gibbs free energy (G), entropy (S), and volume (V). It derives an expression for the partial derivative of H with respect to pressure during isothermal changes, concluding that for ideal gases, this derivative is zero, signifying that enthalpy is independent of pressure.
- 22.5: Thermodynamic Functions have Natural Variables
- This page covers fundamental thermodynamic equations that define internal energy, enthalpy, Gibbs energy, and Helmholtz energy, emphasizing their relationships through differential forms and the first law of thermodynamics. It introduces concepts like chemical potential and Maxwell Relations, linking thermodynamic functions.
- 22.6: The Standard State for a Gas is an Ideal Gas at 1 Bar
- This page explains standard state conditions (SSC) for gases, defined at 1 bar using ideal gas values. The temperature is not included in the SSC definition, and while real gases deviate from ideal behavior, this standardization allows for consistent corrections for non-ideality among various gases.
- 22.7: The Gibbs-Helmholtz Equation
- This page discusses Gibbs energy in ideal gases and solids, exploring its relationship with enthalpy and entropy through the Gibbs-Helmholtz expression. It describes integrating volume over pressure via the gas law and the role of temperature in determining state functions through heat capacities.
- 22.8: Fugacity Measures Nonideality of a Gas
- This page discusses the relationship between Gibbs energy, pressure, and temperature in closed systems, emphasizing fugacity, a concept introduced by Lewis in 1905. It elaborates on how fugacity extends Gibbs energy expressions beyond ideal gases, incorporating the fugacity coefficient (\(\phi\)) to indicate non-ideality. The text further explains the connection between fugacity, Gibbs energy, and chemical potential, enhancing the understanding of thermodynamic equilibrium.