Thermodynamics

Thermodynamics deals with energy changes as a reaction or process occurs, without worrying about how fast it happens or the mechanism — that is the job of chemical kinetics. To study energy change we first mark off the part of the universe we are interested in: the system. Everything else is the surroundings, and the real or imaginary surface separating them is the boundary. Heat and work are exchanged across this boundary.

Systems are classified by what can cross the boundary. An open system can exchange both matter and energy with the surroundings (hot tea in an open cup). A closed system exchanges only energy, not matter (tea in a sealed but conducting flask). An isolated system exchanges neither matter nor energy (tea in a perfect thermos flask). Most laboratory reactions in a stoppered flask are treated as closed systems.

The state of a system is described by macroscopic, measurable properties — pressure, volume, temperature, and amount — called state variables. When these fix the state completely, the system is said to be in a definite state. Any property whose value depends only on the initial and final state, not on how the change was carried out, is a state function (e.g. internal energy, enthalpy). A quantity that depends on the path taken is a path function (e.g. heat, work). Examiners frequently ask you to classify a list of quantities as state or path functions.

Properties of a system fall into two classes based on how they behave when the size of the system changes. An extensive property depends on the amount of matter present in the system — mass, volume, internal energy, enthalpy, entropy, and Gibbs energy are all extensive; double the amount of substance and these quantities double. An intensive property does not depend on the amount of matter — temperature, pressure, density, and molar quantities (like molar heat capacity) are intensive; they stay the same whether you have 1 g or 100 g of the substance.

A useful test: if you divide a system into two equal halves, an extensive property is halved in each part, while an intensive property stays unchanged. This distinction matters when combining thermodynamic quantities — for instance, dividing an extensive property (like total enthalpy) by an extensive property (like moles) always gives an intensive property (molar enthalpy).

Internal energy (U) is the total energy stored in a system — the sum of translational, rotational, vibrational, and bond energies of all the molecules. We can never measure U in absolute terms, only its change, ΔU, between two states; since U depends only on the state, ΔU is the same regardless of the path taken.

Internal energy of a system changes when energy enters or leaves as heat (q) or as work (w). The sign convention used in Class 11 NCERT (IUPAC convention): heat absorbed by the system is positive (q > 0), heat given out is negative (q < 0); work done on the system is positive (w > 0), work done by the system is negative (w < 0).

The first law of thermodynamics is the law of conservation of energy applied to a system: energy can neither be created nor destroyed, though it can change form. Mathematically, ΔU = q + w. This single equation is the working tool of the whole chapter.

Work of expansion: when a gas expands against an external pressure p_ext by a volume ΔV, the work done by the gas is p_extΔV, so the work done on the system is w = −p_extΔV. For an ideal gas expanding into vacuum (free expansion), p_ext = 0, so w = 0. For a reaction at constant volume (ΔV = 0), w = 0, so ΔU = q_V — the heat measured in a bomb calorimeter directly gives ΔU.

For a reversible isothermal expansion of an ideal gas from V₁ to V₂ at temperature T, the gas does maximum work: w = −2.303 nRT log(V₂/V₁) = −2.303 nRT log(p₁/p₂). Since ΔU depends only on T for an ideal gas and T is constant here, ΔU = 0, so q = −w = 2.303 nRT log(V₂/V₁).

An isothermal process occurs at constant temperature (dT = 0); since U of an ideal gas depends only on T, ΔU = 0 for any isothermal change of an ideal gas, so q = −w. An adiabatic process occurs with no heat exchange with the surroundings (q = 0); the boundary is thermally insulated, so ΔU = w — all the work done on/by the system changes its internal energy, and the temperature changes.

In an adiabatic expansion the gas does work at the expense of its own internal energy, so it cools (w is negative, ΔU decreases, T falls). In an adiabatic compression, work is done on the gas, ΔU increases, and T rises. Contrast this with an isothermal expansion, where heat flows in from the surroundings exactly fast enough to keep T — and hence U — constant.

An isochoric process occurs at constant volume (ΔV = 0), so w = 0 and ΔU = q_V. An isobaric process occurs at constant pressure; here q_p = ΔU + pΔV, which is exactly how enthalpy is defined in the next section.

ΔU is measured using a bomb calorimeter: the reaction (usually combustion) is carried out in a sealed, rigid steel vessel immersed in a known mass of water, so the volume is constant. No expansion work is possible (w = 0), so all the heat released or absorbed shows up as a temperature change of the water and calorimeter: q_V = ΔU = −C × ΔT, where C is the heat capacity of the calorimeter assembly. This value is reported per mole after scaling to the amount reacted.

ΔH, the enthalpy change at constant pressure, is measured in a calorimeter open to the atmosphere (like a coffee-cup calorimeter for solution reactions) where the system is free to expand or contract against constant atmospheric pressure: q_p = ΔH.

Most reactions in the laboratory are carried out in open vessels, at constant atmospheric pressure, not constant volume. It is convenient to define a new state function, enthalpy, H = U + pV, so that the heat absorbed at constant pressure equals ΔH directly, without needing to separately track the pV work.

Derivation of ΔH = ΔU + pΔV (at constant pressure): Starting from the first law, ΔU = q_p + w. For expansion work at constant external pressure p, w = −pΔV. So ΔU = q_p − pΔV, which gives q_p = ΔU + pΔV. Since H = U + pV, at constant pressure ΔH = ΔU + pΔV. Comparing the two, q_p = ΔH — the heat absorbed at constant pressure is exactly the enthalpy change.

Relation between ΔH and ΔU for gaseous reactions: for a reaction in which the moles of gas change, using pV = nRT at constant T and p, pΔV = Δn_gRT, where Δn_g is the difference between moles of gaseous products and moles of gaseous reactants. So ΔH = ΔU + Δn_gRT. If Δn_g = 0 (equal moles of gas on both sides, or no gas involved), ΔH = ΔU.

By convention, ΔH is negative for an exothermic reaction (heat is released to the surroundings, enthalpy of products is lower than reactants) and positive for an endothermic reaction (heat is absorbed, enthalpy of products is higher).

The heat capacity of a system is the heat needed to raise its temperature by 1 K (or 1 °C): C = q/ΔT. It depends on the size and nature of the system, and on the conditions of measurement — heat capacity at constant volume, C_V, and heat capacity at constant pressure, C_p, are different because at constant pressure some of the heat also does expansion work.

Relation between C_p and C_V for an ideal gas — derivation: At constant volume, q_V = ΔU = C_VΔT. At constant pressure, q_p = ΔH = C_pΔT. For one mole of an ideal gas, ΔH = ΔU + Δ(pV) = ΔU + RΔT (since pV = RT for one mole). Substituting, C_pΔT = C_VΔT + RΔT. Dividing throughout by ΔT gives C_p − C_V = R. C_p is always greater than C_V because heating at constant pressure must also supply energy for the gas to expand and do work, in addition to raising its internal energy.

Molar heat capacity is the heat capacity per mole, an intensive quantity, with units J K⁻¹ mol⁻¹. Specific heat capacity is heat capacity per gram (J g⁻¹ K⁻¹). Water's high specific heat (4.18 J g⁻¹ K⁻¹) is why it is used as the medium in calorimeters.

The enthalpy of a reaction, Δ_rH, is the enthalpy change when the number of moles of reactants shown in the balanced equation react completely to give products, under standard conditions unless stated otherwise. It equals the sum of the standard enthalpies of formation of the products minus that of the reactants, each multiplied by its stoichiometric coefficient: Δ_rH° = ΣaΔ_fH°(products) − ΣbΔ_fH°(reactants).

Standard conditions (indicated by the ° symbol) mean 1 bar pressure and a specified temperature, usually 298 K, with substances in their standard states (the most stable physical state of the substance at that temperature and 1 bar).

Different named enthalpy changes are simply Δ_rH for a reaction of a specific type:

• Standard enthalpy of formation, Δ_fH° — enthalpy change when 1 mole of a compound is formed from its elements in their standard states. Δ_fH° of an element in its standard state is zero by definition.
• Standard enthalpy of combustion, Δ_cH° — enthalpy change when 1 mole of a substance is completely burnt in excess oxygen; always negative (exothermic).
• Enthalpy of atomization, Δ_aH° — enthalpy needed to break a substance into gaseous atoms, e.g. CH₄(g) → C(g) + 4H(g).
• Enthalpy of sublimation, fusion, and vaporization — enthalpy changes for solid → gas, solid → liquid, and liquid → gas at constant T.
• Enthalpy of solution and dilution — heat change when a solute dissolves in a solvent, or when a solution is further diluted.

Hess's law of constant heat summation states that the total enthalpy change for a reaction is the same whether it occurs in one step or in a series of steps, because enthalpy is a state function and ΔH depends only on the initial and final states, not the path.

This lets us calculate the enthalpy of a reaction that cannot be measured directly by combining the enthalpies of reactions that can be measured, using ordinary algebra: reverse an equation and reverse the sign of its ΔH; multiply an equation by a factor and multiply its ΔH by the same factor; add equations that share common intermediates so they cancel, and add their ΔH values.

Classic example: enthalpy of formation of CO(g) from C(graphite) and O₂(g) cannot be measured directly because burning carbon in a limited supply of oxygen always gives some CO₂ as well. Instead we use: (i) C(graphite) + O₂(g) → CO₂(g), ΔH₁ = −393.5 kJ mol⁻¹, and (ii) CO(g) + ½O₂(g) → CO₂(g), ΔH₂ = −283.0 kJ mol⁻¹. Reversing (ii) and adding to (i): C(graphite) + ½O₂(g) → CO(g), ΔH = ΔH₁ − ΔH₂ = −393.5 − (−283.0) = −110.5 kJ mol⁻¹.

Hess's law numericals are also solved using a Born–Haber-cycle-style energy diagram, drawing the target reaction directly between reactants and products, and the known steps as an alternative path, then equating the sum of ΔH along both routes.

Bond enthalpy, Δ_bondH°, is the enthalpy required to break one mole of a particular covalent bond in a gaseous molecule to give gaseous atoms/fragments. For a diatomic molecule this equals the bond dissociation enthalpy directly. For a polyatomic molecule with several identical bonds (e.g. the four C–H bonds in methane), the mean bond enthalpy is used, since each successive bond-breaking step actually requires a slightly different amount of energy.

Estimating reaction enthalpy from bond enthalpies: Δ_rH° = Σ(bond enthalpies of bonds broken in reactants) − Σ(bond enthalpies of bonds formed in products), because breaking bonds always requires energy (endothermic) and forming bonds always releases energy (exothermic). This method gives an approximate value since it uses average bond enthalpies, not the exact bond enthalpy in that specific molecule.

Enthalpy of lattice formation, Δ_latticeH° — enthalpy change when one mole of an ionic solid is formed from its gaseous ions; it is large and negative because of the strong electrostatic attraction between oppositely charged ions. It cannot be measured directly and is obtained using a Born–Haber cycle, which applies Hess's law to a sequence of steps: sublimation of the metal, ionization, dissociation and electron gain of the non-metal, and lattice formation, connecting them to the directly measurable enthalpy of formation of the compound.

A spontaneous process is one that occurs on its own, without continuous external intervention, once started. It may be fast or slow — spontaneity says nothing about the rate. A spontaneous process cannot reverse on its own; it needs external work to be reversed (e.g. water flows downhill spontaneously; pumping it back up needs work). Non-spontaneous does not mean impossible — it means the process will not occur by itself under the given conditions.

Early thinking assumed all exothermic reactions (ΔH < 0) are spontaneous, since systems tend towards lower energy. But some spontaneous processes are endothermic — ice melting above 0°C, or NH₄Cl dissolving in water, both absorb heat yet occur on their own. So enthalpy alone cannot decide spontaneity; a second factor, entropy, is needed.

Entropy (S) is a state function that measures the degree of randomness or disorder of a system. The second law of thermodynamics states that the total entropy of the universe (system + surroundings) always increases in a spontaneous process: ΔS_total = ΔS_system + ΔS_surroundings > 0 for a spontaneous process, and = 0 at equilibrium.

For a reversible process at temperature T, the entropy change of the system is ΔS = q_rev/T. Entropy generally increases (ΔS > 0) when a solid melts, a liquid vaporizes, a gas expands, moles of gas increase in a reaction, or a solid dissolves to give ions in solution; entropy decreases in the reverse of these changes.

The third law of thermodynamics states that the entropy of a perfectly crystalline substance is taken as zero at absolute zero (0 K), since there is then only one way of arranging the molecules — this gives an absolute reference point for calculating standard molar entropies, unlike enthalpy where only changes can be measured.

Since spontaneity needs the total entropy change of the universe, but we usually only know about the system, Gibbs defined a new state function combining ΔH and ΔS of the system alone: Gibbs energy, G = H − TS. At constant temperature and pressure, ΔG = ΔH − TΔS.

Derivation of the criterion: ΔS_total = ΔS_sys + ΔS_surr. At constant p and T, the surroundings absorb −ΔH_sys of heat reversibly, so ΔS_surr = −ΔH_sys/T. Substituting, ΔS_total = ΔS_sys − ΔH_sys/T. Multiplying through by −T: −TΔS_total = ΔH_sys − TΔS_sys = ΔG_sys. Since ΔS_total ≥ 0 for a spontaneous or equilibrium process and T is positive, ΔG_sys ≤ 0 for the same process.

So at constant temperature and pressure: ΔG < 0 — reaction is spontaneous (exergonic); ΔG > 0 — reaction is non-spontaneous in the forward direction (endergonic), though the reverse reaction is spontaneous; ΔG = 0 — system is at equilibrium.

Because ΔG = ΔH − TΔS, four combinations of sign are possible: ΔH < 0, ΔS > 0 gives ΔG < 0 always (spontaneous at all T); ΔH > 0, ΔS < 0 gives ΔG > 0 always (non-spontaneous at all T); ΔH < 0, ΔS < 0 is spontaneous only at low T (below T = ΔH/ΔS); ΔH > 0, ΔS > 0 is spontaneous only at high T (above T = ΔH/ΔS). This is why some endothermic processes (like ice melting, or many dissolution reactions) are spontaneous — the −TΔS term outweighs the positive ΔH once T is large enough.

Gibbs energy links thermodynamics to chemical equilibrium (studied in the next chapter) through ΔG° = −RT ln K = −2.303 RT log K, where K is the equilibrium constant of the reaction at temperature T. A large negative ΔG° gives a large K (reaction goes almost to completion); a large positive ΔG° gives a very small K (reaction barely proceeds); ΔG° = 0 gives K = 1.

It is important to distinguish ΔG (at the actual, non-equilibrium concentrations, which determines the direction a reaction will shift) from ΔG° (the fixed standard value used to calculate K). Once the system reaches equilibrium, ΔG = 0, even though ΔG° for the reaction is not necessarily zero.