Review

Review#

1. Checklist of Key Concepts#

  1. Origins & Roles of Free Energies

    • Helmholtz Free Energy Defined as

      \[A \;=\; U - T\,S,\]

      with natural variables \(\left(T,V\right)\). At fixed temperature and volume, a process is spontaneous if \(\Delta A<0\).

    • Gibbs Free Energy Defined as

      \[G \;=\; H - T\,S \;=\; U + P\,V - T\,S,\]

      with natural variables \(\left(T,P\right)\). At constant \(T,P\), spontaneity requires \(\Delta G<0\), and \(\Delta G=0\) marks equilibrium.

    • Statistical‐Mechanical Link The partition function \(Q\) underpins both; e.g.,

      \[A = -k_B T \ln Q,\]

      connecting macroscopic free energy to microscopic states.

  2. The Third Law of Thermodynamics

    • Planck’s Statement: As \(T\to0\), the entropy of any pure, defect-free crystalline substance approaches zero.

    • Boltzmann’s View: \(S = k_B\ln\Omega\), where \(\Omega\) is the count of microstates. A perfect crystal has \(\Omega=1\) at 0 K.

    • Residual Entropy: Real crystals often contain equivalent configurations (e.g., orientational disorder in solid CO), giving a nonzero “residual” \(S(0)\). Example: \(S_\text{res}=k_B\ln2\) per molecule.

  3. T- and P-Dependence of \(\Delta G\)

    • Temperature Effects:

      \[\Delta G(T_f)-\Delta G(T_i) = \int_{T_i}^{T_f}\!\Delta C_p\,dT \;-\; T\,\int_{T_i}^{T_f}\!\frac{\Delta C_p}{T}\,dT.\]

      Splits enthalpic and entropic contributions; used to assess ammonia formation from 298 K to 773 K, showing \(\Delta G\) becomes less negative (and eventually positive) at high \(T\).

    • Pressure Effects: At fixed \(T\),

      \[dG = V\,dP,\]

      so for an ideal‐gas reaction with \(\Delta n\) change in moles,

      \[\Delta G(P) = \Delta G^\circ + RT\,\Delta n\,\ln\frac{P}{P^\circ}.\]

      Example: raising \(P\) to ∼319 bar at 773 K makes ammonia synthesis spontaneous again.

2. Checklist of Most Important Equations#

Equation

Variables

Narrative & Use Cases

Caveats/Approximations

\(dU = T\,dS - P\,dV + \sum_i \mu_i\,dN_i\)

\(U\): internal energy
\(S\): entropy
\(P\): pressure
\(V\): volume
\(\mu_i\), \(N_i\): chemical potentials & particle numbers

Fundamental First Law in natural variables. Basis for deriving all thermodynamic potentials.

Omits non‐PV work; assumes simple homogeneous system.

\(H = U + P\,V,\quad dH = T\,dS + V\,dP\)

\(H\): enthalpy

Convenient for constant‐\(P\) processes (e.g., reactions in open vessels).

Requires only PV work.

\(A = U - T\,S,\quad dA = -S\,dT - P\,dV\)

\(A\): Helmholtz free energy

Governs spontaneity under fixed \(T,V\). Maximum non‐PV work equals \(-\Delta A\).

Valid for closed systems at constant \(T,V\).

\(G = H - T\,S,\quad dG = -S\,dT + V\,dP\)

\(G\): Gibbs free energy

Determines whether processes at constant \(T,P\) are spontaneous (\(\Delta G<0\)).

Holds for constant‐\(T,P\); neglects non‐PV work.

\(A = -k_B T \ln Q\)

\(k_B\): Boltzmann constant
\(Q\): partition function

Links macroscopic \(A\) to microscopic energy levels; compute equilibrium constants and thermodynamic functions.

Exact only if full \(Q\) known; often approximated.

\(S = \frac{U}{T} + k_B\ln Q\)

—

Statistical‐mechanical entropy in the canonical ensemble.

Same as above.

\(S = k_B\ln \Omega\)

\(\Omega\): number of microstates

Boltzmann’s principle for isolated systems; underlies the Third Law.

Conceptual; \(\Omega\) intractable in practice.

\(\Delta S = \displaystyle\int_{0}^{T} \frac{C_p(T')}{T'}\,dT'\)

\(C_p\): heat capacity at constant pressure

From Third Law: computes \(S\) from 0 K (where \(S=0\) for a perfect crystal).

Assumes no phase transitions; \(C_p\to0\) as \(T\to0\).

\(\Delta H = \displaystyle\int_{T_i}^{T_f} C_p(T)\,dT\)

—

Temperature corrections to standard‐state enthalpies.

Needs \(C_p(T)\); often treated as constant.

\(dG = -S\,dT + V\,dP\)

—

Basis for Clapeyron relation and pressure/temperature dependence of equilibrium.

Requires known \(S\) and \(V\).

\(G(P) = G^\circ + RT\ln\frac{P}{P^\circ}\)

\(R\): gas constant

Adjusts Gibbs energy for nonstandard pressures in ideal gas approximation; used to find equilibrium pressure.

Ideal‐gas assumption.

\(\displaystyle\Delta G(P,T) = \Delta G^\circ(T) + RT\,\Delta n\,\ln\frac{P}{P^\circ}\)

\(\Delta n\): net change in moles of gas

Solves for \(P\) or \(T\) at which a gas‐phase reaction becomes spontaneous or reaches equilibrium.

Assumes all species ideal gases; single pressure for all components.