Chemistry · Analytical and physical chemistry · Year 1

Thermochemistry and Gibbs energy

Why does a cold pack get icy cold, and why does a lime kiln run at 900 °C?

The situation

If you squeeze a single-use “cold pack” from the pharmacy, the water inside dissolves ammonium nitrate and the pack turns icy cold at once; a calcium chloride “heater”, on the contrary, warms up. In lime and cement works limestone is fired at about 900 °C, while in nitrogen plants such as Navoiyazot ammonia is synthesised in a high-pressure column at 400–500 °C. Which process absorbs heat and which releases it, and how can this be measured? Why does a process that absorbs heat also go spontaneously, and from what temperature does limestone start to decompose?

Topic

Thermochemistry and chemical thermodynamics. lime and cement kilns (CaCO₃ → CaO + CO₂) and the Navoiyazot ammonia synthesis column; a simpler start — the single-use “cold compress” at the pharmacy…

Key ideas

Hess's law
ΔG = ΔH − TΔS
Equilibrium constant

Formulas

Δ_{r} H^{\circ} = \sum ν Δ_{f} H_{prod.}^{\circ} - \sum ν Δ_{f} H_{react.}^{\circ}

A consequence of Hess's law: the heat of reaction from the enthalpies of formation (independent of the path).

Δ H (T_{2}) = Δ H (T_{1}) + \int_{T_{1}}^{T_{2}} Δ C_{p} d T

Kirchhoff's law: the dependence of the heat effect on temperature through the difference of heat capacities.

Δ G = Δ H - T Δ S

Gibbs energy: at constant $p, T$ , $Δ G < 0$ means the process is spontaneous.

Δ G^{\circ} = - R T \ln K

The isotherm equation: the equilibrium constant from the standard Gibbs energy.

\frac{d \ln K_{p}}{d T} = \frac{Δ H^{\circ}}{R T^{2}}

The van 't Hoff isobar: for an exothermic reaction ( $Δ H < 0$ ), K decreases as the temperature rises.

Theory

Fundamentals of thermodynamics and the first law

Thermodynamics began in the 19th century with heat engines (Carnot, Joule, Clausius); Hess (1840), Gibbs and van 't Hoff brought it into chemistry. The basic concepts: a system — isolated (neither matter nor energy is exchanged), closed (energy only) or open; state functions ( $U$ , $H$ , $S$ , $G$ ) — their change depends only on the initial and final states, whereas work and heat depend on the path. The first law: $Δ U = Q - A$ . At constant pressure the heat equals the change in enthalpy: $Q_{p} = Δ H$ , $H = U + p V$ ; when gases take part, $Δ H = Δ U + Δ n R T$ (the ideal gas $p V = n R T$ — in the “Gas under a piston” experiment). In a thermochemical equation $Δ H < 0$ is exothermic (heat is released) and $Δ H > 0$ is endothermic. Standard state: 1 bar (100 kPa), usually 298 K; for the stable form of an element $Δ_{f} H^{\circ} = 0$ .

Hess's and Kirchhoff's laws

Hess's law: the heat of a reaction does not depend on the intermediate steps — only on the initial and final states. Consequences: (1) the heat of reaction = the difference between the enthalpies of formation of the products and of the reactants; (2) it can also be calculated from heats of combustion; (3) quantities that cannot be measured directly can be found from a cycle — for example, the Born–Haber cycle gives the lattice energy of NaCl ( $U$ ≈ −787 kJ/mol), and the heat of formation of a crystal hydrate is found from the difference of two heats of solution. Examples: CaCO₃ → CaO + CO₂ $Δ H^{\circ}$ = +178.3 kJ; N₂ + 3H₂ → 2NH₃ −92.2 kJ; combustion of natural gas CH₄ + 2O₂ → CO₂ + 2H₂O(l) −890.4 kJ. The heat capacity $C_{p} = {(\partial H / \partial T)}_{p}$ depends on temperature ( $C_{p} = a + b T + c^{'} / T^{2}$ or the Shomate equation); Kirchhoff's law $d Δ H / d T = Δ C_{p}$ : in ammonia synthesis $Δ C_{p}$ ≈ −44 J/K, so at 700 K $Δ H$ ≈ −106 kJ.

Calorimetry (lab works)

A calorimeter is a vessel well insulated from its surroundings (a Dewar flask), with a stirrer, an accurate thermometer (0.001–0.01 °C) and an electric heater for calibration. 1. The calorimeter constant: a current is passed through the heater, $Q = U I t$ (6.00 V · 2.00 A · 120 s = 1440 J), and $C = Q / Δ T$ is found; the heat capacity of the vessel itself is $C_{v} = C - m_{w} c$ . 2. Measurement: the experiment has three periods — the initial one (the temperature drift over 5–10 minutes), the main one (the salt dissolves or the reaction proceeds) and the final one. Because of heat exchange with the surroundings, $Δ T$ is corrected from the graph: the straight lines of the initial and final periods are extrapolated to the moment $t^{*}$ inside the main period (≈ 63 % of the change). 3. Calculation: $Q = C Δ T$ , the integral heat of solution is $Δ H = - Q / n$ . Crystal hydrate (L2): the difference between the heats of solution of CuSO₄ (−73.1) and CuSO₄·5H₂O (+6.0 kJ/mol, at infinite dilution) is the heat of hydration, −79.1 kJ/mol (textbooks also give −66.5 and +11.7; the difference is still ≈ −78 kJ/mol). Neutralisation (L3): strong acid + strong alkali is always H⁺ + OH⁻ → H₂O, −55.8 kJ/mol at infinite dilution (the ≈ −57 kJ/mol in textbooks was measured at certain concentrations and includes the heat of dilution). Safety: wear safety goggles and gloves when working with NaOH and acids; NaOH warms up as it dissolves; pour the acid into the water; switch on the heater only when it is immersed in the liquid.

The second law, entropy and Gibbs energy

Statements of the second law: Clausius — heat does not pass spontaneously from a cold body to a hot one; Thomson (Kelvin) — there is no cyclic machine that converts heat taken from a single source entirely into work; Carathéodory — near any state there are states that cannot be reached by an adiabatic path. The Carnot cycle and heat engines are covered in the “Heat engine: the Otto cycle” and “Refrigerator and heat pump” experiments. Entropy is $d S = δ Q_{rev} / T$ , with the statistical meaning $S = k \ln W$ (“Entropy and the arrow of time”); when ideal gases mix, $Δ S = - n R \sum x_{i} \ln x_{i} > 0$ . In an isolated system $Δ S \geq 0$ . At constant $T, p$ the criterion is the Gibbs energy $G = H - T S$ ( $Δ G \leq 0$ ), and at constant $T, V$ the Helmholtz energy $A = U - T S$ ; $G$ is the “free” energy (the part that can be turned into work), $T S$ the “bound” energy. $G$ , $A$ , $H$ , $U$ are thermodynamic potentials (characteristic functions). The chemical potential $μ_{i} = {(\partial G / \partial n_{i})}_{T, p}$ is the Gibbs energy that one mole of a substance brings in; $G$ is the universal one because chemistry works in practice at constant $T$ and $p$ . Additivity (L28): $Δ S^{\circ} = \sum ν S^{\circ}$ ; if a gas forms, entropy increases (CaCO₃: +160.6 J/K), if gas is consumed, it decreases (NH₃ synthesis: −198.8 J/K). The four cases: $Δ H < 0, Δ S > 0$ — always spontaneous; $Δ H > 0, Δ S < 0$ — never; the rest depend on temperature ( $T = Δ H / Δ S$ ).

The third law and the equilibrium constant

Nernst's theorem: as $T \to 0$ , $Δ S \to 0$ for processes in condensed systems; Planck's postulate: the entropy of a perfect crystal is zero at 0 K — which is why standard entropies are absolute (the $S^{\circ}$ tables). At low temperature the heat capacity is quantum in nature: $C \propto T^{3}$ (Debye; “Quantum statistics”). Berthelot's principle (“only exothermic reactions go spontaneously”) is wrong: the dissolving of NH₄NO₃ is endothermic (+25.7 kJ/mol), yet $Δ S$ = +108.7 J/K and $Δ G$ = −6.7 kJ — it goes spontaneously. Equilibrium: $Δ G = Δ G^{\circ} + R T \ln Q$ (the van 't Hoff isotherm), and at equilibrium $Δ G = 0$ : $Δ G^{\circ} = - R T \ln K$ . The isobar $d \ln K_{p} / d T = Δ H^{\circ} / R T^{2}$ and the isochore $d \ln K_{c} / d T = Δ U^{\circ} / R T^{2}$ are the quantitative form of Le Chatelier's principle. $K (T)$ from thermodynamic functions: the Ulich approximation ( $Δ C_{p} = 0$ ), $Δ C_{p}$ = const and the full integral (Temkin–Schwartsman: $Δ G_{T}^{\circ} = Δ H_{298}^{\circ} - T Δ S_{298}^{\circ} - T \sum Δ a_{k} M_{k}$ ). Heterogeneous equilibrium: in CaCO₃ ⇌ CaO + CO₂ the solids do not enter K, $K_{p} = p (C O_{2}) / p^{\circ}$ — at each temperature there is a definite “dissociation pressure”; equilibrium with the CO₂ in air (≈ 0.04 kPa) is reached at ≈ 530 °C, and a pressure of 101 kPa at ≈ 880–900 °C.

Ammonia synthesis, esterification; the model

Ammonia: of the methods of fixing free nitrogen, the main one in industry is the Haber–Bosch synthesis: a purified mixture of N₂ + 3H₂ passes through a column over an iron catalyst at 400–500 °C and high pressure (15–30 MPa in medium-pressure plants), the NH₃ formed is cooled and separated, and the remaining gas is recycled. The reaction is exothermic and the amount of gas decreases: low temperature and high pressure raise the yield, but at low temperature the reaction is very slow — the compromise temperature is chosen with the catalyst (qualitatively — in the “Reversible reactions and equilibrium” experiment). Esterification (L18): CH₃COOH + C₂H₅OH ⇌ CH₃COOC₂H₅ + H₂O, $K$ ≈ 4 (Berthelot and Péan de Saint-Gilles, 1862), $Δ H \approx 0$ — K is almost independent of temperature; from equal moles, 2/3 of the mixture is ester at equilibrium. To find K: take a sample of the equilibrium mixture and titrate the remaining acid with NaOH. The model: solutions are ideal and their heat capacity equals that of water; the heat of solution is taken as independent of concentration (infinite-dilution values); gases are ideal — at 30 MPa the real NH₃ yield is higher than the model (Larson–Dodge: 47 % at 400 °C, model ≈ 43 %); the lime kiln and the column are schematic mock-ups; time is speeded up 10 times.

Did you know?

Hess's law was published in 1840 by the Swiss-born chemist Germain Hess, who worked in St Petersburg — this was before the general formulation of the law of conservation of energy.
Fritz Haber received the Nobel Prize for 1918 for synthesising ammonia from its elements; Carl Bosch received the Nobel Prize in 1931 for high-pressure chemical methods.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Electrochemical analysis — Year 1
Qualitative analysis and gravimetry — Year 2
Titrimetric analysis — Year 2
Optical methods of analysis — Year 2
Extraction and chromatography — Year 2
Spectra: determining structure — Year 2
Alloys: phase diagrams — Year 2
Chemical kinetics and Arrhenius — Year 2
Catalysis and enzymes — Year 2
Colloids and adsorption — Year 2