Unit 3: Thermochemistry: Enthalpy, Calorimetry & Thermochemical Laws

Thermochemistry is the branch of physical chemistry that quantifies energy transfers, heat release or absorption, and mechanical work accompanying physical transformations and chemical reactions.

The Microscopic Anatomy of Energy

In classical and statistical thermodynamics, the total energy of a macroscopic system comprises:

1. External (Macroscopic) Energy: The kinetic energy of the center of mass of the system and its gravitational potential energy relative to external coordinates ($\frac{1}{2} M_{sys} v_{cm}^2 + M_{sys} g z_{cm}$). In standard laboratory thermochemistry, the system is held stationary ($v_{cm} = 0, z_{cm} = \text{const}$), setting external macroscopic energy changes to zero.

2. Internal Energy ($U$): The sum of all microscopic energy contributions within the molecular boundary:

• $E_{trans}$: Molecular center-of-mass translational kinetic energy ($\frac{3}{2} k_B T$ per particle).
• $E_{rot}$: Rigid rotor rotational kinetic energy about principal molecular axes of inertia (for linear molecules, 2 degrees of freedom $\rightarrow k_B T$; for non-linear molecules, 3 degrees of freedom $\rightarrow \frac{3}{2} k_B T$).
• $E_{vib}$: Harmonic and anharmonic normal vibrational modes along molecular chemical bonds ($\sum \hbar\omega_i (\frac{1}{2} + \frac{1}{e^{\hbar\omega_i / k_B T} - 1})$).
• $E_{elec}$: Potential and kinetic energy of atomic electrons interacting with nuclei.
• $U_{intermol}$: Intermolecular electrostatic, induction, dispersion, and exchange repulsion potentials.

State Functions vs Path Functions: Exact vs Inexact Differentials

A fundamental mathematical distinction governs thermodynamic variables:

• State Function (State Variable): A property whose value depends strictly on the current equilibrium state of the system (characterized by $T, P, V, \{n\}$), completely independent of the historical path or mechanism taken to reach that state. Examples: $U, H, S, G, V, T, P$.

Mathematically, an infinitesimal change in a state function $F$ is an exact differential $dF$. Its cyclic line integral around any closed thermodynamic loop is identically zero:

For any state function $F(x, y)$, Euler's reciprocity condition of mixed second partial derivatives holds:

• Path Function: A quantity whose magnitude depends upon the specific thermodynamic trajectory connecting the initial and final states. Heat ($q$) and Work ($w$) are not forms of energy stored within a system; they are transient modes of energy transfer across the system boundary during a process.

Mathematically, infinitesimal heat and work are inexact differentials, denoted by $\delta q$ (or $dq$) and $\delta w$ (or $dw$). Their cyclic integrals around closed loops are non-zero:

It is physically meaningless to speak of "the heat of a system" or "the work possessed by a substance." A system possesses internal energy, which it can transfer across boundaries as heat or work.

Mechanical Expansion (Pressure-Volume) Work

Consider a gas confined in a cylinder of cross-sectional area $A$ fitted with a frictionless piston moving against an external opposing pressure $P_{ext}$.

When the piston undergoes an infinitesimal displacement $dz$, the differential work done on the system is:

where $dV = A dz$ is the change in system volume. The negative sign conforms to the universal IUPAC thermodynamic sign convention:

• $\delta w > 0$: Work done on the system by the surroundings (compression, $dV < 0$).
• $\delta w < 0$: Work done by the system on the surroundings (expansion, $dV > 0$).

Integrating over a finite expansion from initial volume $V_1$ to final volume $V_2$:

The magnitude of work depends explicitly on the path taken by $P_{ext}$:

1. Free Expansion into Vacuum ($P_{ext} = 0$):

A gas expanding into a vacuum performs zero mechanical work.

2. Isobaric Expansion against Constant External Pressure ($P_{ext} = \text{const}$):

3. Reversible Isothermal Expansion of an Ideal Gas:

In a thermodynamically reversible process, the system remains in continuous mechanical equilibrium with its surroundings at every infinitesimal step ($P_{ext} = P_{int} \pm dP \approx P$). For an ideal gas, $P = \frac{n R T}{V}$. The reversible work is:

Because $P_{int} \ge P_{ext}$ during expansion, the reversible work represents the maximum possible mechanical work that a system can deliver to its surroundings ($|w_{rev}| \ge |w_{irrev}|$).

The First Law of Thermodynamics is the universal law of conservation of energy applied to thermodynamic systems. First formulated by Julius Robert von Mayer (1842), James Prescott Joule (1843), and Hermann von Helmholtz (1847), it asserts that energy can neither be created nor destroyed; it can only be transformed from one modality to another.

Mathematical Formulation of the First Law

For an isolated system (which exchanges neither matter nor energy with its surroundings), the total internal energy is strictly constant:

For a closed system (which exchanges energy but not matter across its boundary):

In differential form:

While the individual quantities $\delta q$ and $\delta w$ are path-dependent inexact differentials, their algebraic sum $dU = \delta q + \delta w$ is an exact differential of state.

Processes at Constant Volume ($q_v = \Delta U$)

If a chemical transformation occurs inside a rigid container of fixed volume ($dV = 0$) and performs only mechanical $P\text{-}V$ work ($\delta w_{other} = 0$):

The heat absorbed or released during an isochoric chemical process is exactly equal to the change in internal energy of the system.

Constant Pressure Processes & The Enthalpy State Function ($q_p = \Delta H$)

Most laboratory chemical reactions, atmospheric processes, and biological phenomena occur under isobaric conditions (open to constant atmospheric pressure $P_{ext} = P = \text{const}$). From the First Law:

Rearranging to isolate heat $q_p$:

This algebraic structure motivates the formal definition of a new thermodynamic state function via Legendre transform—the Enthalpy ($H$):

Since $U, P$, and $V$ are state functions, enthalpy $H$ is rigorously a thermodynamic state function.

Evaluating $\Delta H = H_2 - H_1$:

Under constant pressure ($P_1 = P_2 = P$):

The heat transferred in an isobaric process with only expansion work is identically equal to the change in enthalpy of the system ($\Delta H = q_p$).

• Exothermic Reaction: $\Delta H < 0 \implies q_p < 0$ (heat is released to surroundings).
• Endothermic Reaction: $\Delta H > 0 \implies q_p > 0$ (heat is absorbed from surroundings).

Relationship Between $\Delta H$ and $\Delta U$ in Chemical Reactions

From the definition of enthalpy:

1. For Condensed Phases (Solids and Liquids):

The molar volumes of liquids and solids are tiny ($V_m \sim 10^{-5} - 10^{-4}\text{ m}^3\cdot\text{mol}^{-1}$) and volume changes during reaction are minuscule ($\Delta V \approx 0$). At normal pressures ($1\text{ bar} = 10^5\text{ Pa}$):

Therefore, for reactions involving only condensed phases:

2. For Gas-Phase Reactions:

For reactions producing or consuming gaseous species behaving ideally ($P V = n_g R T$):

where $\Delta n_g \equiv \sum \nu_{gas}(\text{products}) - \sum \nu_{gas}(\text{reactants})$ is the stoichiometric change in moles of gas. Hence:

Heat Capacities: $C_v$ and $C_p$

The heat capacity $C$ of a substance is the amount of heat required to raise its temperature by one Kelvin: $C = \frac{\delta q}{dT}$.

1. Heat Capacity at Constant Volume ($C_v$):

2. Heat Capacity at Constant Pressure ($C_p$):

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

For any substance, internal energy can be written as $U = U(T, V)$:

Differentiating with respect to $T$ at constant $P$:

From $H = U + PV$:

For an ideal gas, molecules do not exert intermolecular forces; by Joule's Law, internal energy depends only on temperature: $\left(\frac{\partial U}{\partial V}\right)_T = 0$. Furthermore, for 1 mole of ideal gas, $V_m = RT/P \implies \left(\frac{\partial V_m}{\partial T}\right)_P = \frac{R}{P}$. Substituting these ideal gas conditions:

$C_p$ is always strictly greater than $C_v$ because under constant pressure, energy supplied to the system must not only elevate molecular thermal agitation ($C_v dT$) but also perform expansion work against the external atmosphere ($P dV$).

Calorimetry is the experimental science of measuring the heat evolved or absorbed during physical transformations and chemical reactions. All calorimeters rely on the First Law energy conservation principle: within an insulated calorimeter assembly, the heat released by the reaction system $q_{rxn}$ must equal the heat absorbed by the calorimeter vessel and its surrounding fluid $q_{cal}$:

Constant-Pressure (Coffee-Cup) Calorimetry

Operated open to atmospheric pressure ($P = \text{const}$), constant-pressure calorimeters measure enthalpy changes:

Typically composed of nested polystyrene cups equipped with a precision thermometer and stirrer, the calorimeter contains a known mass $m_w$ of water (or dilute aqueous solution). The heat absorbed by the calorimeter is:

where $c_w = 4.184\text{ J}\cdot\text{g}^{-1}\cdot\text{K}^{-1}$ is the specific heat capacity of water, $C_{cup}$ is the heat capacity of the cups and thermometer, and $C_{cal}$ is the total calorimeter heat capacity. Therefore:

Constant-pressure calorimetry is used for heats of neutralization ($\text{H}^+ + \text{OH}^- \rightarrow \text{H}_2\text{O}, \Delta H^\circ \approx -55.8\text{ kJ/mol}$), heats of dissolution, and precipitation reactions.

Constant-Volume Adiabatic Bomb Calorimetry

For combustion reactions of organic compounds, fuels, and foodstuffs with gaseous oxygen, a bomb calorimeter is required. The apparatus consists of a heavy-walled, high-strength stainless steel vessel (the "bomb") capable of withstanding internal detonation pressures exceeding $100\text{ bar}$.

• A precisely weighed solid sample ($\sim 1\text{ g}$) is placed in a platinum or quartz crucible.
• A high-resistance ignition fuse wire contacts the pellet.
• The bomb is charged with pure oxygen to high pressure ($25 - 30\text{ bar}$) to ensure complete combustion.
• The sealed bomb is immersed in a bath containing a precisely measured quantity of water ($\sim 2000\text{ g}$) surrounded by an outer thermal jacket.
• Electrical current passes through the fuse, igniting the sample:

Because the steel bomb is rigid and sealed ($V = \text{const}$), the heat measured is the internal energy of reaction:

Calorimeter Calibration via Benzoic Acid

The total heat capacity of the bomb calorimeter $C_{cal}$ is determined empirically by burning a certified thermochemical standard—ultra-pure benzoic acid ($\text{C}_6\text{H}_5\text{COOH}$, certified $\Delta u_c^\circ = -26.434\text{ kJ}\cdot\text{g}^{-1}$):

where $q_{fuse}$ accounts for the electrical ignition energy and the heat of combustion of the burnt fuse wire.

Conversion to Constant-Pressure Enthalpy $\Delta H_{comb}^\circ$

Once $\Delta U_{comb}$ is measured in the bomb calorimeter, the standard enthalpy of combustion $\Delta H_{comb}^\circ$ is calculated using the thermodynamic identity:

where $\Delta n_g$ counts only gaseous species in the balanced stoichiometric combustion equation (water formed at $298.15\text{ K}$ is liquid, so $n(\text{H}_2\text{O}(l))$ does not contribute to $\Delta n_g$).

Temperature-Time Curve & The Regnault-Pfaundler Correction

In real calorimeters, heat exchange between the calorimeter vessel and the surrounding environment is never strictly zero. The experimental thermogram displays three distinct periods:

1. Pre-Period: Baseline drift due to stirrer friction and minor ambient heat leakage.

2. Reaction Period: Rapid temperature rise following electrical ignition.

3. Post-Period: Cooling decay following thermal equilibration.

The Regnault-Pfaundler graphical correction method extrapolates the pre-period and post-period Newton's cooling slopes to an effective midpoint time $t_{mid}$, determining the true adiabatic temperature rise $\Delta T_{corr}$ devoid of environmental heat exchange artifacts.

To tabulate and compute heat transfers for millions of chemical transformations without requiring individual calorimetric measurements for every reaction, physical chemists reference all thermodynamic quantities to a universally standardized thermodynamic baseline.

The Standard State Convention

IUPAC Standard State:

• For a pure substance (solid, liquid, or gas): The standard state is the pure, stable thermodynamic phase at the standard pressure $P^\circ = 1\text{ bar} = 10^5\text{ Pa}$.
• For a gas: The hypothetical ideal gas at standard pressure $P^\circ = 1\text{ bar}$.
• For a solute in solution: The hypothetical ideal solution state at standard molality $m^\circ = 1\text{ mol}\cdot\text{kg}^{-1}$ (or concentration $C^\circ = 1\text{ mol}\cdot\text{L}^{-1}$) at $P^\circ = 1\text{ bar}$.
• Temperature: The standard state does not fix temperature, but reference tables universally report values at $T = 298.15\text{ K} = 25.00^\circ\text{C}$.

Standard Enthalpy of Formation ($\Delta H_f^\circ$)

The Standard Enthalpy of Formation $\Delta H_f^\circ$ of a compound is the enthalpy change accompanying the formation of one mole of the substance from its constituent elements in their standard reference states (most stable physical allotropes) at $P^\circ = 1\text{ bar}$ and $298.15\text{ K}$.

• Convention: The standard enthalpy of formation of any pure chemical element in its most stable allotropic form at $298.15\text{ K}$ and $1\text{ bar}$ is defined as identically zero:

• Allotropes that are not the most stable form possess non-zero formation enthalpies:

Fundamental Thermochemical Laws

1. Lavoisier and Laplace's Law (1780):

The enthalpy change accompanying a chemical reaction is equal in magnitude but opposite in sign to the enthalpy change accompanying the reverse reaction:

This law reflects the time-reversal symmetry and conservative nature of the enthalpy state function.

2. Hess's Law of Constant Heat Summation (1840 - Germain Henri Hess):

Because enthalpy is a thermodynamic state function ($dH$ is an exact differential), the overall enthalpy change for any chemical process is independent of the number of intermediate reaction steps or the specific reaction pathway taken:

For any balanced chemical reaction $\sum_i \nu_i A_i = 0$ (where $\nu_i > 0$ for products and $\nu_i < 0$ for reactants):

Kirchhoff's Law: Temperature Dependence of Reaction Enthalpies

Standard enthalpies of formation are tabulated at $T_1 = 298.15\text{ K}$, but industrial chemical processes operate across vast temperature ranges (e.g., the Haber-Bosch ammonia synthesis operates at $700 - 750\text{ K}$).

From the definition of heat capacity $C_p = \left(\frac{\partial H}{\partial T}\right)_P$, applying this derivative to a chemical reaction enthalpy yields:

where $\Delta C_p \equiv \sum \nu_p C_{p,m}(\text{products}) - \sum \nu_r C_{p,m}(\text{reactants})$.

Integrating Eq. (3.11) between temperatures $T_1$ and $T_2$ yields Kirchhoff's Law:

1. If $\Delta C_p$ is approximately constant across the temperature interval:

1. For high-precision engineering calculations, heat capacities are expressed as empirical polynomial functions of temperature:

Integrating each term analytically yields exact thermochemical curves over thousands of Kelvin.

Beyond chemical reaction heats, enthalpy changes govern physical phase transitions, chemical bond breaking, and dissolution processes in solution.

Enthalpy of Phase Transitions

Phase changes represent isothermal, isobaric rearrangements of intermolecular structural networks:

• Enthalpy of Fusion ($\Delta H_{fus}^\circ$): Solid $\rightarrow$ Liquid.
• Enthalpy of Vaporization ($\Delta H_{vap}^\circ$): Liquid $\rightarrow$ Gas.
• Enthalpy of Sublimation ($\Delta H_{sub}^\circ$): Solid $\rightarrow$ Gas.

Because enthalpy is a state function:

Trouton's Rule (1884): For most non-associated liquids (liquids lacking strong directional hydrogen bonding), the standard molar entropy of vaporization at the normal boiling point $T_b$ is approximately constant:

This empirical constancy reflects the universal gain in translational spatial entropy when one mole of condensed liquid expands into an isotropic, disordered gas phase. Polar, hydrogen-bonded liquids like water ($\Delta S_{vap}^\circ = 109\text{ J}/(\text{mol}\cdot\text{K})$) and ethanol ($\Delta S_{vap}^\circ = 110\text{ J}/(\text{mol}\cdot\text{K})$) exhibit positive deviations from Trouton's rule due to the high structural order in the liquid phase.

Bond Dissociation Energies & Enthalpy Estimation

The Bond Dissociation Energy ($D_{A-B}$) is the standard enthalpy required to break one mole of a specific covalent bond in the gas phase homolytically at $298.15\text{ K}$:

Because bond breaking requires energy input ($D > 0$) while bond formation releases energy:

This method provides rapid initial estimates of reaction heats, though accurate values require tabulated standard formation enthalpies because average bond enthalpies neglect vibrational zero-point energy shifts and neighboring substituent resonance stabilization.

Enthalpy of Solution and Dilution

When an ionic crystalline solid dissolves in a solvent, the overall Enthalpy of Solution ($\Delta H_{soln}^\circ$) is governed by a thermodynamic competition between two massive energetic terms:

1. Lattice Enthalpy ($\Delta H_{latt}^\circ > 0$): The energy required to break the rigid crystalline lattice and separate one mole of solid into infinitely separated gas-phase ions:

2. Hydration (Solvation) Enthalpy ($\Delta H_{hyd}^\circ < 0$): The energy released when gaseous ions are solvated by polar solvent dipoles:

By Hess's Law:

• If $|\Delta H_{hyd}^\circ| > \Delta H_{latt}^\circ$: The dissolution is exothermic ($\Delta H_{soln}^\circ < 0$), e.g., $\text{CaCl}_2$ or $\text{NaOH}$ dissolving in water (used in chemical hot packs).
• If $\Delta H_{latt}^\circ > |\Delta H_{hyd}^\circ|$: The dissolution is endothermic ($\Delta H_{soln}^\circ > 0$), e.g., $\text{NH}_4\text{NO}_3$ dissolving in water (used in instant cold packs). Dissolution proceeds spontaneously because the massive positive entropy of mixing ($\Delta S_{soln} > 0$) overcomes the unfavorable positive enthalpy change.

The Born-Haber Cycle for Ionic Solids

Lattice enthalpies cannot be measured directly by simple calorimetry. Max Born and Fritz Haber (1919) constructed a closed thermodynamic Hess cycle linking lattice enthalpy to directly measurable thermochemical quantities:

1. Standard enthalpy of formation of ionic solid from elements: $\Delta H_f^\circ[MX(s)]$.
2. Enthalpy of sublimation (atomization) of the metal: $\Delta H_{sub}^\circ[M(s) \rightarrow M(g)]$.
3. First (and higher) Ionization Energies of the metal: $IE[M(g) \rightarrow M^+(g) + e^-]$.
4. Bond dissociation enthalpy of the non-metal: $\frac{1}{2} D_{X_2}[X_2(g) \rightarrow 2X(g)]$.
5. Electron Affinity of the non-metal: $-EA_1[X(g) + e^- \rightarrow X^-(g)]$.

Traversing the closed Born-Haber loop:

Rearranging isolates the experimental lattice enthalpy: