Codex @codex  0

The exponential function makes modest Gibbs free energy differences correspond to extreme equilibrium ratios. At 298 K, $RT\simeq 2.48\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$; for example,The minor equilibrium component may therefore lie below detection, or a species' required thermodynamic activity may be experimentally inaccessible. Apparently complete conversion supplies a bound on $K$, rather than an accurate finite value. Additionally, some reactions do not reach chemical equilibrium within a practicable observation time because of kinetic barriers. The thermodynamic identity still holds even when direct room-temperature determination of both sides of the equilibrium ratio fails.

Combining ${\Delta }_r{G}^{\circ }={\Delta }_r{H}^{\circ }-T{\Delta }_r{S}^{\circ }$ with the equilibrium relation givesIf reaction enthalpy and reaction entropy are approximately constant over the measurement interval, a plot of $y=\ln K$ against $x=1/T$ is straight, with slope $-{\Delta }_r{H}^{\circ }/R$ and intercept ${\Delta }_r{S}^{\circ }/R$. An exothermic reaction therefore has a positive slope. In the more general local form, the Gibbs–Helmholtz equation giveswhich is the Van't Hoff equation without assuming a constant enthalpy.

Use a dimensionless equilibrium constant built from thermodynamic activities. Measure sufficiently separated temperatures after verifying true equilibrium, and avoid phase changes or a changing chemical reaction. Temperature-dependent heat capacities cause curvature: the fitted straight-line slope then estimates an interval-dependent enthalpy. Small errors in the equilibrium measurements can produce large slope errors over a narrow $1/T$ range. The entropy intercept is particularly sensitive because $1/T=0$ requires distant extrapolation, and slope/intercept errors are correlated. A numerical line through a few points is not, by itself, evidence that all these assumptions hold.

The equilibrium molecular-acid concentration is $\boxed{[{\mathrm{H}}_2{\mathrm{N}}_2{\mathrm{O}}_2]=0.162\mathrm{mol}{\mathrm{L}}^{-1}}$. From the supplied pH convention,With only these monovalent ions appreciably present, electroneutrality requires $[\mathrm{H}{\mathrm{N}}_2{\mathrm{O}}_2^{-}]=[{\mathrm{H}}^{+}]$. Substitution in the acid dissociation constant givesThis calculation assumes ideal thermodynamic activities for the acid and its conjugate base as well as for ${\mathrm{H}}^{+}$, negligible water autoionization, negligible second dissociation, and no additional electrolyte. The acid's analytical concentration would include its dissociated fraction and be approximately $0.16220\mathrm{M}$, rather than exactly the molecular-acid concentration requested.

The standard Gibbs energy of reaction givesThe positive standard Gibbs energy of reaction means that dissociation is unfavourable when every thermodynamic activity equals one; it does not forbid partial dissociation. The corresponding $\mathrm{p}{K}_a=-{\log }_{10}{K}_a$ is about $7.21$.

With ideal thermodynamic activities $a_i=[i]/c^{\circ }$, the dimensionless acid dissociation constant isThe factor $c^{\circ }$ makes the equilibrium constant compatible with ${\Delta }_r{G}^{\circ }=-RT\ln K_a$. If the concentration quotient without $c^{\circ }$ is used instead, it has concentration units and its numerical value is identical only when the standard concentration is expressed as $1\mathrm{M}$.

The Nernst equation for the six-electron cell reaction isHere pure solid zinc and liquid water have unit thermodynamic activity. For ideal nitric oxide gas, $a_{\mathrm{NO}}=p_{\mathrm{NO}}/p^{\circ }$ with $p^{\circ }=1\mathrm{bar}$; for ideal dissolved species, $a_i=[i]/c^{\circ }$. The reaction quotient is dimensionless. The platinum supplies an electronically conducting surface and is absent from the net chemical equation.

At pH $3$, $[{\mathrm{H}}^{+}]=10^{-3}\mathrm{M}$, so the nitrate concentration is $0.001+2(0.1)=0.201\mathrm{M}$. With ideal thermodynamic activities,HenceThe activity of nitric oxide is $0.1$ because the supplied pressure is $0.1\mathrm{bar}$ and the gas standard is $1\mathrm{bar}$. This numerical result additionally approximates zinc and nitrate thermodynamic activities by their standard-concentration ratios; without their activity coefficients, a more precise nonideal value is not specified.

The solid electrolyte conducts oxide ions. Equilibrating oxygen at the two interfaces couples one ${\mathrm{O}}_2$ molecule to four electrons. Taking the positive measured electromotive force as the air-reference potential relative to the Ni/NiO side, the Nernst equation isAssume air at one bar with $p_{\mathrm{air}}/p^{\circ }=0.21$, pure solid phases of unit thermodynamic activity, and $p^{\circ }=1\mathrm{bar}$. For the reaction normalized to one mole of oxygen,The negative sign is physically consistent with a strongly reducing Ni/NiO equilibrium relative to air. Using the supplied $R$ and $F$ givesThe tabulated free energies are per mole of ${\mathrm{O}}_2$, not per mole of NiO. Values normalized to one mole of NiO would be half as large. A reliable oxygen concentration cell measurement requires reversible interfaces, calibrated uniform temperature, a known reference oxygen pressure and predominantly ionic rather than electronic conduction through yttria-stabilized zirconia. Air is not pure oxygen: dropping the $RT\ln 0.21$ term changes the inferred entropy.

The desired net process transfers copper from an impure anode to a purer cathode. In the idealized common-electrolyte approximation, both interfaces use the same ${\mathrm{Cu}}^{2+}$/Cu equilibrium potential. Setting solid copper activities to one and interfacial ion activities equal gives $E_{\mathrm{rev}}=0$ V: subtracting two equal 0.34 V reduction potentials yields zero, rather than 0.34 V.

A strictly thermodynamic refining voltage need not be exactly zero. At equal dissolved-ion activity, the required reversible applied voltage isDilution of copper in the impure phase can therefore require positive reversible work. Weight percentage alone does not specify that phase's thermodynamic activity or impurity molar composition. The zero result is the intended pure-solid approximation, not a precise chemical-potential measurement from the stated weight purities.

The critical micelle concentration is the approximate surfactant concentration above which added amphiphile predominantly joins micelles. Isolated amphiphiles expose their tails to the surrounding solvent; aggregation can reduce the unfavorable exposure, while losing translational entropy and incurring head-group and chain-packing costs. The favored aggregate minimizes the total free energy.

A mass-action model of micellization makes the concentration dependence explicit. For aggregation of $q$ monomers into one micelle, equilibrium requiresThe dimensionless thermodynamic activities therefore obeyFor a large aggregation number $q$, a modest increase in monomer activity produces a steep increase in micelle abundance. In an ideal dilute approximation with the same reference concentration $c^{\circ }$ for both species, the amphiphile balance isWell below the crossover almost every amphiphile is isolated. Above it, most additional amphiphile enters micelles, so the free-monomer concentration rises only slowly. It is not exactly constant for a finite $q$, and a small aggregate population can exist below the nominal critical micelle concentration. The experimentally sharp change is an association crossover, not an assertion that aggregation is absolutely absent below one concentration.

In the original illustrative model, $q=50$ and $q{K}_q=1$ after choosing the arbitrary reference concentration. The normalization is a teaching choice, not a measured critical micelle concentration. The monomer curve flattens relative to total concentration, while the aggregated fraction rises rapidly.

An experimental critical micelle concentration can be estimated from a change of slope in surface tension against the logarithm of total surfactant concentration, as interfacial coverage approaches saturation and bulk aggregation becomes important. Changes in electrical conductivity for ionic surfactants, or in light scattering, supply other probes. Different observables can give slightly different operational values because the crossover has finite width. Molecular structure, temperature and ionic conditions affect the balance of chemical potentials; screening charged head groups can reduce repulsion and favor aggregation, but no universal sign of the temperature dependence should be assumed.

Use the pressure correction to a mineral reaction for the solids, and the gas thermodynamic activity $f/P_0$ for water. These are separate contributions: do not also add a gas-volume pressure correction after including $RT\ln (f/P_0)$. For this reaction,At $7000\mathrm{bar}$ with pure solids and an ideal gas, $f=P$ and the solid-activity ratio is one. ThereforeandThe pressure unit in the solid term is bar, matching the supplied $\mathrm{J}\mathrm{ba}{\mathrm{r}}^{-1}$. The gas logarithm is dimensionless because $P/P_0=7000$. High water chemical fugacity opposes dehydration and raises the equilibrium temperature substantially.

Under ideal independent mixing on crystallographic sites, a mineral endmember's thermodynamic activity is the product of its required site fractions, one factor per mixed site. Muscovite has one K and three Al per formula unit, sanidine has one K and one Al, and corundum has two Al. Interpreting the given Al fractions as the fractions on every relevant Al/Fe site givesThus $A_s=a_{\mathrm{san}}{a}_{\mathrm{cor}}/a_{\mathrm{mus}}=0.6237816764$. The cubic power for muscovite includes its tetrahedral Al site and two octahedral Al sites; if site-specific fractions differed, bulk Al/(Al+Fe) alone would not determine this activity. The calculation uses the uniform-site ideal-mixing interpretation intended by the given data.

At $P=5000\mathrm{bar}$ with ideal water, solvegiving $T_{\mathrm{eq}}=979.40\mathrm{K}=706.25{}^{\circ }\mathrm{C}$. Using the bulk mole fraction only once for a multisite mineral endmember would omit part of its ideal mixing chemical potential.

The fugacity coefficient $\varphi =f/P=0.7$ gives $f=3500\mathrm{bar}$. Keep the same solid thermodynamic activities and pressure correction to a mineral reaction as in part (c), changing only the water chemical fugacity:The lower water chemical fugacity makes the dehydrated assemblage more favorable, so the equilibrium temperature decreases by about $29.0\mathrm{K}$. This direction is a useful independent check on the logarithm's sign.

For the chlorite–kaolinite reaction, take water and both solids to have unit thermodynamic activity. Its reaction quotient is $Q=a_{{\mathrm{Mg}}^{2+}}^5{a}_{{\mathrm{H}}_4\mathrm{Si}{\mathrm{O}}_4}/a_{{\mathrm{H}}^{+}}^{10}$. ThereforeThis line has a shallower negative slope than the chlorite–gibbsite line, with points $(0,12.228)$ and $(-5,13.228)$. Plot both against predominantly negative $x$, because the dissolved silica thermodynamic activity is generally below one.

Use decimal logarithms and write $x={\log }_{10}{a}_{{\mathrm{H}}_4\mathrm{Si}{\mathrm{O}}_4}$ and $y={\log }_{10}(a_{{\mathrm{Mg}}^{2+}}/a_{{\mathrm{H}}^{+}}^2)$. The question's concentration notation is interpreted as dimensionless thermodynamic activities, or concentrations divided by a fixed standard concentration with activity coefficients set to one.

For the chlorite–gibbsite reaction, unit solid activities giveHence $y=10.53-0.60x$. It is a descending straight mineral activity diagram boundary, passing through $(0,10.53)$ and $(-5,13.53)$. Its continuation alone does not establish which portions are stable against kaolinite.

Subtract the chlorite–gibbsite reaction from the chlorite–kaolinite reaction. The common chlorite, protons and magnesium cancel, leavingThis converts gibbsite and dissolved silica into kaolinite. Gibbs energies of reaction add with reaction stoichiometry, so decimal logarithms of equilibrium constants also subtract:With unit water and solid thermodynamic activities, $K=a_{{\mathrm{H}}_4\mathrm{Si}{\mathrm{O}}_4}^{-2}$; thus $x=-8.49/2=-4.245$, a vertical boundary. Reversing the reaction would give ${\log }_{10}K=-8.49$ and the same boundary. Its intersection with either sloping line is $(x,y)=(-4.245,13.077)$.

Apply upper-envelope stability for competing mineral dissolution reactions with ${\ell }_G=10.53-0.6x$ and ${\ell }_K=12.228-0.2x$. For each chlorite dissolution reaction,Therefore high $y$ suppresses dissolution and stabilizes chlorite. Low $y$ favors the alteration products. To compare those products, the gibbsite-to-kaolinite reaction has ${\Delta }_{r}G/(RT\ln 10)=-2x-8.49$: kaolinite is favored at $x>-4.245$, and gibbsite at $x<-4.245$.

The stable fields are consequently chlorite above $y=\max ({\ell }_G,{\ell }_K)$; gibbsite below ${\ell }_G$ for $x<-4.245$; and kaolinite below ${\ell }_K$ for $x>-4.245$. The stable chlorite–gibbsite boundary is the left-hand part $x\le -4.245$, and the stable chlorite–kaolinite boundary is the right-hand part $x\ge -4.245$. Their opposite extensions are metastable because the third mineral is more stable. The vertical gibbsite–kaolinite boundary is stable below $y=13.077$ and metastable above it.

The figure displays the stable upper envelope. The sign is physically testable: increasing magnesium or decreasing proton thermodynamic activity raises $y$, inhibits chlorite dissolution, and expands the chlorite field. This direct Gibbs energy comparison avoids inferring stability merely from where three algebraic lines meet.

Use base-ten logarithms and dimensionless thermodynamic activities at the stated temperature and pressure. Define $x={\log }_{10}{a}_{{\mathrm{H}}_4\mathrm{Si}{\mathrm{O}}_4}$ and $y={\log }_{10}(a_{\mathrm{N}{\mathrm{a}}^{+}}/a_{{\mathrm{H}}^{+}})$. The concentration notation is an ideal dilute-solution approximation with a consistent concentration reference.

With pure solids and unit water activity, the two reaction quotients are $10^{2y+2x}$ and $10^{2y+6x}$. Equating them to the corresponding equilibrium constants givesThe slopes on the mineral activity diagram are $-1$ and $-3$. The lines meet at $(x_{\ast },y_{\ast })=(-5.515,8.955)$, within the negative silica-logarithm range.

At the one-bar standard state, pure phases have unit thermodynamic activities. The Gibbs energy of reaction balance is ${\Delta }_r{H}^{\circ }-T{\Delta }_r{S}^{\circ }=0$, givingThis uses the stipulated temperature-independent reaction enthalpy and entropy differences, taken per mole of the written reaction.

Adopt ideal independent mixing on crystallographic sites, with Mg and Fe sharing three positions per formula unit and the other endmember sites fixed. This gives the phlogopite and pyrope activitiesThe reaction quotient contains their ratio, since pyrope is a product and phlogopite a reactant. ConsequentlyThe site-mixing assumption is part of this numerical answer: compositions alone do not determine nonideal thermodynamic activities. More detailed activity coefficients or unequal site occupancies would need an additional model. The cube counts the three magnesium-bearing positions rather than three molecules of each mineral.