Chemical Thermodynamics

1. Basic Thermodynamic Terms & System Classifications

• System & Surroundings: The system is the specific part of the universe under observation; the remainder of the universe constitutes the surroundings ($\text{Universe} = \text{System} + \text{Surroundings}$).
• Types of Systems:
  1. Open System: Exchanges both matter and energy with surroundings (e.g., reactants in an open beaker).
  2. Closed System: Exchanges energy, but no matter, with surroundings (e.g., reactants in a closed conducting metal flask).
  3. Isolated System: Exchanges neither matter nor energy across its boundary (e.g., reactants in an insulated thermos flask).
• State Functions vs Path Functions:
  • State Functions: Depend strictly on the initial and final states of the system, not on the path traversed ($p, V, T, U, H, S, G$).
  • Path Functions: Depend directly on the route taken between states ($q, w$). Notice that $(q + w) = \Delta U$ is a state function.
• Intensive vs Extensive Properties:
  • Intensive Properties: Independent of the mass or size of the system (temperature, density, pressure, molar heat capacity $C_m$, standard electrode potential $E^\circ$).
  • Extensive Properties: Proportional to the quantity of matter present (mass, volume, internal energy $U$, enthalpy $H$, entropy $S$, Gibbs energy $G$, heat capacity $C$). The ratio of two extensive properties is intensive ($\text{Mass} / \text{Volume} = \text{Density}$).

2. First Law of Thermodynamics, Work & Heat

• First Law Formulation: Energy can neither be created nor destroyed; the energy of an isolated system is constant:

  $$\Delta U = q + w$$

  Where $\Delta U$ is internal energy change, $q$ is heat exchanged, and $w$ is work performed.
• Pressure-Volume Work ($p-V$ Work):

  $$w = -\int p_{\text{ext}} dV$$

  • Irreversible Single-Step Expansion against Constant $p_{\text{ext}}$:

    $$w = -p_{\text{ext}}(V_f - V_i)$$

  • Reversible Isothermal Expansion of an Ideal Gas:

    $$w_{\text{rev}} = -2.303 nRT \log\left(\frac{V_f}{V_i}\right) = -2.303 nRT \log\left(\frac{p_i}{p_f}\right)$$

  • Free Expansion: Expansion into vacuum ($p_{\text{ext}} = 0$) yields $w = 0$, irrespective of whether the expansion is reversible or irreversible.

3. Enthalpy ($H$) & Gaseous Phase Reactions

Enthalpy represents total heat content at constant pressure:

$$\Delta H = q_p$$

At constant volume, $\Delta V = 0 \implies w = 0$, giving $\Delta U = q_v$.

• Fundamental Conversion Formula:

  $$\Delta H = \Delta U + \Delta n_g RT$$

  Where $\Delta n_g = (\text{Moles of gaseous products}) - (\text{Moles of gaseous reactants})$.
  • If $\Delta n_g = 0 \implies \Delta H = \Delta U$ (e.g., $\text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2\text{HI}(g)$).
  • If $\Delta n_g > 0 \implies \Delta H > \Delta U$ (e.g., $\text{PCl}_5(g) \rightarrow \text{PCl}_3(g) + \text{Cl}_2(g)$).
  • If $\Delta n_g < 0 \implies \Delta H < \Delta U$ (e.g., $\text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g)$).

4. Heat Capacity ($C$) & Mayer's Relation

• Heat Capacity Definitions:
  • Specific Heat Capacity ($c$): $q = m \cdot c \cdot \Delta T$.
  • Molar Heat Capacity ($C_m$): $q = n \cdot C_m \cdot \Delta T$.
• Molar Heat Capacities at Constant Volume and Pressure:

  $$C_v = \left(\frac{\partial U}{\partial T}\right)_v \quad \text{and} \quad C_p = \left(\frac{\partial H}{\partial T}\right)_p$$

• Mayer's Equation for One Mole of Ideal Gas:

  $$C_p - C_v = R$$

• Atomicity & Adiabatic Index ($\gamma = C_p / C_v$):
  • Monoatomic Gas ($\text{He}, \text{Ne}, \text{Ar}$): $C_v = \frac{3}{2}R,\; C_p = \frac{5}{2}R \implies \gamma = 1.67$.
  • Diatomic Gas ($\text{H}_2, \text{O}_2, \text{N}_2$): $C_v = \frac{5}{2}R,\; C_p = \frac{7}{2}R \implies \gamma = 1.40$.
  • Non-linear Polyatomic Gas ($\text{H}_2\text{O}, \text{CH}_4$): $C_v = 3R,\; C_p = 4R \implies \gamma = 1.33$.

5. Thermochemistry, Hess's Law & Standard Enthalpies

• Hess's Law of Constant Heat Summation: If a reaction occurs in several intermediate stages, the overall standard reaction enthalpy equals the sum of the standard enthalpies of the intermediate steps:

  $$\Delta_r H = \Delta H_1 + \Delta H_2 + \Delta H_3 + \dots$$

• Standard Enthalpy of Formation ($\Delta_f H^\circ$):
  The enthalpy change for forming 1 mole of a compound from its constituent elements in their reference standard states ($25^\circ\text{C}, 1\text{ bar}$).

  $$\Delta_f H^\circ(\text{graphite}) = 0, \quad \Delta_f H^\circ(\text{O}_2, g) = 0, \quad \Delta_f H^\circ(\text{S}_{\text{rhombic}}) = 0$$

  $$\Delta_r H^\circ = \sum a_i \Delta_f H^\circ(\text{products}) - \sum b_i \Delta_f H^\circ(\text{reactants})$$

• Bond Enthalpy Calculations (Gas Phase Only):

  $$\Delta_r H^\circ = \sum \text{BE}(\text{Reactants}) - \sum \text{BE}(\text{Products})$$

• Born-Haber Cycle for Lattice Enthalpy ($\Delta_{\text{lattice}}H^\circ$):

  $$\Delta_f H^\circ(\text{NaCl}) = \Delta_{\text{sub}}H^\circ(\text{Na}) + \Delta_i H^\circ(\text{Na}) + \frac{1}{2}\Delta_{\text{bond}}H^\circ(\text{Cl}_2) + \Delta_{\text{eg}}H^\circ(\text{Cl}) + \Delta_{\text{lattice}}H^\circ$$

6. Entropy ($S$) & The Second Law of Thermodynamics

Entropy quantifies microscopic disorder or thermal randomness. It is a state function:

$$\Delta S = \frac{q_{\text{rev}}}{T} \quad (\text{Units: }\text{J K}^{-1}\text{ mol}^{-1})$$

• Spontaneity Criterion:

  $$\Delta S_{\text{total}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} > 0$$

  Where $\Delta S_{\text{surr}} = -\frac{\Delta H_{\text{sys}}}{T}$ at constant temperature and pressure.
• Entropy Trends:
  • $\text{Solid} \rightarrow \text{Liquid} \rightarrow \text{Gas}$: Entropy increases sharply.
  • Increase in temperature elevates molecular vibrational and translational freedom ($\Delta S > 0$).
  • Reactions generating more gaseous moles have positive entropy change ($\Delta n_g > 0 \implies \Delta S > 0$).

7. Gibbs Free Energy ($G$) & Chemical Equilibrium

Gibbs energy defines maximum net work extractable from a system under constant temperature and pressure:

$$G = H - TS \quad \implies \quad \Delta G = \Delta H - T\Delta S$$

Spontaneity Truth Table

Equilibrium Constant Link

$$\Delta_r G^\circ = -RT \ln K = -2.303 RT \log K$$

At dynamic equilibrium: $\Delta_r G = 0$ and $T_{\text{eq}} = \frac{\Delta H^\circ}{\Delta S^\circ}$.