Thermodynamics

In thermodynamics, the universe is partitioned into two distinct conceptual domains:

• System: The specific portion of the physical universe selected for thermodynamic study or observation (e.g., a reaction mixture inside a beaker or a gas confined in a cylinder with a piston).
• Surroundings: The rest of the universe outside the system that can interact with it by exchanging energy (heat/work) or matter. In practice, only the immediate environment affected by the system constitutes the operational surroundings.
• Boundary: The real or imaginary surface separating the system from its surroundings. Boundaries may be:
  • Diathermic: Conducting boundary permitting heat transfer between system and surroundings.
  • Adiabatic: Thermally insulating boundary preventing all heat transfer ($q = 0$).
  • Rigid vs. Movable: Rigid boundaries prevent volume changes ($w = 0$), whereas movable boundaries allow mechanical expansion or compression work.

Classification of Thermodynamic Systems:

The thermodynamic state of a macroscopic system is uniquely defined by specifying its macroscopic state variables ($P, V, T, n$):

• State Functions (State Variables): Properties whose numerical values depend solely on the current thermodynamic state of the system, completely independent of the path or method used to reach that state. Examples: Pressure ($P$), Volume ($V$), Temperature ($T$), Internal Energy ($U$), Enthalpy ($H$), Entropy ($S$), Gibbs Free Energy ($G$). $$\oint dX = 0 \quad (\text{Cyclic integral of any state function is identically zero})$$ The change in a state function depends only on initial and final states: $\Delta X = X_{\text{final}} - X_{\text{initial}}$.
• Path Functions: Quantities whose values depend explicitly on the thermodynamic path or mechanism traversed during the transformation between states. Examples: Work ($w$) and Heat ($q$). They are inexact differentials ($\delta w, \delta q$), and their cyclic integrals are generally non-zero ($\oint \delta w \neq 0$).

• Extensive Properties: Properties that depend directly on the mass, volume, or quantity of matter present in the system. Examples: Mass ($m$), Volume ($V$), Total Internal Energy ($U$), Enthalpy ($H$), Entropy ($S$), Gibbs Free Energy ($G$), Heat Capacity ($C$).
  Additive Rule: If a system is divided into two halves, each half possesses half the original value of an extensive property.
• Intensive Properties: Properties that are entirely independent of the amount or size of matter present in the system. Examples: Temperature ($T$), Pressure ($P$), Density ($d$), Molar Volume ($V_m$), Specific Heat Capacity ($c$), Molar Enthalpy, Refractive Index, Surface Tension, Viscosity, Electromotive Force ($E^\circ$).
  Ratio Rule: The ratio of two extensive properties is always an intensive property (e.g., $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$, $\text{Molar Volume} = \frac{V}{n}$).

A thermodynamic process occurs when a system transitions from one equilibrium state to another:

• Isothermal Process: Temperature remains strictly constant throughout ($\Delta T = 0$, $dT = 0$). Diathermic walls permit heat flow. For an ideal gas, $\Delta U = 0$ and $\Delta H = 0$.
• Adiabatic Process: System is completely insulated from surroundings so no heat is exchanged ($q = 0$, $\delta q = 0$). Expansion causes cooling; compression causes heating.
• Isobaric Process: Pressure remains constant throughout ($\Delta P = 0$, $dP = 0$). Most open bench chemistry reactions are isobaric ($P = 1 \; \text{atm}$).
• Isochoric Process: Volume remains constant throughout ($\Delta V = 0$, $dV = 0$). No expansion work is possible ($w = 0$). Reactions in rigid bomb calorimeters are isochoric.
• Cyclic Process: A sequence of transformations that ultimately returns the system to its precise initial state. For any cyclic process: $\Delta U_{\text{cycle}} = 0, \; \Delta H_{\text{cycle}} = 0, \; \Delta S_{\text{cycle}} = 0$.

Internal Energy ($U$): The total microscopic energy possessed by a thermodynamic system, encompassing molecular translational, rotational, and vibrational kinetic energies, alongside electronic, nuclear, and intermolecular potential energies. $$U = E_{\text{trans}} + E_{\text{rot}} + E_{\text{vib}} + E_{\text{electronic}} + E_{\text{bonding}} + E_{\text{intermolecular}}$$ The absolute value of $U$ cannot be measured directly, but the change $\Delta U = U_2 - U_1$ is an experimentally measurable state function.

Heat ($q$): Energy transferred across a boundary solely as a result of a temperature difference between the system and surroundings. Heat flows spontaneously from higher to lower temperature until thermal equilibrium is attained.

Consider a frictionless, weightless piston of cross-sectional area $A$ confining a gas inside a cylinder. The gas expands against an external pressure $P_{\text{ext}}$ through an infinitesimal displacement $dl$:

Integrating from initial volume $V_1$ to final volume $V_2$ gives the general expression for mechanical expansion work:

Special Cases of Work:

1. Irreversible Expansion against Constant External Pressure: $$w = -P_{\text{ext}} \int_{V_1}^{V_2} dV = -P_{\text{ext}} (V_2 - V_1) = -P_{\text{ext}} \Delta V$$
2. Free Expansion into Vacuum: $$P_{\text{ext}} = 0 \implies w = 0 \quad (\text{No work is done during free expansion, whether reversible or irreversible})$$
3. Isothermal Reversible Expansion of an Ideal Gas:

   In a reversible expansion, the internal gas pressure balances external pressure at every infinitesimal step ($P_{\text{ext}} = P_{\text{gas}} - dP \approx P_{\text{gas}}$). From the ideal gas law, $P = \frac{nRT}{V}$:

   $$w_{\text{rev}} = -\int_{V_1}^{V_2} \frac{nRT}{V} dV = -nRT \ln\left(\frac{V_2}{V_1}\right) = -2.303 nRT \log_{10}\left(\frac{V_2}{V_1}\right)$$

   Since temperature is constant, Boyle's Law dictates $P_1 V_1 = P_2 V_2 \implies \frac{V_2}{V_1} = \frac{P_1}{P_2}$:

   $$w_{\text{rev}} = -2.303 nRT \log_{10}\left(\frac{P_1}{P_2}\right)$$

   Note: For expansion ($V_2 > V_1$), $\log_{10}(V_2/V_1) > 0$, so $w < 0$ (work done BY system). For compression ($V_2 < V_1$), $w > 0$ (work done ON system). Furthermore, $|w_{\text{rev}}| > |w_{\text{irrev}}|$ for expansion.

The First Law is the universal thermodynamic statement of the Law of Conservation of Energy: Energy can neither be created nor destroyed, although it can be transformed from one form into another. The total energy of an isolated system remains constant.

IUPAC Sign Conventions (Mandatory for Chemistry):

• Heat absorbed by the system from surroundings: $q > 0$ ($+q$, endothermic).
• Heat released by the system to surroundings: $q < 0$ ($-q$, exothermic).
• Work done ON the system by surroundings (compression): $w > 0$ ($+w$).
• Work done BY the system on surroundings (expansion): $w < 0$ ($-w$).

For a closed system undergoing a process at constant volume ($\Delta V = 0$):

This demonstrates that the heat exchanged by a system at constant volume ($q_v$) is exactly equal to the change in its internal energy $\Delta U$. Because $U$ is a state function, $q_v$ becomes independent of path for an isochoric transformation.

Most chemical laboratory and industrial processes occur under constant atmospheric pressure ($P_{\text{ext}} = P = \text{constant}$) rather than constant volume. From the First Law:

To quantify this constant-pressure heat exchange, the thermodynamic state function Enthalpy ($H$), also called heat content, is defined:

The change in enthalpy $\Delta H$ is precisely equal to the heat absorbed or evolved by the system at constant pressure ($q_p$). Because $U, P,$ and $V$ are all state functions, Enthalpy $H$ is strictly a state function and an extensive property.

For any process involving ideal gases, $PV = nRT$. Substituting into $\Delta H = \Delta U + \Delta(PV)$ at constant temperature $T$:

where $\Delta n_g$ is the stoichiometric difference between moles of gaseous products and gaseous reactants:

Physical Implications:

• If $\Delta n_g = 0$ (e.g., $\text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2\text{HI}(g)$): $\Delta H = \Delta U$.
• If $\Delta n_g > 0$ (e.g., $\text{PCl}_5(g) \rightleftharpoons \text{PCl}_3(g) + \text{Cl}_2(g)$): $\Delta H > \Delta U$ (expansion work is performed against the atmosphere).
• If $\Delta n_g < 0$ (e.g., $\text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g)$): $\Delta H < \Delta U$ (compression work is done by the atmosphere on the system).
• For reactions involving only liquids and solids: $\Delta V \approx 0 \implies \Delta H \approx \Delta U$.

Heat Capacity ($C$): The quantity of heat required to raise the temperature of a given mass of substance by one degree ($1 \; \text{K}$ or $1^\circ\text{C}$): $C = \frac{q}{\Delta T} = \frac{\delta q}{dT}$.

• Molar Heat Capacity at Constant Volume ($C_v$): $$C_v = \left(\frac{\partial U}{\partial T}\right)_v \implies dU = n C_v dT \implies \Delta U = n C_v \Delta T$$
• Molar Heat Capacity at Constant Pressure ($C_p$): $$C_p = \left(\frac{\partial H}{\partial T}\right)_p \implies dH = n C_p dT \implies \Delta H = n C_p \Delta T$$

Derivation of Mayer's Relation ($C_p - C_v = R$):

For 1 mole of an ideal gas: $H = U + PV = U + RT$. Differentiating with respect to temperature $T$:

Physical Rationale for $C_p > C_v$: At constant volume, all supplied heat is utilized exclusively to raise molecular kinetic energy (increase temperature). At constant pressure, the gas expands against the atmosphere; hence, supplied heat must not only raise internal kinetic energy but also supply external expansion work ($P \Delta V = R \Delta T$).

Heat Capacity Ratio ($\gamma = C_p / C_v$):

• Monoatomic gas ($\text{He, Ne, Ar}$): $C_v = \frac{3}{2}R, \; C_p = \frac{5}{2}R \implies \gamma = \frac{5}{3} \approx 1.67$.
• Diatomic gas ($\text{N}_2, \text{O}_2, \text{HCl}$ at room temp): $C_v = \frac{5}{2}R, \; C_p = \frac{7}{2}R \implies \gamma = \frac{7}{5} = 1.40$.
• Triatomic / Polyatomic non-linear gas ($\text{H}_2\text{O, SO}_2$): $C_v = 3R, \; C_p = 4R \implies \gamma = \frac{4}{3} \approx 1.33$.

In an adiabatic process, $q = 0$. From the First Law, $\Delta U = w_{\text{ad}}$. Therefore:

Integrating yields the Poisson relations for reversible adiabatic processes:

Thermochemistry deals with the heat energy changes accompanying chemical reactions and phase transformations. The Standard State of a substance at a specified temperature (usually $298.15 \; \text{K} = 25^\circ\text{C}$) is its purest and most stable physical form at a pressure of exactly $1 \; \text{bar}$ ($10^5 \; \text{Pa}$).

• Standard Enthalpy of Formation ($\Delta_f H^\circ$): The enthalpy change accompanying the formation of 1 mole of a compound directly from its constituent elements in their standard reference states.
  By IUPAC Convention: The standard enthalpy of formation of every pure element in its most stable reference physical state is arbitrarily assigned as zero (e.g., $\Delta_f H^\circ[\text{C}_{\text{graphite}}] = 0$, $\Delta_f H^\circ[\text{O}_2(g)] = 0$, $\Delta_f H^\circ[\text{S}_{\text{rhombic}}] = 0$, $\Delta_f H^\circ[\text{P}_{\text{white}}] = 0$, $\Delta_f H^\circ[\text{Br}_2(l)] = 0$). $$\Delta_r H^\circ = \sum \nu_p \Delta_f H^\circ(\text{products}) - \sum \nu_r \Delta_f H^\circ(\text{reactants})$$
• Standard Enthalpy of Combustion ($\Delta_c H^\circ$): The enthalpy change occurring when 1 mole of a substance undergoes complete oxidation/combustion in excess oxygen under standard conditions (always exothermic, $\Delta_c H^\circ < 0$). $$\Delta_r H^\circ = \sum \nu_r \Delta_c H^\circ(\text{reactants}) - \sum \nu_p \Delta_c H^\circ(\text{products})$$
• Enthalpy of Neutralization: The heat liberated when 1 gram-equivalent of an acid is completely neutralized by 1 gram-equivalent of a base in dilute aqueous solution. For any strong acid and strong base: $$\text{H}^+(aq) + \text{OH}^-(aq) \longrightarrow \text{H}_2\text{O}(l) \quad \Delta_{\text{neut}} H^\circ = -57.1 \; \text{kJ/mol} \; (-13.7 \; \text{kcal/mol})$$ For weak acids or weak bases (e.g., $\text{CH}_3\text{COOH}$ or $\text{NH}_4\text{OH}$), $|\Delta_{\text{neut}} H^\circ| < 57.1 \; \text{kJ}$ because part of the evolved heat is consumed in the endothermic ionization of the weak electrolyte.

Formulated empirically by Germain Hess in 1840, this law is a direct manifestation of the First Law of Thermodynamics and the state function nature of enthalpy:

Applications: Determination of standard enthalpies of formation for compounds that cannot be synthesized directly from elements (e.g., $\text{CO}, \text{CH}_4, \text{C}_2\text{H}_2$), calculation of lattice enthalpies via the Born-Haber cycle, and determination of resonance energies.

• Bond Dissociation Enthalpy: The enthalpy required to break 1 mole of a specific covalent bond in a gaseous molecule into isolated gaseous atoms or radicals (e.g., $\text{H}_2(g) \to 2\text{H}(g), \; \Delta_{\text{bond}} H = +435.8 \; \text{kJ/mol}$).
• Mean (Average) Bond Enthalpy: In polyatomic molecules like methane ($\text{CH}_4$), breaking successive $\text{C-H}$ bonds requires different energies due to changing electronic environments. The mean bond enthalpy is the average: $$\epsilon(\text{C-H}) = \frac{1}{4} \Delta_a H^\circ[\text{CH}_4(g)] = \frac{1665}{4} = 416.25 \; \text{kJ/mol}$$
• Calculating Reaction Enthalpy from Bond Enthalpies:
  $$\Delta_r H^\circ = \sum \text{B.E.}(\text{reactants broken}) - \sum \text{B.E.}(\text{products formed})$$
  Critical Examination Warning: Notice that for bond energies, reactants come FIRST (energy absorbed to break bonds minus energy released when new bonds form). Do not confuse this with enthalpies of formation where products come first!

The variation of reaction enthalpy with temperature at constant pressure is governed by Kirchhoff's Law:

where $\Delta C_p = \sum \nu_p C_p(\text{products}) - \sum \nu_r C_p(\text{reactants})$. Similarly, at constant volume: $\Delta_r U_{T_2} - \Delta_r U_{T_1} = \Delta C_v (T_2 - T_1)$.

The First Law of Thermodynamics affirms the conservation of energy during any process, but it fails completely to answer two fundamental questions:

1. Why do natural processes proceed spontaneously in one direction only (e.g., heat flows from hot to cold, gas expands into vacuum, water flows downhill)?
2. Is a decrease in enthalpy (exothermicity, $\Delta H < 0$) a sufficient criterion for spontaneity? No! Many endothermic processes proceed spontaneously at room temperature, such as the dissolution of ammonium nitrate in water ($\Delta H > 0$), evaporation of water, and the melting of ice above $0^\circ\text{C}$. Conversely, many exothermic reactions do not proceed without external initiation.

Rudolf Clausius (1865) introduced the thermodynamic state function Entropy ($S$) as a quantitative measure of the degree of randomness, molecular disorder, or spatial dispersal of matter and thermal energy in a system.

where $\delta q_{\text{rev}}$ is the heat exchanged reversibly at absolute thermodynamic temperature $T$. The SI unit of entropy is $\text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$.

Physical Characteristics of Entropy:

• Entropy is a state function and an extensive property.
• Entropy increases upon phase transitions from ordered to disordered states: $$S_{\text{solid}} < S_{\text{liquid}} \ll S_{\text{gas}}$$
• Entropy increases when the number of gaseous molecules increases during a chemical reaction ($\Delta n_g > 0$).
• Entropy increases with increasing temperature, expansion of volume, and dissolution of crystalline solutes in solvents.

The Second Law can be stated in several equivalent formulations:

• Clausius Statement: It is impossible to construct a cyclic machine whose sole effect is the transfer of heat from a cooler body to a hotter body without external work.
• Kelvin-Planck Statement: It is impossible to construct a heat engine operating in a cycle that absorbs heat from a reservoir and converts it completely into equivalent mechanical work without discharging heat to a cold reservoir.
• Entropy Formulation (Universal Criterion for Spontaneity): In any natural, spontaneous (irreversible) process, the total entropy of the universe (system plus surroundings) always increases:
  $$\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} > 0 \quad (\text{Spontaneous / Irreversible})$$ $$\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} = 0 \quad (\text{Reversible / Equilibrium})$$ $$\Delta S_{\text{universe}} < 0 \quad (\text{Non-spontaneous / Impossible})$$

1. Phase Transitions at Constant Temperature:
   • Entropy of Fusion: $\Delta S_{\text{fus}} = \frac{\Delta H_{\text{fus}}}{T_f}$ (where $T_f$ is melting point in Kelvin).
   • Entropy of Vaporization: $\Delta S_{\text{vap}} = \frac{\Delta H_{\text{vap}}}{T_b}$ (where $T_b$ is boiling point in Kelvin).
   • Trouton's Rule: For most non-polar, non-associated liquids (like benzene, $\text{CCl}_4$, hexane), the entropy of vaporization at normal boiling point is nearly constant: $\Delta S_{\text{vap}} \approx 88 \; \text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$ (or $10.5 \; R$). Water and ethanol deviate due to hydrogen bonding.
2. Entropy Change of an Ideal Gas:

   Combining the First Law ($dq = dU + P dV$) and ideal gas relations ($dU = n C_v dT$, $P = nRT/V$):

   $$dS = \frac{dU + P dV}{T} = n C_v \frac{dT}{T} + n R \frac{dV}{V}$$ $$\Delta S = n C_v \ln\left(\frac{T_2}{T_1}\right) + n R \ln\left(\frac{V_2}{V_1}\right) = n C_p \ln\left(\frac{T_2}{T_1}\right) - n R \ln\left(\frac{P_2}{P_1}\right)$$
   • For an Isothermal Process ($T_1 = T_2$): $\Delta S = n R \ln\left(\frac{V_2}{V_1}\right) = 2.303 n R \log_{10}\left(\frac{V_2}{V_1}\right) = 2.303 n R \log_{10}\left(\frac{P_1}{P_2}\right)$.
   • For an Isochoric Process ($V_1 = V_2$): $\Delta S = n C_v \ln\left(\frac{T_2}{T_1}\right) = 2.303 n C_v \log_{10}\left(\frac{T_2}{T_1}\right)$.
   • For an Isobaric Process ($P_1 = P_2$): $\Delta S = n C_p \ln\left(\frac{T_2}{T_1}\right) = 2.303 n C_p \log_{10}\left(\frac{T_2}{T_1}\right)$.

While the Second Law establishes $\Delta S_{\text{universe}} > 0$ as the ultimate test of spontaneity, evaluating $\Delta S_{\text{surroundings}}$ is practically inconvenient. Josiah Willard Gibbs (1873) resolved this by defining a thermodynamic potential that depends entirely on system variables at constant temperature and pressure:

Because $H, T,$ and $S$ are state functions, Gibbs Free Energy ($G$) is a state function and extensive property. For an isothermal change:

At constant temperature and pressure, the heat transferred to surroundings is $q_{\text{surr}} = -q_p = -\Delta H_{\text{sys}}$. Thus:

Multiplying through by $-T$ yields:

Physical Significance: $-\Delta G$ represents the maximum useful (non-expansion) work obtainable from a system at constant temperature and pressure: $-\Delta G = w_{\text{useful}} = w_{\text{non-}PV}$ (e.g., electrical work generated by a galvanic cell: $\Delta G = -n F E_{\text{cell}}$).

The sign of $\Delta G = \Delta H - T \Delta S$ depends on the algebraic signs of $\Delta H$ and $\Delta S$ and the magnitude of $T$:

For any reversible reaction $a\text{A} + b\text{B} \rightleftharpoons c\text{C} + d\text{D}$, the actual free energy change $\Delta_r G$ is related to the standard free energy change $\Delta_r G^\circ$ and the reaction quotient $Q$ by:

At dynamic chemical equilibrium, $\Delta_r G = 0$ and $Q = K$ (the equilibrium constant). Substituting these conditions:

• If $\Delta_r G^\circ < 0$: $K > 1$, equilibrium heavily favors products (reaction proceeds to near completion).
• If $\Delta_r G^\circ = 0$: $K = 1$, reactants and products are present in equal effective thermodynamic concentrations.
• If $\Delta_r G^\circ > 0$: $K < 1$, equilibrium heavily favors reactants (reaction barely proceeds).

Formulated by Walther Nernst (Nernst Heat Theorem, 1906) and generalized by Max Planck:

Significance of the Third Law:

• Unlike enthalpy or internal energy (where only changes $\Delta H$ and $\Delta U$ are measurable), the Third Law enables the determination of Absolute Standard Molar Entropies ($S^\circ$) for any substance at temperature $T$: $$S_T^\circ = \int_0^T \frac{C_p}{T} dT$$
• Standard entropy of reaction is then directly calculated: $\Delta_r S^\circ = \sum \nu_p S^\circ(\text{products}) - \sum \nu_r S^\circ(\text{reactants})$.
• Residual Entropy: Certain substances do not possess zero entropy at $0 \; \text{K}$ because their molecules freeze into disordered orientations in the crystal lattice. Examples include $\text{CO}$ (dipole orientation disorder: $\text{C-O}$ vs $\text{O-C}$), $\text{N}_2\text{O}$, $\text{NO}$, and ice (hydrogen bond orientation disorder). The residual entropy of $\text{CO}$ is $S_{\text{res}} = R \ln 2 \approx 5.76 \; \text{J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$.