21: Entropy & the Third Law of Thermodynamics
- Page ID
- 62255
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 21.1: Entropy Increases With Increasing Temperature
- This page explores the relationship between entropy and energy flow, emphasizing reversible and irreversible processes. It elucidates the connection between entropy (S) and internal energy (U) using equations related to constant volume (C_V) and constant pressure (C_P) heat capacities.
- 21.2: The 3rd Law of Thermodynamics Puts Entropy on an Absolute Scale
- This page discusses entropy treatment in relation to the Third Law of Thermodynamics, highlighting that entropy of perfect crystals approaches zero at absolute zero. It addresses challenges in achieving perfect crystals due to defects, leading to residual entropy, and explores phase transitions and entropy calculations with examples like sulfur and carbon monoxide.
- 21.3: The Entropy of a Phase Transition can be Calculated from the Enthalpy of the Phase Transition
- This page discusses phase transitions, including melting, which occur under constant temperature and pressure, leading to enthalpy and entropy changes. It highlights first-order transitions, characterized by discontinuities in first-order derivatives, and notes that all phase transitions involve changes in entropy. Additionally, it mentions that second-order derivatives, such as heat capacity, exhibit singularities at transition points.
- 21.4: The Debye Function is Used to Calculate the Heat Capacity at Low Temperatures
- This page discusses the Debye function, which models the heat capacity of nonmetallic crystals at low temperatures using the equation \( \bar{C}_P = aT^3 \), where \(a\) ensures continuity with existing data. Developed by chemist Peter Debye, it also addresses metallic crystals with a modified equation \( \bar{C}_P = aT + bT^3 \), which includes an additional constant \(b\).
- 21.5: Practical Absolute Entropies Can Be Determined Calorimetrically
- This page explains the calculation of absolute entropy for substances, using carbon dioxide as an example. It outlines two key equations for entropy change, applicable at constant pressure and during phase transitions, utilizing calorimetric data. The process involves integrating heat capacities and considering phase changes to determine absolute entropy at various temperatures, with a specific calculation for CO2 at 300 K included.
- 21.6: Practical Absolute Entropies of Gases Can Be Calculated from Partition Functions
- This page discusses the computation of entropy in a system using the partition function \(Q\) with the formula \(S=\frac{U}{T}+k_B \ln{Q}\). It highlights how internal energy \(U\) can be derived from \(Q\), leading to a comprehensive entropy formulation. Additionally, it relates the third law of thermodynamics, stating that a perfect crystal's entropy is zero at absolute zero, thus affirming the consistency of this formulation with thermodynamic principles.
- 21.7: Standard Entropies Depend Upon Molecular Mass and Structure
- This page explains the relationship between entropy and microstates in particles, highlighting that increased molecular mass results in lower energy state spacing and higher entropy. It notes that noble gases exemplify this trend, as their molar entropy rises with mass. Furthermore, it mentions that molecules with more atoms tend to have greater energy dispersion and entropy due to additional degrees of freedom for energy states.
- 21.8: Spectroscopic Entropies sometimes disgree with Calorimetric Entropies
- This page discusses the measurement of gas entropy, highlighting that inconsistencies known as residual entropy can emerge in certain substances. This occurs in materials with multiple configurations at absolute zero, challenging the third law of thermodynamics which asserts that entropy should be zero at that temperature. Examples like glass, ice, and carbon monoxide showcase residual entropy due to their imperfect crystal structures.
- 21.9: Standard Entropies Can Be Used to Calculate Entropy Changes of Chemical Reactions
- This page discusses the calculation of entropy as a state function for chemical reactions, specifically demonstrating the change in entropy (\(\Delta_{rxn}S^\circ\)) in the combustion of methane. It explains how to determine \(\Delta_{rxn}S^\circ\) by summing the standard entropies of products and subtracting those of reactants. Standard entropies for the relevant substances are included, with the combustion of methane resulting in an entropy change of \(-5.
- 21.E: Entropy and the Third Law of Thermodynamics (Exercises)
- This page explains how to calculate the change in entropy for cooling 200 g of water from 70.0°C to 20.0°C. It involves converting mass to moles and using the molar heat capacity along with the temperature change to determine the total entropy change for the cooling process.