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The acid dissociation constant quantifies the chemical equilibrium for loss of one proton from a chemical acid. In water, write the actual proton transfer as $\mathrm{HA}+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{H}}_3{\mathrm{O}}^{+}+{\mathrm{A}}^{-}$. Its full dimensionless equilibrium constant isThe usual aqueous acid dissociation constant uses the solvent convention with water thermodynamic activity approximated by one:Here hydrated hydrogen ions are understood; ${\mathrm{H}}^{+}$ is shorthand, not an isolated bare proton in the chemical solution. All thermodynamic activities and $K_a$ are dimensionless quantities, using consistent standard states. If water thermodynamic activity differs appreciably from one, retain it in the full equilibrium constant instead of silently mixing conventions.

With activity coefficients ${\gamma }_i$ and reference concentration $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$, put $a_i={\gamma }_i[i]/c^{\circ }$ for each solute. ThenThe last expression is the dilute ideal concentration approximation, not the definition in a nonideal chemical solution. Omitting $c^{\circ }$ instead produces a quotient with units of concentration; its numerical value depends on the chosen units. Do not take a logarithm of that dimensional quotient. Conditional concentration quotients can change with ionic strength even at fixed temperature, whereas the thermodynamic acid dissociation constant retains the activity coefficient corrections. The acid dissociation constant entry distinguishes these conventions.

At the same temperature, solvent and standard states, a larger acid dissociation constant, equivalently a smaller pKa, indicates a stronger tendency to donate a proton. It does not specify an analytical concentration, a pH, or a reaction rate. Composition and the weak acid mass and charge balances are needed to predict pH; chemical kinetics addresses how quickly the chemical equilibrium is approached.

ATP is a nucleotide consisting of adenosine, itself adenine joined to ribose, carrying three phosphate groups. From the ribose outward, these groups are labelled $\alpha$, $\beta$, and $\gamma$. ATP connects energy-supplying processes to energy coupling for synthesis and active transport; the same carrier can be regenerated and reused. ^[ref].

The net hydrolysis commonly used in energy coupling isHere ${\mathrm{P}}_i$ denotes inorganic phosphate. At buffered pH $7$, this biochemical shorthand groups the appropriate protonation states into pools and leaves proton exchange with the buffer solution implicit. An explicit, charge-balanced choice is ${\mathrm{ATP}}^{4-}+{\mathrm{H}}_2\mathrm{O}⟶{\mathrm{ADP}}^{3-}+{\mathrm{HP}{\mathrm{O}}_4}^{2-}+{\mathrm{H}}^{+}$, without implying these are the only chemical species present. See the IUBMB recommendations for biochemical thermodynamics.

The quoted textbook biochemical standard state value is approximatelyThis convention uses fixed pH $7$, temperature $298.15\mathrm{K}$ ($25{}^{\circ }\mathrm{C}$), reference pressure $100\mathrm{kPa}$ ($1\mathrm{bar}$), unit reference thermodynamic activities for dissolved reactants and reaction products, and water thermodynamic activity approximated by one. The prime indicates the fixed-pH biochemical convention. See OpenStax Biology 2e, 6.4: standard ATP hydrolysis value and Biology 2e, 6.2: biochemical standard conditions. This is a reference value, rather than a universal value inside a biological cell; precise tables must also specify chemical solution conditions and any magnesium ion binding convention. Different reference conventions and data treatments give different tabulated values, as discussed in Li et al., 2010: a database of thermodynamic quantities for biochemical reactions.

An exergonic reaction has a negative change in Gibbs free energy, so the overall ATP hydrolysis is favourable under those reference conditions. Breaking a chemical bond by itself costs energy; forming new chemical bonds can release energy. The net change compares all reactants with all reaction products, including their interactions with water and the entropy contribution. Calling the phosphate connection a "high-energy bond" describes a large favourable change in Gibbs free energy on hydrolysis, rather than energy released by bond breaking alone. See 7.5 Strengths of Ionic and Covalent Bonds and Cooper, The Cell: ATP and metabolic energy.

Composition changes the driving force. At the same fixed pH, temperature, pressure, and binding convention, write the molar reaction change asHere $R$ is the molar gas constant, $T$ the absolute temperature, and $a$ a dimensionless thermodynamic activity for a biochemical pool. Constant water thermodynamic activity is absorbed into the reference term; fixed pH accounts for buffered protons. The shorthand $\Delta G$ means transformed molar reaction Gibbs free energy change, ${\Delta }_r{G}^{\prime }$, rather than a finite sample's total change. The natural logarithm acts on a dimensionless reaction quotient, never on concentrations with units. See the IUBMB biochemical thermodynamics recommendations and the general equilibrium constant relation.

For a stipulated ideal dilute chemical solution, let $a_i=[i]/c^{\circ }$, with $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$. Choose $[\mathrm{ATP}]=5\mathrm{mmol}{\mathrm{L}}^{-1}$, $[\mathrm{ADP}]=0.5\mathrm{mmol}{\mathrm{L}}^{-1}$, and $[{\mathrm{P}}_i]=1\mathrm{mmol}{\mathrm{L}}^{-1}$, ignoring activity coefficient corrections and magnesium binding. These are illustrative inputs, not a claim about a particular biological cell. ThenandIncreasing $Q$ tenfold raises $\Delta G$ by $RT\ln 10\approx 5.708\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ at this temperature. Thus the ATP to ADP ratio and available inorganic phosphate matter, even with the same standard state value. These are local molar predictions at the stipulated composition, rather than a constant total yield as an isolated sample reacts.

For a nonideal chemical solution, write its thermodynamic activities on a mole fraction basis as $a_A=x{\gamma }_A$ and $a_B=(1-x){\gamma }_B$. Use composition-independent pure-component reference chemical potentials at the same temperature and pressure:The activity coefficients describe departure from the ideal solution under that reference convention.^[ref]

At fixed temperature and pressure, insert these expressions into the Gibbs-Duhem equation. The ideal solution terms cancel because $xd\ln x+(1-x)d\ln (1-x)=dx-dx=0$. What remains isThus two independent fits to measured activity coefficients can each look smooth while failing to describe one possible Gibbs free energy. This is a necessary consistency condition; satisfying it alone does not establish experimental accuracy or thermodynamic stability.

For an original algebraic example, suppose at one fixed temperature and pressure that $\ln {\gamma }_A=\alpha (1-x{)}^2$, with dimensionless constant $\alpha$. The constraint requires $d\ln {\gamma }_B/dx=2\alpha x$. With the pure-B normalization ${\gamma }_B(0)=1$, integration givesFor $\alpha =1/2$ and $x=1/4$, the derivatives of the two logarithms are $-3/4$ and $1/4$; their weighted sum is $-3/16+3/16=0$. Choosing $\ln {\gamma }_B=\alpha x$ instead would give a weighted sum $3/16$ at that composition, contradicting the Gibbs-Duhem equation. These are invented model functions, with no claim to fit a particular mixture.

The fixed-temperature, fixed-pressure restriction matters. Along a path with changing temperature or pressure, the general equation contains $SdT-Vdp$, and the reference chemical potentials and $RT$ also change. Across phase coexistence, apply the relation separately within each smooth thermodynamic phase; do not differentiate a discontinuous switch between phases as if it were a single composition curve.

Stipulate pKa $5$ and $K_w=10^{-14}$; these are model inputs, not measurements of an identified substance. At pH $5$ the pair contribution is $0.575646\dots C$. For analytical concentrations $C=0.010$ and $0.100\mathrm{mol}{\mathrm{L}}^{-1}$, respectively, it isThe free hydrogen ion and hydroxide contribution is the same at this pH, ${\beta }_{\mathrm{water}}=(\ln 10)(10^{-5}+10^{-9})\mathrm{mol}{\mathrm{L}}^{-1}=0.0000230282\dots \mathrm{mol}{\mathrm{L}}^{-1}$ per pH. The full buffer capacities are therefore approximately $0.00577949$ and $0.0575877\mathrm{mol}{\mathrm{L}}^{-1}$ per pH. Identical pH fixes the free hydrogen ion thermodynamic activity; it does not fix the inventory of weak acid and conjugate base available for neutralization.

At one unit of pH on either side of pKa, $r=10$ or $0.1$, giving ${\beta }_{\mathrm{pair}}/{\beta }_{\mathrm{pair},\max }=40/121\approx 0.331$. The partners still form a buffer solution, but its local resistance is only about one third of the maximum for that analytical concentration. At fixed pH, the pair contribution scales exactly with $C$; the full buffer capacity contains the additional $C$-independent water contribution. Consequently, extreme dilution does not justify dropping self-ionization of water.

A proposed modern measurement extends the weak acid dissociation measurement by estimating the local slope of an acid-base titration. Transfer a measured sample with a pipette into a stirred vessel; clamp a combination glass electrode/reference electrode clear of the stirring magnet and place a temperature probe nearby. A calibrated burette delivers a chemical base of independently standardized concentration $c_T$, such as sodium hydroxide. Record the initial sample volume $V_0$, cumulative delivered volume $v$, stable pH, and temperature after each small addition. Use a separate fresh portion for an acidward series. These are proposed measurement conditions, not reported experimental results.

For the instrument calibration, use fresh standard buffer solutions bracketing the measured pH range, with values appropriate to the actual temperature. The Hanna manual specifies separate rinse and calibration portions, gentle stirring, and approximately $4\mathrm{cm}$ immersion of the probe tip. It calls for stable readings; the sensing bulb and reference electrode junction must remain immersed. Rinse between standards and samples and avoid carrying rinse water into the measured portion.^[ref]

The readout pairs each added amount of substance with the measured pH. With negligible added volume, neighboring readings give the finite difference estimateassigned to the interval's midpoint. High buffer capacity appears as a small change of pH for a measured reagent dose. Choose increments small enough to resolve changing buffer capacity, but large enough that the change of pH exceeds measurement noise. Repeat with a fresh portion and different increment sizes; remeasure a standard afterward to check drift. A water blank can reveal contamination, but its pH must not be subtracted from the sample pH. These controls are proposed tests of the measurement and model assumptions.

If additions appreciably increase volume, even the analytical concentration changes: $C(v)=C_0{V}_0/(V_0+v)$. Merely dividing the added moles by the current volume does not restore the constant-volume derivative. One alternative is explicitly to report a changing-volume convention. For $c(v)=c_{T}v/(V_0+v)$, defineThis definition includes the reagent's changing concentration and differs from the constant-volume buffer capacity defined above.^[ref]

To compare with the constant-volume theory, fit the changing-volume chemical equilibrium model instead of treating the observed slope as $\beta$. In the same ideal monoprotic acid system, define $S_0=[{\mathrm{A}}^{-}{]}_0+[{\mathrm{OH}}^{-}{]}_0-[{\mathrm{H}}^{+}{]}_0$. For baseward additions, the electric charge balance at each reading isUse a minus sign before $c_{T}v$ for the corresponding acidward series. From the model parameters, evaluate the constant-volume buffer capacity at the chosen analytical concentration and pH. The fit also tests whether one acid dissociation constant describes the observations.

Limitations include incomplete mixing, slow glass electrode response, uncertain reagent concentration, and changes in temperature or ionic strength. Dissolved atmospheric carbon dioxide introduces an additional chemical acid system; comparable exposure and freshly prepared samples help distinguish it from the intended monoprotic acid response. Additional reacting chemical species and nonideal activity coefficients invalidate the simple formula. Especially when pH increments are tiny, measurement noise can dominate the inferred derivative; reporting the dose, pH interval, volume convention, and repeatability is part of reporting a meaningful buffer capacity.

A buffer solution contains appreciable quantities of a weak acid and its conjugate base, and resists changes in pH caused by small quantities of added chemical acid or chemical base. The conjugate base accepts added protons; the weak acid supplies protons when a chemical base consumes them. Consuming most of either partner undermines the buffering action. See Flowers et al., Chemistry 2e, buffers.

For a separate toy calculation, stipulate pKa $5$ for a weak acid $\mathrm{HA}$; this is an input, not measured material data. Initially $1\mathrm{L}$ of buffer solution contains $0.10\mathrm{mol}$ of $\mathrm{HA}$ and $0.10\mathrm{mol}$ of its conjugate base ${\mathrm{A}}^{-}$. In the ideal concentration approximation, the Henderson-Hasselbalch equation gives initial pH $5$. Add $0.010\mathrm{mol}$ of hydroxide with negligible volume change. The dominant neutralization has one-to-one stoichiometry:Each mole of consumed hydroxide removes one mole of weak acid and forms one mole of conjugate base and water. Thus the inventories after neutralization areThe shared volume cancels from the concentration ratio. The Henderson-Hasselbalch equation therefore predictsThis small increase illustrates how conversion between abundant partners absorbs added chemical base. The calculation neglects the tiny free hydronium and hydroxide inventories, changes during re-establishment of chemical equilibrium, self-ionization of water, and activity coefficient corrections. The displayed inventories approximate the final chemical species distribution; they are not an exact solution of all chemical equilibrium conditions.

A buffer solution has a finite buffer capacity: its weak acid inventory supplies protection against added chemical base, and its conjugate base inventory supplies protection against added chemical acid. Here an addition approaching $0.10\mathrm{mol}$ of hydroxide uses up the available $\mathrm{HA}$ in the dominant stoichiometry calculation. Close to that limit, the neglected chemical equilibrium contributions matter; once $\mathrm{HA}$ is exhausted, inserting a zero denominator into the Henderson-Hasselbalch equation cannot predict the actual pH. Excess hydroxide and the remaining chemical species require a fresh chemical equilibrium calculation. This is why a buffer solution stabilizes conditions for chemical reactions over a finite range of additions. See Flowers et al., Chemistry 2e, buffer capacity and added base.

For the pair $\mathrm{HA}/{\mathrm{A}}^{-}$, rearranging the acid dissociation constant gives the activity relationWith thermodynamic activity of water approximated by one and ideal dilute thermodynamic activities for both partners, this yields the familiar Henderson-Hasselbalch equation:The concentrations are at chemical equilibrium; using initial values requires negligible equilibration changes. Below, also neglect self-ionization of water and competing chemical reactions.

Stipulate $\mathrm{p}{K}_a=4.76$ for an illustrative weak acid, without identifying it with a measured substance. A conjugate base to weak acid concentration ratio of two givesEqual concentrations give a ratio of one, whose logarithm is zero, hence pH $4.76$. Multiplying the ratio by ten raises pH by one: ratios $0.1$, $1$, and $10$ give pH values $3.76$, $4.76$, and $5.76$. The ratio controls the ideal prediction, while the absolute quantities control how much added chemical acid or chemical base the buffer solution can accommodate.

The common-ion effect is the change in a chemical equilibrium caused by adding an ion already participating in it. In the fixed-activity coefficient AB model, adding the common B anion lowers the molar solubility. The shared ion participates regardless of which salt supplied it.^[ref]

The preceding solubility product and material balance give a short proof without a qualitative slogan. Taking a derivative of $s_{\ast }^2+c_0{s}_{\ast }=K_{\mathrm{sp},c}$ at fixed $K_{\mathrm{sp},c}$ givesThus the molar solubility decreases continuously as $c_0$ increases. At $c_0=0$, the derivative is $-1/2$; for a very large common-ion background, the rationalized expression approaches $K_{\mathrm{sp},c}/c_0$. The solubility product is fixed within this model even while the molar solubility changes.

Neglecting $s_{\ast }$ beside $c_0$ replaces $s_{\ast }(c_0+s_{\ast })=K_{\mathrm{sp},c}$ by $c_0{s}_{\mathrm{app}}=K_{\mathrm{sp},c}$. This approximation always overestimates the exact molar solubility, since it neglects the B anions supplied by AB itself. For $c_0>0$, its relative error has an exact simple form:At $c_0=0.030\mathrm{M}$, $s_{\mathrm{app}}=0.003333333\mathrm{M}$, approximately $10.1\%$ too high. At $c_0=0.100\mathrm{M}$, $s_{\ast }=0.000990195\mathrm{M}$ and $s_{\mathrm{app}}=0.001000000\mathrm{M}$, approximately $0.990\%$ too high. A small solubility product by itself does not justify the approximation: the common-ion background must be large compared with the resulting molar solubility.

For a desired fractional relative error at most $\epsilon >0$, substituting $s_{\ast }/c_0\le \epsilon$ into the exact balance gives the equivalent criterionFor this model, a $5\%$ tolerance requires $c_0\ge 0.0436436\mathrm{M}$. This bound concerns only the algebraic approximation. In a real chemical solution, changes in ionic strength and activity coefficients, or competing chemical reactions, can invalidate a constant $K_{\mathrm{sp},c}$ even when this bound is met.

In an ideal aqueous chemical solution, set $h=[{\mathrm{H}}_3{\mathrm{O}}^{+}]/c^{\circ }=10^{-\mathrm{pH}}$, with $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$, and approximate activity coefficients by one. Put $K_1=10^{-2.35}$ and $K_2=10^{-9.78}$ for the stipulated dimensionless acid dissociation constants. The Henderson-Hasselbalch equation gives the adjacent-state ratiosWith analytical concentration $C_T=[G^{+}]+[G^0]+[G^{-}]$, normalization gives the three fractions and mean electric charge in units of the elementary charge:All three fractions are nonnegative and sum to one. At pH $7$, the model gives about $99.832\%$ zwitterion, $0.00224\%$ cation and $0.1657\%$ anion; $\bar{z}\approx -0.00163$. Predominantly zwitterionic therefore need not mean exactly at the isoelectric point.

Setting $\bar{z}=0$ requires $h^2=K_1{K}_2$. Taking the logarithm yieldsAt this model isoelectric point, ${\alpha }_{+}={\alpha }_{-}\approx 0.0001927$ and ${\alpha }_0\approx 0.9996146$: about $0.0193\%$ of each charged form remains. At either pKa, the relevant adjacent fractions are equal, but the third fraction is not mathematically zero. This three-state model neglects the minor uncharged form ${\mathrm{H}}_2\mathrm{N}-\mathrm{C}{\mathrm{H}}_2-\mathrm{COOH}$; macroscopic pKa values alone do not determine its fraction separately from the zwitterion.

The Henderson-Hasselbalch equation estimates buffer solution pH using a weak acid's acid dissociation constant and the conjugate base to weak acid concentration ratio.

Its activity form is an exact rearrangement of the chosen chemical equilibrium relation. Write the acid dissociation step as $\mathrm{HA}\rightleftharpoons {\mathrm{H}}^{+}+{\mathrm{A}}^{-}$, with hydrated hydrogen ions understood in water and the solvent convention incorporated into the acid dissociation constant. For consistent standard states,Here each $a_i$ is a dimensionless thermodynamic activity at chemical equilibrium. Exactness refers to this specified acid dissociation constant relation and the chosen single-ion thermodynamic activity convention, not to identifying the ratio with quantities initially mixed. If water is instead written explicitly as a reactant, its thermodynamic activity must be treated consistently. See Hippler and Metcalfe, Using activities to correct the Henderson-Hasselbalch equation (2020), equations 3–4.

For a toy weak acid with stipulated pKa $5$, use the ideal dilute concentration approximation and put $r=[{\mathrm{A}}^{-}]/[\mathrm{HA}]$. The Henderson-Hasselbalch equation becomes $\mathrm{pH}\approx 5+{\log }_{10}r$:More conjugate base relative to weak acid means higher pH. Each tenfold increase in the ratio contributes exactly one to the calculated pH. The stipulated pKa is not a tabulated value for an identified substance.

The distinction between thermodynamic activity and concentration is explicit when $a_i={\gamma }_i[i]/c^{\circ }$, where ${\gamma }_i$ is an activity coefficient and $c^{\circ }$ is the reference concentration:Neglecting the final term gives the usual approximate Henderson-Hasselbalch equation. It is justified when the ratio of activity coefficients is sufficiently close to one, as in a suitable dilute ideal approximation. Ionic strength, including contributions from other dissolved ions, affects activity coefficients; a nominally dilute weak acid alone does not establish that all corrections are small. The ratio must use chemical equilibrium concentrations. Replacing these by prepared amounts or post-neutralization inventories is a further approximation, suitable only when subsequent redistribution is small. See Hippler and Metcalfe, concentration, activity, and analytical-concentration approximations.

The short derivation is to solve the acid dissociation constant relation for $a_{{\mathrm{H}}^{+}}$, take a negative base-ten logarithm, and use the definitions of pH and pKa. Its useful beauty is that a chemical equilibrium becomes a simple rule about a ratio: select a weak acid with pKa near the desired pH, then adjust the partner ratio. Scaling both partner concentrations equally leaves the ideal pH prediction unchanged, while changing buffer capacity. In practical buffer solutions, temperature, ionic strength, and the available inventories still matter.

An ideal solution is a chemical solution whose component chemical potentials depend on composition throughHere $n_i$ is an amount of substance, $x_i$ its mole fraction, $R$ the molar gas constant, and ${\mu }_i^{\ast }$ the pure-component liquid reference at the same temperature $T$ and pressure $P$. This model applies to every component across the composition range; it is stronger than an approximation valid only near a pure solvent.^[ref]

With this pure-liquid standard state convention, the thermodynamic activity satisfies ${\mu }_i={\mu }_i^{\ast }+RT\ln a_i$, so an ideal solution has $a_i=x_i$, or activity coefficient ${\gamma }_i=a_i/x_i=1$.^[ref] This statement concerns the liquid. Assuming that a coexisting vapor is an ideal gas mixture is an additional approximation. Its mole fractions $y_i$ generally differ from the liquid mole fractions $x_i$.

For an original two-component example, stipulate $T=300\mathrm{K}$ and pure-component vapor pressures $p_A^{\ast }=80\mathrm{kPa}$ and $p_B^{\ast }=20\mathrm{kPa}$. A and B are invented distinguishable components; these numbers are chosen model inputs, not material data. Combine an ideal solution with an ideal gas vapor, neglect the pure-liquid reference's pressure correction, and exclude other gases. At liquid mole fractions $x_A=0.4$ and $x_B=0.6$, Raoult's law and addition of partial pressures giveThus the coexisting vapor contains about $72.73\%$ A by amount of substance, although the liquid contains only $40\%$. The total pressure is neither pure-component vapor pressure, and $x_A$ cannot be substituted for $y_A$. This enrichment is the basis of distillation.

The usefulness of the ideal solution is that one composition law connects chemical potentials, vapor-liquid equilibrium, entropy of mixing and Gibbs free energy of mixing. Its liquid components can interact strongly: the defining condition is their composition dependence, rather than the absence of all interactions as in an ideal gas model.

The Mixture measurements and the Gibbs-Duhem constraint article describes the inspected acetone–chloroform preparation, apparatus, paired readouts, controls and sampling limits.^[ref]

That existing apparatus account supplies the experimental controls and limitations. Its refractive index calibration yields mass fractions, which must be converted to mole fractions. For a measured component-A mass fraction $w_A$ and component molar masses $M_A,M_B$,Apply this conversion separately to the flask sample and to completely condensed vapor to obtain $x_A$ and $y_A$. Comparing $y_{i}P$ with $x_i{p}_i^{\ast }(T)$ tests Raoult's law at the measured temperature; using the changing temperature at each composition is essential. The chosen A/B values above are unrelated to the protocol's substances.

The more general diagnostic, retaining an ideal gas vapor and negligible pure-liquid pressure correction, isThis follows by replacing $\ln x_i$ with $\ln (x_i{\gamma }_i)$ in the chemical potential subtraction used for Raoult's law. A reproducible discrepancy from ${\gamma }_i=1$ can expose failure of the ideal solution approximation, but interpreting it requires checking measurement uncertainty, representative paired sampling and the ideal gas approximation. Apparent agreement within uncertainty supports an approximation over the tested conditions; it cannot establish exact ideality at every composition.

Positive or negative deviations of component partial pressures require activity coefficients different from one under these assumptions. An azeotrope is also absent from the chosen unequal-vapor pressure model: the explicit inequality in ideal binary vapor composition gives $y_A>x_A$ throughout $0<x_A<1$. An interior equality would require $p_A^{\ast }=p_B^{\ast }$; in that degenerate case every composition has $y_A=x_A$, rather than an isolated enrichment reversal. Neither an azeotrope nor liquid phase separation should be inferred from the ideal curves.

Consider a neutral monoprotic acid $\mathrm{HA}$ dissolved in water, with no added salts or other acids. Assume dilute ideal thermodynamic activities, fixed temperature, water thermodynamic activity one, and negligible self-ionization of water. Write $C_T$ for its analytical concentration, $c=C_T/c^{\circ }$ with $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$, and $\alpha$ for its weak acid dissociation fraction. Conservation of acid units and electric charge givesHere ${\mathrm{A}}^{-}$ is the conjugate base, and ${\mathrm{H}}^{+}$ is conventional shorthand for hydrated hydrogen ions, represented by hydronium in this model. Substitution into the dimensionless acid dissociation constant givesSolving this quadratic equation and rejecting its negative root of a polynomial gives an exact result within the stated model:The second form avoids subtracting nearly equal numbers. Unlike the approximation $\alpha \approx \sqrt{K_a/c}$, it retains depletion of neutral weak acid and always satisfies $0<\alpha <1$ for positive $K_a,c$. The broader weak acid mass and charge balances restore self-ionization of water when that omitted contribution matters.

For an original worked example, stipulate $K_a=10^{-5}$ without identifying a real substance. At $C_T=10^{-2}\mathrm{mol}{\mathrm{L}}^{-1}$, the weak acid dissociation fraction is approximately $0.0311267$, or $3.11267\%$, and $[{\mathrm{H}}_3{\mathrm{O}}^{+}]=3.11267\times 10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$. A hundredfold dilution to $C_T=10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$ raises the fraction to $0.270156$, or $27.0156\%$, but lowers $[{\mathrm{H}}_3{\mathrm{O}}^{+}]$ to $2.70156\times 10^{-5}\mathrm{mol}{\mathrm{L}}^{-1}$. A completely ionized strong acid at $C_T=10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$ would have $[{\mathrm{H}}_3{\mathrm{O}}^{+}]\approx 10^{-4}\mathrm{mol}{\mathrm{L}}^{-1}$: less than the more concentrated weak acid in the first calculation, despite its greater converted fraction.

The directions of both changes follow without relying on the plot. Put $x=[{\mathrm{H}}_3{\mathrm{O}}^{+}]/c^{\circ }=c\alpha$. Rearranging the chemical equilibrium relation givesThus smaller $c$ forces smaller $x$, which in turn makes $\alpha$ larger. The derivative proves that total hydronium concentration and converted fraction move in opposite directions during dilution. In this model, adding solvent also makes additional acid units ionize at fixed total acid amount of substance, even though their ions are spread over more volume. That distinction explains why “more ionized” alone does not predict greater electrical conductivity.

Do not extend the no-background model to unlimited dilution: its prediction $[{\mathrm{H}}_3{\mathrm{O}}^{+}]\to 0$ omits water's own ions. The weak acid mass and charge balances include that background. Added salts, other acids or bases, and nonideal activity coefficients also change the relation between analytical concentration, converted fraction and hydronium concentration.

The concentration-based ionic strength is $I_c=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$, where $c_i$ is each dissolved ion's concentration and $z_i$ its signed electric charge number. It has the same units as $c_i$; some formulas instead use the dimensionless ratio $I_c/c^{\circ }$. All dissolved ions contribute, including those that balance the electric charge of a buffer solution's conjugate base. Ionic strength helps describe the interactions represented by activity coefficients. See Hippler and Metcalfe, activity coefficients and dimensionless ionic strength.

The ion product evaluates the same dissolved-ion thermodynamic activity product as the solubility product, but at the current composition rather than necessarily at chemical equilibrium. For AB, $Q_{\mathrm{sp}}=a_{\mathrm{A}}{a}_{\mathrm{B}}$; its concentration counterpart is $Q_c=c_{\mathrm{A}}{c}_{\mathrm{B}}$. Compare $Q_c$ with $K_{\mathrm{sp},c}$ only under compatible activity coefficient assumptions.^[ref]

For other stoichiometries, use the dissolution coefficients as powers: ${\mathrm{AB}}_2(s)\rightleftharpoons {\mathrm{A}}^{2+}(aq)+2{\mathrm{B}}^{-}(aq)$ gives $Q_{\mathrm{sp}}=a_{\mathrm{A}}{a}_{\mathrm{B}}^2$. A solubility product is not itself the molar concentration dissolved; even in the ideal AB model in pure water, that concentration is $\sqrt{K_{\mathrm{sp},c}}$.

A concrete motivating experiment is the acetone–chloroform mixture in Truman State University's Liquid-Vapor Equilibrium of a Binary System, Procedure and Figure 1. A two-neck flask feeds a distillation head with a water-cooled laboratory condenser, condensate sampling arm and temperature probe. Starting with one pure liquid, the procedure adds the other component successively and waits for stable temperature with returning condensate. Small paired samples from the flask and condensate are read on an Abbé refractometer; a tabulated refractive index–composition relation gives the two compositions. Controls include three-point thermometer calibration, corrected atmospheric pressure readings, and agreement of the initial pure-liquid and condensate refractive indices with the reference value. Small withdrawals and little retained condensate reduce disturbance of composition. Sample evaporation can bias the readout. The inferred paired mole fractions, temperature and pressure describe vapor-liquid equilibrium, subject to the conversion from mass fraction to mole fraction and to representative sampling. This teaching protocol reports no measured results of a particular student run.^[ref]

How does such a measurement expose activity coefficients? Let $x_i$ and $y_i$ be the mole fractions in the liquid and gas thermodynamic phases, respectively, and let $p_i^{\ast }(T)$ be the pure-component vapor pressure. Assume the gas thermodynamic phase is an ideal gas mixture and neglect the change of the pure-liquid reference chemical potential between its saturation pressure and the experimental pressure. Equality of each component's chemical potential between the two thermodynamic phases then givesThis turns the paired composition readings into estimates of the activity coefficients. The approximation needs refinement when the gas departs appreciably from ideal gas behavior or the neglected liquid pressure effect matters.

The apparatus measures a boiling-composition series in which temperature changes as the mixture changes. Its readings therefore cannot be inserted directly into the fixed-temperature, fixed-pressure activity coefficient constraint. They motivate the more general task: test whether the measured mixture properties can arise from a common Gibbs free energy while accounting for changing state variables. A failure can reveal incompatible property fits or experimental bias; a pass supplies a consistency check, not proof that every reading is correct.

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.

An amphiphile has a solvent-affine part and a solvent-averse part within one molecule. In water these are commonly called hydrophilic head and hydrophobic tail. Two nonbiological examples are sodium dodecyl sulfate, with a twelve-carbon tail and an anionic sulfate head, and cetyltrimethylammonium bromide, with a sixteen-carbon tail and a cationic quaternary-ammonium head. Their counterions balance the ionic head-group charge; the head and tail do not have interchangeable interactions with water.

Aggregation reduces unfavorable exposure of hydrocarbon tails to water but loses molecular translational entropy. Interfacial free energy, head-group solvation and electrostatic repulsion, chain packing and conformational entropy determine the preferred aggregate. The hydrophobic effect is a solvent-mediated free-energy balance, rather than a literal attractive bond between tails.

The amphiphile packing parametercompares tail volume $v$, effective head-group area $a_0$ and maximum tail length ${\ell }_c$. A spherical core of radius $R$ obeys $v/a_0=R/3$, and a long cylindrical core obeys $v/a_0=R/2$. The condition $R\le {\ell }_c$ gives the conventional spherical window $P\lesssim 1/3$ and cylindrical window $1/3\lesssim P\lesssim 1/2$. Less strongly curved bilayers become compatible for $1/2\lesssim P\lesssim 1$, with planar packing near one; values above one can favor inverse curvature in an appropriate nonpolar surrounding medium. These windows are geometric guides rather than exact phase boundaries.

A larger hydrated or electrostatically repelling head increases $a_0$ and favors positive curvature. Screening of charged heads or a smaller head can lower $a_0$ and favor rods or sheets. Multiple tails increase the tail volume relative to head area and can favor bilayers. Tail length, branching, temperature and solvent selectivity alter both geometry and interaction free energy, so one cannot predict a unique aggregate from hydrophobicity alone.^[ref]

In the supplied chemical-potential expression, $\mu$ is the equilibrium chemical potential per amphiphile, ${\mu }_{\alpha }^0$ the standard free energy per amphiphile within an aggregate of size $\alpha$, and $x_{\alpha }$ the dimensionless molecular fraction belonging to that size class. The aggregate molecular fraction versus aggregate number fraction distinction explains the logarithm: the aggregate count fraction is $x_{\alpha }/\alpha$, and dividing its translational chemical potential by $\alpha$ gives a per-molecule value. ThusHere $k_B$ is the Boltzmann constant, $T$ absolute temperature, and $\alpha$ a positive integer. The logarithm acts on a dimensionless ideal activity; a concentrated nonideal solution would require activity coefficients.

equilibrium exchange of amphiphiles gives the same per-molecule $\mu$ for every size. For monomers, $\mu ={\mu }_1^0+k_{B}T\ln x_1$. Substitution yields the finite-micelle aggregation-number distributionThis formula by itself does not prescribe a preferred micelle size: the size dependence of ${\mu }_{\alpha }^0$ must include the competing packing and interaction terms. If their balance favors a finite size ${\alpha }_{\ast }$, a monomer population and a peak around that size can coexist. Dividing by $\alpha$ would instead plot the distribution of aggregate counts.

At very low $x_{\mathrm{tot}}$, the factors $x_1^{\alpha }$ suppress large aggregates and most molecules remain monomers. As $x_1$ rises, a favorable aggregate class grows extremely rapidly because of this high power. The critical micelle concentration from standard chemical potentials has the large-size estimateMore precisely, the illustrative crossover condition $x_{{\alpha }_{\ast }}=x_1$ givesAdded material above the crossover enters aggregates predominantly. Monomer concentration then increases slowly rather than being exactly pinned: its logarithm enters $\mu$, while aggregate populations respond with a large factor ${\alpha }_{\ast }$. For finite micelles this is a sharp association crossover, not an absolute prohibition on aggregates below one exact concentration.

The drawing uses a declared illustrative standard-free-energy model favoring aggregates near $\alpha =40$. It is not an inferred distribution or critical concentration for either named surfactant. The separate monomer peak is at $\alpha =1$; a distribution of molecular fractions weights each aggregate by its number of molecules.

For the general physics of a nearly planar fluid sheet, let $h(x,y)$ be a small-slope displacement, $\kappa$ membrane bending rigidity and $\sigma$ mechanical tension. At fixed topology and symmetric reference curvature, the quadratic Helfrich energy isThe bending term is the small-slope curvature-elastic energy; additional spontaneous-curvature or boundary terms require extra information.^[ref]

For Fourier amplitudes $h_{\mathbf{q}}=A^{-1}{\int }_{A}h(\mathbf{x})e^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$, equipartition gives the membrane fluctuation spectrum under tension,Higher temperature increases fluctuations, while greater membrane bending rigidity or tension suppresses them. Short-wavelength modes are predominantly bending-controlled; longer wavelengths can be tension-controlled. Finite sheet size, supports, confinement and imposed area set boundary conditions and infrared cutoffs; continuum elasticity also needs a short-distance cutoff. solvent viscosity affects relaxation rates, although it does not enter this equilibrium amplitude at fixed elastic parameters.

TODO: user-deferred biology. The requested importance of these fluctuations for bodily function is deferred. The preceding result covers only the independent continuum and thermal physics.

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.

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.

The compositions alone do not specify a unique thermodynamic activity model. Adopt the usual ideal, independent mixing on crystallographic sites approximation: three Mg–Fe positions per formula unit, with the other sites fixed at their stated endmember compositions. Then the mineral endmember activities areThe exponent counts the three mixing cations; treating each whole mineral formula as a single mixing site would be a different model. In particular, the magnesian biotite and garnet components here are phlogopite and pyrope.

For the written reaction, the reaction quotient iswith the other solid activities equal to one. The Gibbs energy of reaction balance therefore givesThe smaller product-endmember activity lowers the reaction chemical potential and lowers the equilibrium temperature in this model. Nonideal activity coefficients or unequal occupancies of crystallographic sites would require further information rather than a revised universal numerical answer.^[ref]