Chemical Thermodynamics

System: Part of universe under study. Surroundings: Everything else. Boundary separates them.

Types of System:

State Function: Property depending only on state, not on path (e.g., $P, V, T, U, H, S, G$). Path Function: Depends on path taken ($q, W$).

Extensive Properties: Depend on amount ($V, n, U, H, S$). Intensive Properties: Independent of amount ($T, P, d, \chi$).

Internal Energy ($U$): Total energy (KE + PE + bond + nuclear) of system. Absolute $U$ not measurable; only $\Delta U$ measurable.

Work (W): $W = -P_{\text{ext}}\Delta V$ for $P$-$V$ work.

Sign Convention (IUPAC):

• $q > 0$: heat absorbed by system; $q < 0$: heat released
• $W > 0$: work done on system; $W < 0$: work done by system

First Law of Thermodynamics: Energy is conserved. $$\Delta U = q + W$$

For different processes:

Reversible vs Irreversible:

• Reversible: infinitely slow, equilibrium maintained
• Irreversible: real, finite time, equilibrium not maintained

Isothermal reversible work (ideal gas): $W = -nRT\ln(V_2/V_1) = -nRT\ln(P_1/P_2)$.

Adiabatic reversible work: $W = (P_2V_2 - P_1V_1)/(\gamma - 1) = nC_V\Delta T$.

Enthalpy ($H$): State function defined as $H = U + PV$.

For change at constant $P$: $\Delta H = \Delta U + P\Delta V = q_P$.

Relation: $\Delta H = \Delta U + \Delta n_g RT$, where $\Delta n_g$ = (moles of gaseous products) - (moles of gaseous reactants).

Exothermic: $\Delta H < 0$ (heat released). Endothermic: $\Delta H > 0$ (heat absorbed).

Standard Enthalpy Changes:

Standard States are pure substance at $1$ bar pressure (specific $T$, usually $298$ K). $\Delta_f H^\ominus$ of element in standard state = $0$.

Hess's Law of Constant Heat Summation: Enthalpy change for a reaction depends only on initial and final states, not on the pathway. Allows calculation of $\Delta H$ for reactions hard to measure directly.

$$\Delta H_{\text{rxn}} = \sum\nu \Delta_f H^\ominus_{\text{products}} - \sum\nu \Delta_f H^\ominus_{\text{reactants}}$$

Bond Enthalpy (Bond Dissociation Energy): Energy needed to break $1$ mole of a particular bond in gaseous state.

$$\Delta H_{\text{rxn}} = \sum(\text{Bond enthalpies of reactants}) - \sum(\text{Bond enthalpies of products})$$

(Note: works only for gaseous species; mean bond enthalpies are used for poly-atomic molecules.)

Lattice Enthalpy: $\Delta H$ when 1 mole of an ionic solid dissociates into gaseous ions: $$\text{NaCl}(s) \to \text{Na}^+(g) + \text{Cl}^-(g); \quad \Delta_L H = +788 \text{ kJ/mol}$$

Kirchhoff's Equation: $\Delta H_2 - \Delta H_1 = \Delta C_P(T_2 - T_1)$.

Entropy ($S$): Measure of disorder/randomness. State function. SI unit: J/(mol·K).

Definition: For reversible process at temperature $T$: $dS = dq_{\text{rev}}/T$.

Second Law of Thermodynamics: For an isolated system, entropy never decreases. $$\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surr}} \geq 0$$

• Spontaneous (irreversible): $\Delta S_{\text{universe}} > 0$
• Reversible (equilibrium): $\Delta S_{\text{universe}} = 0$

Entropy Changes:

Trends:

• $S(\text{gas}) > S(\text{liquid}) > S(\text{solid})$
• $\Delta S$ positive for: number of gaseous moles increases, mixing, dissolving solids
• $\Delta S$ negative for: gas → liquid → solid, gas moles decrease

Standard Entropy of Reaction: $$\Delta S^\ominus_{\text{rxn}} = \sum\nu S^\ominus_{\text{products}} - \sum\nu S^\ominus_{\text{reactants}}$$

Surroundings: Heat lost by system is gained by surroundings. $$\Delta S_{\text{surr}} = -\Delta H_{\text{sys}}/T$$ (for constant $T, P$).

Third Law of Thermodynamics: Entropy of a perfectly crystalline solid at absolute zero ($0$ K) is zero. $$S(0 \text{ K}) = 0 \text{ (perfect crystal)}$$ Allows absolute entropy values to be defined for substances.

Entropy for ideal gas (isothermal): $\Delta S = nR\ln(V_2/V_1) = -nR\ln(P_2/P_1)$.

For temperature change at const $V$: $\Delta S = nC_V \ln(T_2/T_1)$. At const $P$: $\Delta S = nC_P \ln(T_2/T_1)$.

Gibbs Free Energy ($G$): Combined criterion of spontaneity at constant $T, P$: $$G = H - TS$$ $$\Delta G = \Delta H - T\Delta S$$

Spontaneity Criteria at constant $T, P$:

$\Delta G$ Analysis Based on Signs:

Temperature of equilibrium (for case where $\Delta G$ changes sign): $$T = \Delta H/\Delta S$$

Relation with Equilibrium Constant: $$\Delta G^\ominus = -RT\ln K = -2.303 RT\log K$$

If $K > 1$: $\Delta G^\ominus < 0$ (favors products); $K < 1$: $\Delta G^\ominus > 0$ (favors reactants); $K = 1$: $\Delta G^\ominus = 0$.

Free Energy Change Under Non-standard Conditions: $$\Delta G = \Delta G^\ominus + RT\ln Q$$ where $Q$ = reaction quotient.

Relation with Electrochemistry: $\Delta G^\ominus = -nFE^\ominus_{\text{cell}}$, where $n$ = electrons, $F$ = Faraday constant ($96500$ C/mol).

Maximum useful work (other than P-V): $W_{\text{useful}} = -\Delta G$ (at constant $T, P$, reversible).

Standard $\Delta G^\ominus$ of formation: Like $\Delta_f H^\ominus$, $\Delta_f G^\ominus$ of element in standard state = $0$. For a reaction: $$\Delta G^\ominus_{\text{rxn}} = \sum\nu\Delta_f G^\ominus_{\text{products}} - \sum\nu\Delta_f G^\ominus_{\text{reactants}}$$