20: Entropy and The Second Law of Thermodynamics
- Page ID
- 62254
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 20.1: Energy Does not Determine Spontaneity
- This page explores the concept of spontaneity in natural events, highlighting that it is not just linked to heat. It emphasizes the importance of entropy as a state function, revealing its role in understanding the connection between work and heat. The discussion transitions from macroscopic to molecular perspectives on entropy, examining energy distribution in systems.
- 20.2: Nonequilibrium Isolated Systems Evolve in a Direction That Increases Their Energy Dispersal
- This page examines spontaneous processes in isolated systems and the conflict between energy minimization and dispersal. It analyzes relationships involving internal energy, volume work, and heat for an ideal gas, revealing that while heat is not a state function, entropy (S) emerges as a state function when considering the integration factor 1/T. This establishes dS = δq_rev/T as an exact differential, thereby quantifying energy dispersal and the number of microstates in the system.
- 20.3: Unlike heat, Entropy is a State Function
- This page discusses entropy as a state function that integrates to zero over circular paths returning to initial conditions. It examines a path composed of isotherm, isochore, and adiabat segments, emphasizing that although heat and work are path-dependent, entropy remains consistent. The analysis demonstrates that despite varying heat transfer values, the change in entropy remains the same, reinforcing its nature as a state function, in contrast to heat.
- 20.4: The Second Law of Thermodynamics
- This page discusses the behavior of isolated systems, which do not exchange heat, mass, or radiation, akin to a small universe. It explains the zero law process where heat flows from hot to cold metal blocks until thermal equilibrium is achieved, illustrating that entropy must increase (dS > 0) for spontaneity. In non-isolated systems, entropy changes involve heat exchange, complicating the reliability of entropy as a criterion for spontaneity, given the possibility of irreversible processes.
- 20.5: The Famous Equation of Statistical Thermodynamics is S=k ln W
- This page discusses entropy as a measure of chaos and energy dispersal, highlighting Ludwig Boltzmann's contributions in deriving a statistical method to calculate it. His expression connects entropy to the logarithm of microstates, illustrating the concept with a calculation of entropy for a carbon monoxide crystal using his formula.
- 20.6: We Must Always Devise a Reversible Process to Calculate Entropy Changes
- This page discusses the second law of thermodynamics and the increasing entropy in isolated systems, particularly in the universe. It highlights recent debates regarding entropy growth influenced by gravity and black holes. Additionally, the text contrasts irreversible and reversible isothermal expansions of ideal gases, emphasizing the distinctions in energy and heat.
- 20.7: Thermodynamics Provides Insight into the Conversion of Heat into Work
- This page discusses energy transfer via heat and work, highlighting the second law of thermodynamics which restricts heat-to-work conversion. It describes the Carnot cycle, detailing its four stages and efficiency formula \( \eta = 1 - \frac{T_c}{T_h} \). It also covers a circular reversible process on a PV diagram and emphasizes the role of temperature in determining efficiency.
- 20.8: Entropy Can Be Expressed in Terms of a Partition Function
- This page explains how to calculate the entropy of a system using Boltzmann's definition, deriving the ensemble entropy from the partition function and the state probabilities. It presents the formula for entropy, \(S_{system} = - k \sum_j p_j \ln p_j\), and arrives at the final expression \(S = \frac{U}{T} + k \ln Q\), where \(U\) represents internal energy and \(T\) is temperature.
- 20.9: The Statistical Definition of Entropy is Analogous to the Thermodynamic Definition
- This page discusses the relationship between entropy (\(S\)) and microstates (\(W\)) in an ensemble, deriving ensemble entropy using Stirling's approximation. It simplifies the entropy derived from the ensemble and examines the impact of microstate probabilities on entropy. The text concludes by relating changes in entropy to reversible heat transfer, culminating in the expression \(dS = \frac{\delta q_{rev}}{T}\).