4: Putting the First Law to Work
- Page ID
- 84315
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: Prelude to Putting the First Law to Work
- The text discusses how important thermochemical quantities, specifically molar heat capacities, are expressed through partial derivatives. These properties, measurable experimentally, help calculate changes since they represent the slope of enthalpy or internal energy surfaces along specific paths of constant pressure or volume.
- 4.2: Total and Exact Differentials
- This page discusses the concepts of internal energy as a function of volume and temperature, emphasizing the significance of state variables in defining a system's state. It explains how changes in internal energy (U) can be mathematically represented by total differentials, highlighting the importance of understanding the slopes in these path directions through integration.
- 4.3: Compressibility and Expansivity
- This page discusses the properties of isothermal compressibility and isobaric thermal expansivity. Isothermal compressibility quantifies how a substance's volume changes with pressure at constant temperature, while isobaric thermal expansivity measures volume change with temperature at constant pressure.
- 4.4: The Joule Experiment
- The text explores the concept of changes in internal energy, considering as a function of volume and temperature. It relates to the constant volume heat capacity and introduces "internal pressure". The work of James Prescott Joule is discussed, who demonstrated that ????? should be zero based on his experiments.
- 4.5: The Joule-Thomson Effect
- The page explains the Joule-Thomson experiment and its significance in understanding gas cooling during expansion, which influenced refrigerator design. It also details how not all gases cool upon expansion, like hydrogen and helium, which can warm up. The Joule-Thomson coefficient (\(\mu_{JT}\)) determines this temperature change, generally calculated through the change of properties like pressure and enthalpy.
- 4.6: Useful Definitions and Relationships
- This chapter outlines several important thermodynamic definitions and relationships, such as heat capacities, coefficient of thermal expansion, and isothermal compressibility. It demonstrates how these can be used to derive useful expressions and solve problems, like determining changes in enthalpy during isothermal compression.
- 4.E: Putting the First Law to Work (Exercises)
- The page contains a series of thermodynamics questions involving derivations, calculations, and proofs related to concepts like the internal energy of an ideal gas, exact differentials, van der Waals gas properties, volume changes under pressure, isobaric expansivity, Joule-Thomson coefficient, changes in internal energy, enthalpy, and heat capacities.
- 4.S: Putting the First Law to Work (Summary)
- This chapter focuses on thermodynamics, detailing learning objectives such as expressing total differentials of thermodynamic functions, utilizing Euler relations, and deriving partial differential transformations. Key concepts include isobaric thermal expansivity, isothermal compressibility, and internal pressure.