7: Mixtures and Solutions
- Page ID
- 84336
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Up until this point, we have conserved single-component systems which do not change in composition. By and large, nature consists of much more complicated systems, containing many components and continually undergoing changes in composition through phase changes or chemical reactions or both! In order to expand our thermodynamic toolbox, we will begin by discussing mixtures.
- 7.1: Thermodynamics of Mixing
- The document discusses the mixing of two ideal gases, beginning with their initial state in separate partitions of a container. Once mixed isothermally, partial pressures drop and volumes double, yet total pressure remains consistent. The enthalpy of mixing \(\Delta H_{mix}\) is zero due to ideal behavior, meaning no molecular interactions are considered. The entropy of mixing is positive since mixing increases disorder.
- 7.2: Partial Molar Volume
- The page discusses the concept of partial molar volume in a mixture of two compounds, A and B. It explains how the change in volume for the mixture can be described using the total differential and by integrating terms related to each component's amount and partial molar volume.
- 7.3: Chemical Potential
- The page discusses the concept of chemical potential, defined as the partial molar Gibbs function change for a compound in a mixture, which indicates how the Gibbs function changes with composition. The Gibbs function's total change is expressed with various thermodynamic functions (\(U\), \(H\), \(A\), \(G\)), with the chemical potential integrated into each expression.
- 7.4: The Gibbs-Duhem Equation
- The page explains the Gibbs-Duhem equation, which imposes a compositional constraint on changes in the chemical potential of a mixture at equilibrium under constant temperature and pressure. It derives the equation by examining the partial molar Gibbs function, expressing the total Gibbs energy, and equating derivative forms.
- 7.5: Non-ideality in Gases - Fugacity
- The document explains the modification of the chemical potential formula for real gases, which do not behave ideally. It introduces fugacity, a factor that accounts for deviations from ideal gas behavior, replacing pressure in the chemical potential equation. It discusses the relationship between fugacity and pressure, presenting equations and derivations to show this connection.
- 7.6: Colligative Properties
- This page discusses colligative properties, which describe how the addition of solute impacts the solvent's properties in a solution. It focuses on three primary effects: freezing point depression, boiling point elevation, and vapor pressure lowering, all of which depend solely on the solvent. The page provides detailed explanations and mathematical derivations for each of these phenomena, highlighting their causes, implications, and calculations involved.
- 7.7: Solubility
- The document explains how to determine the maximum solubility of a solute using its chemical potential in a liquid solution compared to a pure solid solute. It describes achieving saturation when both potentials are equal. The solubility is initially expressed using the Gibbs function change and later related to the enthalpy of fusion via the Gibbs-Helmholtz equation.
- 7.8: Non-ideality in Solutions - Activity
- The bulk of the discussion in this chapter dealt with ideal solutions. However, real solutions will deviate from this kind of behavior. So much as in the case of gases, where fugacity was introduced to allow us to use the ideal models, activity is used to allow for the deviation of real solutes from limiting ideal behavior.
- 7.E: Mixtures and Solutions (Exercises)
- This page contains a series of chemistry problems related to various thermodynamic and physical chemistry concepts, such as calculating the fugacity constant, estimating vapor pressure, determining molar mass from freezing point depression, evaluating vapor pressure over mixed liquids, using osmotic pressure to find molar mass, calculating Henry's Law constant, and finding changes in enthalpy, entropy, and Gibbs free energy for mixing substances.
- 7.S: Mixtures and Solutions (Summary)
- This page outlines learning objectives focused on understanding the thermodynamics of mixing in ideal solutions, including calculating changes in enthalpy, entropy, and Gibbs free energy. It covers defining chemical potential, calculating colligative properties, determining solubility from chemical potential equality, and understanding fugacity and activity concepts. Key vocabulary includes activity coefficient, Raoult???s Law, and osmotic pressure, among others.