13: Expansion, Compression and the TdS Equations
- Page ID
- 7297
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 13.1: Coefficient of Expansion
- This page of the textbook covers coefficients of thermal expansion (linear, area, volume) and their applications in materials, explaining how these coefficients vary with temperature and differ among solids, liquids, and gases. It discusses the relationship between thermal expansion and atomic vibrations due to anharmonicity. Additionally, it presents a practical example and emphasizes that thermal expansion is a tensor quantity in anisotropic crystals, affecting shape based on direction.
- 13.2: Compression
- This page covers isothermal compressibility (κ), detailing its role in measuring volume reduction under pressure at constant temperature and distinguishing it from adiabatic compressibility. It introduces the isothermal bulk modulus as the reciprocal of κ. Additionally, it encourages readers to learn the SI units for these properties and includes exercises on the compressibility of an ideal gas and determining the bulk modulus of air at atmospheric pressure.
- 13.3: Pressure and Temperature
- This page explores the interrelationship among pressure, temperature, and heat capacities (CP and CV) of materials, noting that CP typically exceeds CV due to expansion work at constant pressure, except for certain anomalies like water near 2°C. It covers thermal expansion coefficients and isothermal compressibility, explaining that the difference CP − CV can be derived independently of the state equation.
- 13.4: The TdS Equations
- This page explains the three TdS equations in thermodynamics, which are crucial for calculating entropy changes in reversible processes using different thermodynamic variables (P, V, T). It highlights the derivation of these equations, emphasizing the connections through Maxwell relations and their relationship with internal energy and enthalpy. The page concludes by noting that these equations facilitate the calculation of entropy changes between states, given the system's equation of state.
- 13.5: Expansion, Compression and the TdS Equations
- This page examines the interplay between pressure, volume, temperature, and entropy in thermodynamics, centering on reversible adiabatic processes. It presents equations linking these properties and discusses isothermal and adiabatic compressibilities, along with heat capacities and their ratio γ. The insights extend to sound speed in gases, referencing Newton's early measurement challenges.
- 13.7: Rigidity Modulus (Shear Modulus)
- This page explores adiabatic and isothermal bulk moduli, noting that adiabatic bulk modulus typically increases with pressure but may decrease in a vacuum. It presents a relationship involving volume expansion coefficient and specific heat. Additionally, it explains that unlike bulk moduli, rigidity moduli do not distinguish between adiabatic and isothermal conditions, as they are unaffected by changes in volume or length.
- 13.8: Volume, Temperature and the Grüneisen Parameter
- This page explores the relationship between temperature and volume changes in materials under adiabatic and reversible conditions, focusing on isentropic compression where temperature rises as volume decreases. It introduces the Grüneisen parameter, which quantifies this relationship using dimensionless derivatives.