Review

1. Checklist of Key Concepts¶

Section 5.1: Free Energy¶

1. Fundamental Inequality

   • Combining the First Law with the Clausius inequality $dS \ge \delta q/T$ and the work bound $\delta w \ge -P\,dV$ gives

   $$dU \le T\,dS - P\,dV,$$

   (1)

   with equality for reversible changes. This single inequality encodes the direction of spontaneous evolution for a closed system with $PV$-only work.

2. Extremum Principles for Thermodynamic Potentials

   • Under different experimental constraints, a particular potential is minimized at equilibrium:

     • Constant $S, V$: $dU \le 0$ ⇒ $U$ minimized.

     • Constant $S, P$: $dH \le 0$ ⇒ $H$ minimized.

     • Constant $T, V$: $dA \le 0$ ⇒ $A$ minimized.

     • Constant $T, P$: $dG \le 0$ ⇒ $G$ minimized.

   • The $T,V$ and $T,P$ cases cover the vast majority of laboratory conditions.

3. Legendre Transforms and Natural Variables

   • Each potential is obtained from $U$ by a Legendre transform that swaps an extensive variable for its conjugate intensive variable, matching the natural variables to the experimentally controllable ones.

   • Summary of natural variables and equilibrium differentials:

   potential	definition	differential	natural variables
   $U$	—	$dU = T\,dS - P\,dV$	$S, V$
   $H$	$U + PV$	$dH = T\,dS + V\,dP$	$S, P$
   $A$	$U - TS$	$dA = -S\,dT - P\,dV$	$T, V$
   $G$	$U + PV - TS$	$dG = -S\,dT + V\,dP$	$T, P$

   • Defining $A$ and $G$ extends the strategy used in Section 3.3 to define $H$.

4. Conjugate Variables as Derivatives

   • At equilibrium, intensive variables are read off as slopes of a potential along its natural-variable axes. For example:

   $$T = \left(\frac{\partial U}{\partial S}\right)_V,\qquad P = -\left(\frac{\partial U}{\partial V}\right)_S,$$

   (2)

   with analogous identities for $H$, $A$, and $G$.

5. Euler Relation (fixed $N$)

   • For a simple system with fixed composition, $U(S,V)$ is homogeneous of degree 1 in its extensive arguments. Euler’s theorem gives

   $$U = TS - PV.$$

   (3)

   • Allowing $N$ to vary adds a $\mu N$ term; this will be developed when the chemical potential is introduced.

6. Helmholtz Free Energy as a Master Potential

   • The bridge to statistical mechanics from Section 4.3 is

   $$A = -k_{\mathrm B}T\ln Q.$$

   (4)

   • Combined with $dA = -S\,dT - P\,dV$, every equilibrium property follows by differentiation:

   $$S = -\left(\frac{\partial A}{\partial T}\right)_V,\quad P = -\left(\frac{\partial A}{\partial V}\right)_T = k_{\mathrm B}T\!\left(\frac{\partial\ln Q}{\partial V}\right)_T,$$

   (5)

   with $U = A + TS$ and $C_V = (\partial U/\partial T)_{N,V}$ following in turn.

7. Ideal Gas from $A$

   • For a monatomic ideal gas, $Q = V^N/(N!\,\Lambda^{3N})$ and Stirling’s approximation give

   $$A \approx -Nk_{\mathrm B}T\!\left[\ln\!\left(\frac{V}{N\Lambda^3}\right) + 1\right].$$

   (6)

   • Differentiating yields the Sackur–Tetrode entropy $S = Nk_{\mathrm B}[\ln(V/N\Lambda^3) + 5/2]$, the equation of state $PV = Nk_{\mathrm B}T$, and the internal energy $U = \tfrac{3}{2}Nk_{\mathrm B}T$ — all from a single function.

8. Gibbs Free Energy and Non-$PV$ Work

   • When non-$PV$ work is present, the First Law gains a $\delta w_{\text{non-}PV}$ term. At constant $T$ and $P$:

   $$dG\big|_{T,P} = \delta w_{\text{non-}PV,\text{rev}}.$$

   (7)

   • $\Delta G$ measures the reversible non-$PV$ work available from a process — the “useful” work beyond unavoidable $PV$ work against the atmosphere. For irreversible processes, the useful work output is strictly less than $|\Delta G|$.

Section 5.2: Third Law¶

1. Planck’s Statement of the Third Law

   • As $T \to 0\,\mathrm{K}$, the entropy of any pure crystalline substance approaches a constant. For an ideal crystal (no defects, impurities, or disorder), that constant is zero:

   $$S(0\,\mathrm{K}) = 0 \quad\text{(ideal crystal)}.$$

   (8)

   • This fixes the otherwise-undetermined integration constant in $\Delta S = \int \delta q_{\mathrm{rev}}/T$.

2. Statistical-Mechanical Justification

   • From Boltzmann’s formula $S = k_{\mathrm B}\ln\Omega$ (Section 4.3), a unique ground state has $\Omega_0 = 1$, so $S(0) = k_{\mathrm B}\ln 1 = 0$. The third law is the statement that a system with a nondegenerate ground state has zero entropy at absolute zero.

3. Residual Entropy

   • If the ground state is degenerate or the system has frozen-in disorder ($\Omega_0 > 1$), entropy approaches a nonzero constant:

   $$S_{\mathrm{res}} = k_{\mathrm B}\ln\Omega_0.$$

   (9)

   • This does not violate the third law: Planck’s statement says $S \to$ a constant, not necessarily zero.

   • Examples:

     • Solid CO (two nearly isoenergetic orientations per molecule): $\Omega_0 = 2^N$, giving $S_{\mathrm{res,m}} = R\ln 2 \approx 5.76\ \mathrm{J\,mol^{-1}\,K^{-1}}$ (experiment: $\approx 4.6$, indicating partial ordering).

     • Ice (Pauling’s model with ice rules): $\Omega_0 \approx (3/2)^N$, giving $S_{\mathrm{res,m}} = R\ln(3/2) \approx 3.37\ \mathrm{J\,mol^{-1}\,K^{-1}}$ (experiment: $\approx 3.41$).

4. Heat Capacities Must Vanish as $T \to 0$

   • For the integral $S(T) = S(0) + \int_0^T (C_P/T')\,dT'$ to converge, $C_P$ (and similarly $C_V$) must go to zero at $T = 0$.

   • Classical equipartition gives a temperature-independent $C_V$ and therefore violates the third law. Quantum statistical mechanics resolves the contradiction: the Einstein and Debye heat capacities (Section 2.6) vanish as $T \to 0$ because vibrational modes freeze out.

5. Unattainability of Absolute Zero (Nernst form)

   • No finite sequence of thermodynamic operations can cool a system to exactly $T = 0$. Each successive cooling step removes less entropy than the previous one; the process converges but never reaches zero.

6. Absolute (Third-Law) Entropies

   • With $S(0) = 0$ fixed for ideal crystals, absolute entropies are computed by integrating $C_P/T$ from 0 to $T$ and adding phase-transition contributions:

   $$S^\circ(T) = \int_0^T \frac{C_P(T')}{T'}\,dT' + \sum_{\text{transitions}} \frac{\Delta H_{\mathrm{trs}}}{T_{\mathrm{trs}}}.$$

   (10)

   • Tabulated $S^\circ(298.15\ \mathrm{K})$ values (NIST WebBook, JANAF) are the basis for computing reaction entropies and Gibbs energies.

Section 5.3: Ammonia Formation¶

1. Thermodynamics vs. Kinetics

   • $\Delta G$ determines which direction is favored at equilibrium; activation barriers determine how fast equilibrium is reached. These are independent questions.

   • For $\mathrm{N_2 + 3\,H_2 \to 2\,NH_3}$ at 298.15 K: $\Delta H_r^\circ \approx -92\ \mathrm{kJ\,mol^{-1}}$, $\Delta S_r^\circ \approx -198\ \mathrm{J\,K^{-1}\,mol^{-1}}$, $\Delta G_r^\circ \approx -33\ \mathrm{kJ\,mol^{-1}}$. Thermodynamically spontaneous but kinetically inert because of the $\mathrm{N\equiv N}$ triple bond.

   • A catalyst lowers the activation barrier without changing $\Delta G_r^\circ$ or the equilibrium constant — it accelerates forward and reverse reactions equally.

2. Temperature Effect for $\Delta S_r^\circ < 0$

   • From $\Delta G_r^\circ = \Delta H_r^\circ - T\,\Delta S_r^\circ$, a negative $\Delta S_r^\circ$ means $-T\,\Delta S_r^\circ > 0$ grows with $T$, driving $\Delta G_r^\circ$ upward.

   • At high enough $T$, $\Delta G_r^\circ$ changes sign and the reaction becomes non-spontaneous at 1 bar.

3. Temperature Corrections to $\Delta H$ and $\Delta S$

   • From constant-$P$ integration:

   $$H(T_f) - H(T_i) = \int_{T_i}^{T_f} C_P\,dT,\qquad S(T_f) - S(T_i) = \int_{T_i}^{T_f} \frac{C_P}{T}\,dT.$$

   (11)

   • With constant $C_P$: $\Delta H = C_P\,\Delta T$ and $\Delta S = C_P\ln(T_f/T_i)$.

4. Ideal-Gas $C_P$ from Active Degrees of Freedom

   • Using $f$ translational + rotational degrees of freedom (rigid-rotor, no vibrations):

   $$C_P = \left(\frac{f}{2} + 1\right)R.$$

   (12)

   • Linear $\mathrm{N_2, H_2}$: $f = 5$, $C_P = \tfrac{7}{2}R$.

   • Nonlinear $\mathrm{NH_3}$: $f = 6$, $C_P = 4R$.

   • For the ammonia reaction: $\Delta C_{P,r} = 2(4R) - (7R/2) - 3(7R/2) = -6R$.

5. Result at 500 °C

   • Applying constant-$C_P$ corrections from 298.15 K to 773.15 K gives $\Delta H_r^\circ(773) \approx -115.6\ \mathrm{kJ\,mol^{-1}}$, $\Delta S_r^\circ(773) \approx -245.4\ \mathrm{J\,K^{-1}\,mol^{-1}}$, and

   $$\Delta G_r^\circ(773) \approx +74.1\ \mathrm{kJ\,mol^{-1}}.$$

   (13)

   • The reaction is non-spontaneous at 1 bar — the central dilemma of Haber–Bosch.

6. Pressure Dependence of $G$ (Ideal Gas)

   • From $dG\big|_T = V\,dP$ with $V = RT/P$:

   $$G(T, P) - G^\circ(T) = RT\ln\!\left(\frac{P}{P^\circ}\right).$$

   (14)

   • For a reaction with stoichiometric change $\Delta\nu$ in moles of gas:

   $$\Delta G_r(T, P) \approx \Delta G_r^\circ(T) + \Delta\nu\,RT\ln\!\left(\frac{P}{P^\circ}\right).$$

   (15)

   • For $\mathrm{N_2 + 3\,H_2 \to 2\,NH_3}$: $\Delta\nu = 2 - 4 = -2$, so increasing $P$ lowers $\Delta G_r$ (Le Châtelier: high pressure favors the side with fewer gas molecules).

7. Restoring Spontaneity with Pressure

   • Setting $\Delta G_r = 0$ and solving for $P$:

   $$P = P^\circ\exp\!\left(-\frac{\Delta G_r^\circ(T)}{\Delta\nu\,RT}\right).$$

   (16)

   • At 500 °C: $P \approx 320\ \mathrm{bar}$. At 400 °C: $P \approx 87\ \mathrm{bar}$. The Haber–Bosch operating window (400–500 °C, 150–350 bar) is the engineering compromise predicted by these estimates.

8. Engineering Knobs Summary

   • Catalyst: speeds up both directions; does not change $\Delta G_r^\circ$ or $K$.

   • Temperature: raises rate (Arrhenius) but pushes $\Delta G_r^\circ$ upward when $\Delta S_r^\circ < 0$.

   • Pressure: compensates for the temperature penalty when $\Delta\nu < 0$.

2. Checklist of Most Important Equations¶

Below is a unified list of the major equations from Sections 5.1–5.3.

A. Fundamental Inequality

• Applicability: closed system with $PV$-only work. Equality for reversible processes; strict inequality for spontaneous irreversible processes.

B. Thermodynamic Potentials and Their Differentials (fixed $N$, $PV$-only work)

• Natural variables: $U(S,V)$, $H(S,P)$, $A(T,V)$, $G(T,P)$.

C. Extremum Principles (equilibrium conditions)

• The potential whose natural variables match the imposed constraints is minimized at equilibrium.

D. Helmholtz Free Energy as a Master Potential

• Applicability: canonical ensemble. All equilibrium thermodynamic quantities follow by differentiation of a single function $A(T,V,N)$.

E. Pressure from the Partition Function

• Applicability: canonical ensemble. Recovers $PV = Nk_{\mathrm B}T$ for the monatomic ideal gas.

F. Sackur–Tetrode Entropy (monatomic ideal gas)

• Applicability: ideal monatomic gas in the classical (high-$T$, low-density) limit where Stirling’s approximation and Boltzmann statistics apply.

G. Gibbs Free Energy and Non-$PV$ Work

• Applicability: reversible process at constant $T$ and $P$. $\Delta G$ equals the reversible non-$PV$ work; for irreversible processes, $|\Delta G|$ is an upper bound on useful work output.

H. Third Law

• Fixes the entropy reference point; residual entropy reflects ground-state multiplicity, not a failure of the third law.

I. Absolute (Third-Law) Entropy

• Applicability: any substance whose heat capacity and phase-transition enthalpies are known from 0 to $T$; supplies tabulated $S^\circ(298.15\ \mathrm{K})$ values.

J. Temperature Corrections to Reaction Thermodynamics (constant $C_P$)

• Applicability: reaction data from a reference temperature $T_i$ to a target temperature $T_f$, assuming temperature-independent heat capacities over the interval.

K. Ideal-Gas Pressure Dependence of $G$

• Applicability: pure ideal gas at constant $T$. For a reaction with stoichiometric gas change $\Delta\nu$:

• For $\Delta\nu < 0$, higher pressure lowers $\Delta G_r$ and favors products.