Second Law of Thermodynamics — Macroscopic and Microscopic Foundations

Scope: rigorous derivation and engineering application of the second law, unifying macroscopic statements (Clausius, Kelvin–Planck, Carnot, exergy) with microscopic/statistical mechanics (Boltzmann, Gibbs ensembles, partition functions). Includes control-mass and control-volume balances, entropy generation, availability, and links to information entropy.

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1. Macroscopic Statements and Their Equivalence

1.1 Kelvin–Planck Statement

No device operating in a cycle can receive heat from a single reservoir and produce an equivalent amount of work with no other effect.

1.2 Clausius Statement

No process is possible whose sole result is the transfer of heat from a colder body to a hotter body.

1.3 Equivalence Sketch

• If a Clausius-violating refrigerator moves heat from cold to hot with no work input, couple it to a heat engine to produce net work from a single reservoir → violates Kelvin–Planck.
• If a Kelvin–Planck-violating engine produces work from a single reservoir, use work to drive a refrigerator to move heat cold→hot with no other effect → violates Clausius. Hence the two statements are equivalent.

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2. Carnot Cycle and Absolute Temperature

2.1 Reversible Heat Engine Model

A reversible engine (Carnot) operating between two reservoirs at temperatures $T_H$ and $T_C$ executes: isothermal expansion at $T_H$, adiabatic expansion to $T_C$, isothermal compression at $T_C$, adiabatic compression to $T_H$.

2.2 Carnot Theorems

1. No engine operating between the same two reservoirs is more efficient than a reversible engine.
2. All reversible engines operating between $T_H, T_C$ have the same efficiency, independent of working fluid.

2.3 Efficiency and Absolute Temperature

From reversibility and cyclic integrals:

$$\eta_{rev} = 1 - \frac{Q_C}{Q_H} = 1 - \frac{T_C}{T_H}.$$

This defines the absolute temperature scale up to a constant that is fixed by assigning the triple point of water to 273.16 K.

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3. Clausius Inequality and Entropy Definition

3.1 Clausius Inequality

For any closed system undergoing a cycle:

$$\oint \frac{\delta Q}{T_b} \le 0,$$

where $T_b$ is the boundary temperature at the heat interaction location. Equality holds for fully reversible cycles.

3.2 Entropy as a Property

Consider two states 1→2. Construct a reversible path and use path-independence of state functions to define entropy $S$:

$$\boxed{\Delta S = \int_{1}^{2} \frac{\delta Q_{rev}}{T}}.$$

Because $\oint \delta Q_{rev}/T = 0$ for any reversible cycle, $dS$ is an exact differential.

3.3 Entropy Balance for Closed Systems

General process with internal irreversibility $S_{gen}\ge 0$:

$$\boxed{\Delta S = \int_{1}^{2} \frac{\delta Q}{T_b} + S_{gen}}, \qquad S_{gen} \ge 0.$$

3.4 Entropy Rate Balance for Control Volumes

For control volume (CV) with mass flow and heat at boundary segments $k$ with temperature $T_k$:

$$\boxed{\frac{dS_{CV}}{dt} = \sum_k \frac{\dot Q_k}{T_k} + \sum_{in} \dot m\, s - \sum_{out} \dot m\, s + \dot S_{gen}}, \qquad \dot S_{gen}\ge 0.$$

Steady state: $dS_{CV}/dt=0$.

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4. Useful Differential Identities and Maxwell Relations

From the fundamental relation of a simple compressible system:

$$\boxed{dU = T\,dS - P\,dV}.$$

Legendre transforms give:

$$\begin{aligned} dH &= T\,dS + V\,dP,\dA &= -S\,dT - P\,dV,\dG &= -S\,dT + V\,dP. \end{aligned}$$

Exactness yields Maxwell relations, e.g. from $dG$: $\left(\partial V/\partial T\right)_P = -\left(\partial S/\partial P\right)_T$. These are used to derive property relations and calibrate equations of state.

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5. Irreversibilities and Entropy Generation

5.1 Sources

Friction, viscous dissipation, finite $\Delta T$ heat transfer, mixing, chemical reaction, diffusion, inelastic deformation, electrical resistance, mass transfer across finite chemical potential differences.

5.2 Quantification Near Equilibrium (Continuum Thermodynamics)

Clausius–Duhem inequality for local specific entropy $s$:

$$\rho\,\dot s + \nabla\!\cdot\!\left(\frac{\mathbf q}{T}\right) - \frac{\rho r}{T} \ge 0,$$

where $\mathbf q$ is heat flux, $r$ volumetric heat source. With Fourier and Newtonian laws, the local entropy production rate density is

$$\sigma = \mathbf q\!\cdot\! abla\! act\! act\left(\frac{1}{T}\right) + \frac{\boldsymbol\tau: \nabla \mathbf u}{T} + \sum_i \mathbf J_i\!\cdot\! abla\! act\! act\left(\frac{\mu_i}{T}\right) + \frac{\dot{\xi}\,\mathcal A}{T} \ge 0,$$

with viscous stress $\boldsymbol\tau$, species fluxes $\mathbf J_i$, chemical affinity $\mathcal A$, and reaction rate $\dot{\xi}$.

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6. Exergy (Availability) and the Gouy–Stodola Theorem

6.1 Environment and Dead State

Define environment (superscript 0) at $T_0, P_0$. The dead state is the state in mutual equilibrium with the environment.

6.2 Specific Flow Exergy and Nonflow Exergy

• Nonflow exergy (per mass):

$$\boxed{\psi = (u - u_0) + P_0(v - v_0) - T_0(s - s_0)}.$$

• Flow exergy (per mass) includes kinetic and potential terms:

$$\boxed{\Psi = (h - h_0) - T_0(s - s_0) + \frac{v^2}{2} + gz}.$$

6.3 Exergy Balance for Control Volumes

At steady state with heat at boundary segments $k$ at temperature $T_k$:

$$\boxed{\dot W_{useful} = \sum_k \left(1 - \frac{T_0}{T_k}\right)\,\dot Q_k + \sum_{in} \dot m\,\Psi - \sum_{out} \dot m\,\Psi - \dot X_{dest}},$$

where $\dot X_{dest} = T_0\,\dot S_{gen}$ is exergy destruction.

6.4 Gouy–Stodola

For any steady device:

$$\boxed{\dot X_{dest} = T_0\,\dot S_{gen}}.$$

This gives a direct cost of irreversibility and is the basis of entropy generation minimization in design.

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7. Engineering Efficiencies and Bounds

• Heat engine: $\eta = W_{out}/Q_{in} \le 1 - T_C/T_H$.
• Refrigerator: $\mathrm{COP}_R = Q_L/W_{in} \le T_L/(T_H - T_L)$.
• Heat pump: $\mathrm{COP}_{HP} = Q_H/W_{in} \le T_H/(T_H - T_L)$.
• Turbine/Compressor isentropic efficiency: $\eta_t = (h_1 - h_{2s})/(h_1 - h_2),\; \eta_c = (h_{2s} - h_1)/(h_2 - h_1)$.

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8. Statistical Mechanics Foundations

8.1 Microstate, Macrostate, and Postulate of Equal A Priori Probabilities

A microstate specifies positions and momenta of all particles. A macrostate is defined by macroscopic constraints ($U,V,N$). For an isolated system at equilibrium, all accessible microstates are equally probable.

8.2 Boltzmann Entropy (Microcanonical Ensemble)

For an isolated system with energy $U$ and multiplicity $\Omega(U,V,N)$:

$$\boxed{S(U,V,N) = k_B \ln \Omega(U,V,N)}.$$

Temperature arises from

$$\boxed{\frac{1}{T} = \left(\frac{\partial S}{\partial U}\right)_{V,N}}.$$

Pressure and chemical potential follow from derivatives of $S$:

$$\frac{P}{T} = \left(\frac{\partial S}{\partial V}\right)_{U,N}, \qquad -\frac{\mu}{T} = \left(\frac{\partial S}{\partial N}\right)_{U,V}.$$

8.3 Canonical Ensemble via Weak Coupling to a Reservoir

System S weakly coupled to a large reservoir R at total energy $U_{tot}$. Probability of S in microstate $i$ with energy $\varepsilon_i$:

$$P_i \propto \Omega_R(U_{tot}-\varepsilon_i) \approx \exp\left[\frac{1}{k_B}\left(S_R(U_{tot}) - \frac{\varepsilon_i}{T}\right)\right] \Rightarrow P_i = \frac{e^{-\beta \varepsilon_i}}{Z},$$

with $\beta = 1/(k_BT)$ and partition function

$$\boxed{Z(T,V,N) = \sum_i e^{-\beta \varepsilon_i}}.$$

8.4 Thermodynamic Potentials from $Z$

Helmholtz free energy:

$$\boxed{A(T,V,N) = -k_BT \ln Z}.$$

From $A$:

$$S = -\left(\frac{\partial A}{\partial T}\right)_{V,N}, \quad P = -\left(\frac{\partial A}{\partial V}\right)_{T,N}, \quad \mu = \left(\frac{\partial A}{\partial N}\right)_{T,V}.$$

Energy:

$$U = -\frac{\partial}{\partial \beta} \ln Z = \sum_i P_i\,\varepsilon_i.$$

Fluctuations: $C_V = (\partial U/\partial T)_{V} = k_B \beta^2 (\langle \varepsilon^2\rangle - \langle \varepsilon\rangle^2)$.

8.5 Gibbs and Grand Canonical Ensembles

• Isothermal–isobaric (Gibbs) ensemble: partition function $\Delta = \sum_{states} e^{-\beta (\varepsilon + PV)}$, gives $G = -k_BT \ln Δ$.
• Grand canonical ensemble: $\Xi = \sum_{N} e^{\beta \mu N} Z_N$, gives grand potential $\Phi = -k_BT \ln Ξ = -pV$. These connect directly to engineering free energies and equations of state.

8.6 Gibbs Entropy and Information Form

For discrete microstates with probabilities {$p_i$}:

$$\boxed{S = -k_B \sum_i p_i \ln p_i}.$$

Maximizing $S$ under constraints of normalization and mean energy yields the canonical distribution. This mirrors Shannon information entropy H = -\sum_i p_i g_2 p_i.

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9. Micro–Macro Consistency: Recovering Clausius

For a quasi-static heat transfer at temperature $T$ while the system remains canonical,

$$\delta Q_{rev} = dU - \delta W = dU + P\,dV.$$

From $dA = -S\,dT - P\,dV$, at constant $T$: $\delta Q_{rev} = T\,dS$. Thus,

$$\boxed{dS = \frac{\delta Q_{rev}}{T}},$$

matching the macroscopic definition. For any real process, convexity and Liouville dynamics imply $\Delta S_{total} \ge 0$, reproducing Clausius inequality.

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10. Entropy of Ideal Gases and Mixing

10.1 Ideal Monatomic Gas (Outline)

The Sackur–Tetrode equation (quantum-corrected) for molar entropy:

$$\bar S = R\left[\ln\!\left(\frac{V}{N}\left(\frac{4\pi m U}{3Nh^2}\right)^{3/2}\right) + \frac{5}{2}\right].$$

This recovers classical results and resolves the Gibbs paradox via indistinguishability ($1/N!$ in $\Omega$).

10.2 Entropy Change for Ideal Gas (Engineering Form)

For any ideal gas with temperature-dependent $c_p(T)$:

$$\Delta s(T,P) = \int_{T_1}^{T_2} \frac{c_p(T)}{T} dT - R\ln\!\left(\frac{P_2}{P_1}\right),$$ $$\Delta s(T,V) = \int_{T_1}^{T_2} \frac{c_v(T)}{T} dT + R\ln\!\left(\frac{V_2}{V_1}\right).$$

10.3 Entropy of Mixing (Ideal Gas Mixtures)

For isothermal, isobaric mixing of ideal gases:

$$\Delta S_{mix} = -R \sum_i n_i \ln y_i,$$

where $y_i$ are mole fractions.

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11. Chemical Equilibrium, Affinity, and $K(T)$

11.1 Gibbs Free Energy and Spontaneity

At constant $T,P$: $\Delta G < 0$ for spontaneous change; equilibrium when $\Delta G = 0$.

11.2 Chemical Potential and Reaction Progress

For reaction $\sum_i \nu_i A_i = 0$ with extent $\xi$:

$$\left(\frac{\partial G}{\partial \xi}\right)_{T,P} = \sum_i \nu_i \mu_i = -\mathcal A,$$

where $\mathcal A$ is the chemical affinity. Equilibrium: $\mathcal A=0$.

11.3 Equilibrium Constant

For ideal gases or solutions with standard state $\mu_i = \mu_i^\circ + RT\ln a_i$ (activity $a_i$):

$$\boxed{\Delta_r G^\circ(T) = -RT\ln K(T)},\qquad K(T) = \prod_i a_i^{\nu_i}.$$

Temperature dependence via van ’t Hoff:

$$\boxed{\frac{d\ln K}{dT} = \frac{\Delta_r H^\circ}{RT^2}}.$$

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12. Non-Equilibrium and Finite-Rate Effects

12.1 Finite Heat Transfer

Entropy generation for heat flow $Q$ between reservoirs at $T_H>T_C$: $S_{gen}=Q\left(\tfrac{1}{T_C}-\tfrac{1}{T_H}\right)$.

12.2 Viscous Dissipation

For Newtonian fluid with viscosity $\mu$ and strain-rate tensor $\mathbf D$: $\Phi=2\mu\,\mathbf D:\mathbf D$ is the dissipation rate density. Local $\sigma_{visc}=\Phi/T$.

12.3 Diffusion and Heat Conduction Coupling

Onsager reciprocal relations near equilibrium link fluxes and forces; symmetry of phenomenological coefficients reduces modeling burden.

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13. Worked Engineering Examples

Example 1: Steam Turbine, Entropy Balance and Exergy Loss

Given inlet $p_1, T_1$, outlet $p_2$, mass flow $\dot m$, and negligible heat transfer. Steps:

1. From property tables compute $h_1, s_1$. For isentropic outlet, get $h_{2s}$ at $p_2, s_2=s_1$.
2. Measured outlet has $h_2 > h_{2s}$. Isentropic efficiency $\eta_t=(h_1-h_2)/(h_1-h_{2s})$.
3. Entropy generation rate: $\dot S_{gen}=\dot m(s_2-s_1)$.
4. Exergy destruction: $\dot X_{dest}=T_0\,\dot S_{gen}$. This is the avoidable power loss upper bound.

Example 2: Refrigerator COP Upper Bound

For a refrigerator operating between $T_L=263\,\text{K}$ and $T_H=303\,\text{K}$: $\mathrm{COP}_{Carnot}=T_L/(T_H-T_L)=263/40=6.575$. Any real design must have COP $<6.575$ at these terminal temperatures.

Example 3: Ideal Gas Entropy Change with Variable $c_p(T)$

Use a polynomial fit $c_p/R = a + bT + cT^2 + dT^{-2}$. Integrate to obtain $\Delta s$ and compare to tabulated $s^\circ(T)$.

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14. Summary Equations

• Clausius inequality: $\displaystyle \oint \delta Q/T \le 0$.
• Entropy definitions: $dS=\delta Q_{rev}/T$; $S=-k_B\sum p_i\ln p_i$; $S=k_B\ln Ω$.
• Control volume entropy rate: $\displaystyle dS_{CV}/dt = \sum \dot Q/T + \sum_{in} \dot m s - \sum_{out} \dot m s + \dot S_{gen}$.
• Exergy rate balance: $\displaystyle \dot W = \sum (1-T_0/T_k)\,\dot Q_k + \sum_{in} \dot m\Psi - \sum_{out} \dot m\Psi - T_0\dot S_{gen}$.
• Canonical ensemble: $Z=\sum e^{-\beta\varepsilon_i},\; A=-k_BT\ln Z,\; U=-\partial_{\beta}\ln Z,\; S=-(\partial A/\partial T)_{V,N}$.

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15. References and Standards

• H. B. Callen, Thermodynamics and an Introduction to Thermostatistics.
• R. K. Pathria and P. D. Beale, Statistical Mechanics.
• A. Bejan, Advanced Engineering Thermodynamics.
• Moran, Shapiro, et al., Fundamentals of Engineering Thermodynamics.
• ASME PTCs and IAPWS for water/steam property standards.

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16. Cross-Links

• See First Law: ../first-law.md for energy balances and definitions of $U, H, A, G$.
• See Fluid Dynamics/01_Control_Volumes.md for Reynolds Transport Theorem and local balances.
• See Appendix C for mathematical identities and integral transforms useful in entropy generation minimization.