🔥 Chemical Thermodynamics & Thermochemistry

Thermodynamics dictates the absolute feasibility and energetic limits of all chemical transformations. In competitive exams, mastery requires clear separation between state functions vs. path functions, mastery of exact work differentials in irreversible expansions, and multi-parameter free energy calculations ($Delta{G}^{c}irc=-RTlnK$).

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1. ⚙️ First Law of Thermodynamics

$$\mathrm{\Delta }U=q+w(\text{IUPAC Sign Convention: }w>0\text{ if work done ON system})$$$$dw=-P_{\text{ext}}dV$$

1.1 Work Done in Various Processes (Ideal Gas)

Process	Reversible Path	Irreversible Path (against constant $P_{\text{ext}}$)
Isothermal ($T=\text{const},\mathrm{\Delta }U=0$)	$w_{\text{rev}}=-nRT\ln \left(\frac{V_2}{V_1}\right)=-nRT\ln \left(\frac{P_1}{P_2}\right)$	$w_{\text{irrev}}=-P_{\text{ext}}(V_2-V_1)=-P_{\text{ext}}(\frac{nRT}{P_2}-\frac{nRT}{P_1})$
Adiabatic ($q=0,\mathrm{\Delta }U=w$)	$T{V}^{\gamma -1}=\text{const},P{V}^{\gamma }=\text{const}$ $w_{\text{rev}}=\frac{nR(T_2-T_1)}{\gamma -1}=\frac{P_2{V}_2-P_1{V}_1}{\gamma -1}$	$w_{\text{irrev}}=n{C}_v(T_2-T_1)=-P_{\text{ext}}(V_2-V_1)$ Solve for $T_2$: $n{C}_v(T_2-T_1)=-P_{\text{ext}}(\frac{nR{T}_2}{P_2}-\frac{nR{T}_1}{P_1})$
Isochoric ($V=\text{const}$)	$w=0,q_v=\mathrm{\Delta }U=n{C}_v\mathrm{\Delta }T$	$w=0,q_v=\mathrm{\Delta }U=n{C}_v\mathrm{\Delta }T$
Isobaric ($P=\text{const}$)	$w=-P(V_2-V_1)=-nR\mathrm{\Delta }T$ $q_p=\mathrm{\Delta }H=n{C}_p\mathrm{\Delta }T$	$w=-P_{\text{ext}}(V_2-V_1)$ $q_p=\mathrm{\Delta }H=n{C}_p\mathrm{\Delta }T$

Crucial Exam Comparison: Reversible vs Irreversible Adiabatic Expansion

In adiabatic expansion starting from identical initial states $(P_1,V_1,T_1)$ to the same final pressure $P_2$:

$$|w_{\text{rev}}|>|w_{\text{irrev}}|⟹T_{2,\text{rev}}<T_{2,\text{irrev}}⟹V_{2,\text{rev}}<V_{2,\text{irrev}}$$

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2. 🧪 Thermochemistry & Enthalpy Relations

$$H=U+PV⟹\mathrm{\Delta }H=\mathrm{\Delta }U+\mathrm{\Delta }(PV)=\mathrm{\Delta }U+\mathrm{\Delta }{n}_{g}RT$$

where $\mathrm{\Delta }{n}_g=\sum n_{\text{gaseous products}}-\sum n_{\text{gaseous reactants}}$.

2.1 Hess's Law & Enthalpies of Reaction

1. Standard Enthalpy of Formation ($\mathrm{\Delta }{H}_f^{\circ }$):$$\mathrm{\Delta }{H}_{\text{rxn}}^{\circ }=\sum \mathrm{\Delta }{H}_f^{\circ }(\text{Products})-\sum \mathrm{\Delta }{H}_f^{\circ }(\text{Reactants})$$Note: $\mathrm{\Delta }{H}_f^{\circ }$ is strictly zero for elements in their standard reference states ($O_{2(g)},N_{2(g)},C_{\text{graphite}},B{r}_{2(l)},I_{2(s)},S_{\text{rhombic}},P_{\text{white}}$).
2. Bond Dissociation Enthalpy (BDE) (all species in gaseous state):$$\mathrm{\Delta }{H}_{\text{rxn}}^{\circ }=\sum \text{BDE}(\text{Bonds Broken in Reactants})-\sum \text{BDE}(\text{Bonds Formed in Products})$$
3. Resonance Energy:$$\text{Resonance Energy}=\mathrm{\Delta }{H}_{f,\text{experimental}}^{\circ }-\mathrm{\Delta }{H}_{f,\text{calculated (theoretical)}}^{\circ }$$
4. Kirchhoff's Law (Temperature Dependence of $\mathrm{\Delta }H$):$$\mathrm{\Delta }{H}_{T_2}=\mathrm{\Delta }{H}_{T_1}+{\int }_{T_1}^{T_2}\mathrm{\Delta }{C}_{p}dT\approx \mathrm{\Delta }{H}_{T_1}+\mathrm{\Delta }{C}_p(T_2-T_1)$$

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3. 🌀 Second & Third Laws: Entropy & Spontaneity

3.1 Entropy ($S$) Calculus

$$dS=\frac{d{q}_{\text{rev}}}{T}$$

For an ideal gas undergoing a state change $(T_1,V_1,P_1)\to (T_2,V_2,P_2)$:

$$\mathrm{\Delta }{S}_{\text{system}}=n{C}_v\ln \left(\frac{T_2}{T_1}\right)+nR\ln \left(\frac{V_2}{V_1}\right)=n{C}_p\ln \left(\frac{T_2}{T_1}\right)-nR\ln \left(\frac{P_2}{P_1}\right)$$

• Surroundings Entropy: $\mathrm{\Delta }{S}_{\text{surr}}=-\frac{q_{\text{system}}}{T_{\text{surr}}}$
• Clausius Criterion of Spontaneity:$$\mathrm{\Delta }{S}_{\text{universe}}=\mathrm{\Delta }{S}_{\text{system}}+\mathrm{\Delta }{S}_{\text{surroundings}}\ge 0$$
  • $\mathrm{\Delta }{S}_{\text{univ}}>0$: Spontaneous irreversible process.
  • $\mathrm{\Delta }{S}_{\text{univ}}=0$: Reversible process at dynamic equilibrium.

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4. 💎 Gibbs Free Energy ($G$) & Chemical Equilibrium

$$G=H-TS⟹\mathrm{\Delta }G=\mathrm{\Delta }H-T\mathrm{\Delta }S(\text{at constant }T,P)$$

4.1 Temperature Dependence of Spontaneity

$\mathrm{\Delta }H$	$\mathrm{\Delta }S$	Spontaneity Condition	Examples
$-$	$+$	Spontaneous at ALL temperatures ($DeltaG<0$ always)	$2O_{3(g)}\to 3O_{2(g)}$
$+$	$-$	Non-spontaneous at ALL temperatures ($DeltaG>0$ always)	$3O_{2(g)}\to 2O_{3(g)}$
$-$	$-$	Spontaneous at LOW temperatures ($T<\mathrm{\Delta }H/\mathrm{\Delta }S$)	Gas condensation, Solidification
$+$	$+$	Spontaneous at HIGH temperatures ($T>\mathrm{\Delta }H/\mathrm{\Delta }S$)	Decomposition ($CaC{O}_{3(s)}\to Ca{O}_{(s)}+C{O}_{2(g)}$)

4.2 Van 't Hoff Reaction Isobar

$$\mathrm{\Delta }{G}^{\circ }=-RT\ln K_{\text{eq}}⟹\ln K_{\text{eq}}=-\frac{\mathrm{\Delta }{H}^{\circ }}{RT}+\frac{\mathrm{\Delta }{S}^{\circ }}{R}$$$$\ln \left(\frac{K_2}{K_1}\right)=\frac{\mathrm{\Delta }{H}^{\circ }}{R}(\frac{1}{T_1}-\frac{1}{T_2})$$

• For Endothermic reactions ($Delta{H}^{c}irc>0$): $K_{\text{eq}}$ increases with increasing $T$.
• For Exothermic reactions ($Delta{H}^{c}irc<0$): $K_{\text{eq}}$ decreases with increasing $T$.