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Osmolality counts dissolved particles per mass of solvent; it characterizes a chemical solution, rather than the total amount of water in an organism.

In the dilute ideal particle-count description, osmolarity is the total amount of substance of independently dissolved solute particles per volume of chemical solution: $c_{\mathrm{osm}}={\sum }_j{n}_j/V$. It includes both membrane-permeating and retained solutes; specifying a particular membrane is unnecessary to count particles. A neutral undissociated solute contributes one particle per molecule, whereas separated ions are counted individually. Common units are $\mathrm{osmol}{\mathrm{L}}^{-1}$ or $\mathrm{mOsm}{\mathrm{L}}^{-1}$, with one osmole denoting one mole of the counted particles. Osmolality uses mass of solvent instead of volume of chemical solution; the two numerical values should not be identified without the appropriate mass-to-volume conversion.^[ref]

This ideal counting definition states its scope: interactions can require a thermodynamic activity description of osmotic pressure. Tonicity asks a different question, namely how that surrounding chemical solution changes a specified biological cell's volume through its particular cell membrane.

Osmotic pressure is the excess pressure needed on a chemical solution to maintain osmotic thermodynamic equilibrium with pure solvent across a semipermeable membrane. For a nearly incompressible solvent, equality of chemical potential gives $\Pi \simeq -(RT/{\bar{V}}_w)\ln a_w$. It is an equilibrium property; an observed initial swelling rate is a different measurement.^[ref]

For a dilute ideal chemical solution of retained particles,Here $n$ is the amount of substance of independently dissolved solute particles, measured in moles; $V$ is the final volume of chemical solution, not the starting volume of pure solvent. Count the resulting ions separately if a solute dissociates; do not multiply twice if $n$ already counts them. With $c_{\mathrm{osm}}$ in $\mathrm{mol}{\mathrm{m}}^{-3}$ and $R$ in $\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, $RT{c}_{\mathrm{osm}}$ has units $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{Pa}$. This resembles the ideal gas law, although the particles here are dissolved in a liquid.

The assumptions are uniform temperature, a dilute ideal chemical solution, a nearly incompressible solvent, and a semipermeable membrane that retains the counted solutes while passing solvent. Association, incomplete dissociation and interactions change the relation between added material and independent particles; nonideal thermodynamic activities invalidate simple particle counting. The formula does not describe a concentrated brine merely because the apparatus contains a membrane.

A short derivation explains why particles are counted. In an ideal mixture, $a_w=x_w=n_w/(n_w+n)$, with $n_w$ the amount of substance of water. If $n\ll n_w$, then $-\ln x_w=\ln (1+n/n_w)\simeq n/n_w$ and $V\simeq n_w{\bar{V}}_w$. Substitution gives $\Pi \simeq RTn/V$. This is a dilute-limit approximation, not a claim that water molecules stop moving or that dissolved particles are a gas.

If the ideal-chain segment count and segment length remain fixed, the entropic spring constant is proportional to absolute temperature. ThereforeThis is an increase by about $11.7\%$. The factor $k_{B}T$ multiplies the free energy penalty for the loss of chain configurations; a larger force is required to impose the same normalized weak extension at higher temperature.

Not degrading does not guarantee that a real protein's folded state, effective spring constant or solvent interactions remain unchanged. Thermal unfolding and changes in persistence length can occur without covalent degradation. Those changes modify the experimental curve and may invalidate a fixed-$N,b$ comparison. The boxed value is the explicit ideal-FJC prediction under fixed segment geometry, not an unconditional thermal-stability prediction for the protein.

On releasing the tensile load, the extended chain first retracts towards less aligned conformations because they have much greater configurational multiplicity. Viscous friction and internal relaxation determine the time course; stored free energy is dissipated as heat. An unconstrained ideal chain does not shrink to a point: its mean vector is zero but its root mean square end-to-end length is $b\sqrt{N}\simeq 2.37\mathrm{nm}$ for the supplied ideal-chain parameters.

A real polypeptide also has sequence-dependent interactions. If the original native structure is the stable state in the prevailing solvent and the molecule remains intact and capable of folding, it can subsequently refold, gaining favourable intramolecular and solvent free energy. The folded state is not selected by chain entropy alone, and formation of native contacts need not be instantaneous. Kinetic traps, proline isomerization, protein aggregation, misfolding, surface attachments or chemical damage may prevent complete recovery or produce hysteresis. Thus reversible refolding is a physically motivated possibility, not guaranteed simply by removal of force. This distinction connects the recoil of a stretched polymer to the thermodynamics discussed in Question 6(c).

A stable native conformation is favoured when its Gibbs free energy, including the protein and its surrounding solvent, is lower than that of competing conformational ensembles under the specified conditions:Folding greatly reduces the chain's conformational entropy, an unfavourable contribution. It can nevertheless gain favourable hydrogen-bonding and packing interactions, van der Waals contacts, screened electrostatic interactions and solvent free energy. In aqueous globular proteins, burying nonpolar surface reduces the cost of exposing it to water; the hydrophobic effect is not an extra covalent attraction between all hydrophobic residues. Hydrogen bonds and ionic contacts must be judged relative to the interactions those groups made with solvent before folding, so merely counting bonds in the folded structure does not determine stability.

The protein primary structure determines which contacts and protein secondary structures can cooperate. A free-energy landscape biased towards compatible contacts permits many routes towards the native ensemble, rather than an exhaustive random search through every conformation. Folding funnels describe this bias, while barriers explain metastability and folding rates. Reversible denaturation/refolding experiments show that some proteins can recover native structure and activity when suitable conditions are restored, supporting a sequence-encoded thermodynamic tendency without asserting that every protein folds unaided.

“Stable” means an equilibrium preference and a useful lifetime, not one perfectly rigid shape. Native proteins fluctuate and can have several functional conformations; the unfolded ensemble remains populated with relative weight depending on its full free energy. Changes of temperature, pH, salt, solvent, ligands or pressure alter the balance, and molecular chaperones can prevent protein aggregation or assist kinetic access without supplying a different primary sequence. Membrane proteins require their appropriate environment, and intrinsically disordered proteins need not possess one stable folded state in isolation.

Thus a released stretched chain can gain configurational entropy on first recoiling yet later favour a more ordered native structure because the total free energy balance, including solvent, is different. Equilibrium stability, barriers to unfolding and resistance to irreversible protein aggregation are distinct properties. The existence of a native minimum alone does not guarantee rapid, reversible folding under every experimental condition.

The strongest donor belongs to the acid, ${\alpha }_1=3.5$, and the strongest acceptor to the aminopyridine, ${\beta }_2=8.5$. The cocrystal pairs the strongest donor with the strongest acceptor, while the reciprocal amino-NH···carbonyl contact closes a complementary two-contact motif.

Using the simple favorable-contact score $\alpha \beta$, one heterodimer givesThe self-associated comparison, normalized to the same two molecules and two hydrogen bonds, is half an acid dimer plus half an aminopyridine dimer:ThusIn a model where bond free energy contains $-\alpha \beta$, this is a favorable energy change. No bonds have been artificially added to one side of the comparison: both alternatives contain two contacts per acid/aminopyridine pair.

The calculation rationalizes the hydrogen-bond preference. It is not a complete crystal free energy: packing, additional contacts, entropy and possible solvent effects also matter. Without a specified calibration, the numerical score should not be promoted to a measured cocrystallization free energy.

In water, separated benzene molecules require separate hydration environments around nonpolar surface. Bringing them together buries part of that surface and reorganizes the surrounding water. The resulting solvent contribution favours association relative to cyclohexane, where benzene was already well solvated by dispersion. This solvent-mediated contribution is the hydrophobic effect, not formation of a new covalent bond or a claim that benzene has become a conventional hydrogen-bond acceptor like oxygen.

The supplied constants imply a thirtyfold enhancement. With matched standard states,at an illustrative 298 K. The water value is still slightly below $1{\mathrm{M}}^{-1}$, so enhancement relative to cyclohexane does not mean a negative standard association free energy for a 1 M standard state, nor does it imply that dilute benzene is all dimer. No universal “entropy only” explanation is needed: solvent enthalpy and entropy both contribute and depend on temperature.

Association in solution includes replacement of solute–solvent contacts, not just formation of the isolated dimer. cyclohexane solvates the benzene surface well through dispersion. Much of the London dispersion gained on bringing two benzene molecules together replaces comparable benzene–cyclohexane contacts, leaving only a modest net gain before the translational and orientational entropy penalty. cyclohexane cannot provide the complementary donor/acceptor interactions available when two waters hydrogen-bond, so water association gains a more distinctive interaction relative to its separated solvated monomers.

Using the same concentration standard state, $K^{\circ }=K_c{c}^{\circ }$ and $\Delta G_{\mathrm{assoc}}^{\circ }=-RT\ln K^{\circ }$. For illustration at 298 K and $c^{\circ }=1\mathrm{mol}{\mathrm{L}}^{-1}$, the ratio of the two supplied association constants corresponds toThus the water-molecule pair can associate more favourably in this solvent even though the stipulated isolated-pair binding energies are similar. Gas-phase potential energy alone omits both differential solvation and the entropy of association.

water achieves substantial binding through a localized O–H···O hydrogen bond; benzene achieves a comparable total through a larger set of weaker contacts. In the water dimer, one water is the hydrogen-bond donor and the other presents an oxygen lone pair region as acceptor. The interaction includes electrostatics, induction interaction, dispersion and short-range exchange repulsion; a hydrogen bond is not a separate fundamental force that can simply be added to all those terms again.

For benzene, draw a T-shaped edge-to-face pair or a slipped parallel pair. Its larger, more polarizable electron clouds provide appreciable London dispersion. Its quadrupolar charge distribution also makes the energy orientation dependent, despite each molecule having no permanent dipole by symmetry. An edge C–H approaching the other ring's pi region is therefore preferable to a model of two structureless, noninteracting dipole-free objects. Dispersion attracts while Pauli repulsion limits close overlap. Several distributed contacts can sum to the scale of one water–water hydrogen bond; counting only the number of conventional hydrogen bonds would miss them.

The comparison concerns the given gas-phase pair interaction energies, not an equality of every solvent-dependent association free energy. Correlated primary calculations explicitly identify both T-shaped and parallel-displaced benzene-dimer minima, rather than a uniquely preferred perfectly eclipsed sandwich.^[ref]

water stabilizes the planar amide more strongly relative to the twisted rotational transition region, increasing the free-energy barrier. nitrogen lone-pair donation into the carbonyl gives the C–N bond partial double-bond character through amide resonance, represented byamide bond rotation between the two amide conformations requires loss of favourable overlap. Near a strongly twisted geometry the nitrogen lone pair is less conjugated with C=O, and the charge-separated resonance in chemistry contribution is reduced. Polar water, a hydrogen bond donor, preferentially stabilizes the planar starting structures, including their polar carbonyl/resonance distribution, compared with this less conjugated rotational region. chloroform provides less of that differential stabilization. What controls the rate is the difference between the starting minimum and the transition state, not the absolute stabilization of either one in isolation.

For a first-order interpretation of the half-lives, the characteristic rate constants areThus the interconversion is 25 times slower in water. With comparable Eyring equation prefactors at the same temperature,which is about $8.0\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ at $298\mathrm{K}$. This temperature is illustrative because none is specified for the measurements.

The two endpoint conformations remain minima separated by a twisted barrier in each solvent. The profiles are schematic, with arbitrary absolute barriers; the half-life ratio supports the relative kinetic comparison. If the quoted half-lives describe relaxation of a reversible cis/trans mixture, $k_{\mathrm{obs}}=k_f+k_r$, so the ratio directly concerns relaxation rates. Assigning the full $RT\ln 25$ difference to one microscopic forward barrier additionally requires comparable equilibrium populations or directional-rate information.

B is oxidative addition; D is alkyne insertion into Ni–aryl followed by carbonyl-oxygen coordination; E is carbonyl insertion into Ni–alkenyl. The distinction is visible from the bonds made and from consistent electron counting.

For B the depicted Ni(0) starting complex has two phosphine and two MeCN donors, giving $10+4+4=18$ electrons. Addition across the aryl C–Br bond produces two X-type ligands and Ni(II). The drawn product has lost the two solvent donors, giving $8+4+2+2=16$ electrons. Intrinsic oxidative addition to the unsolvated 14-electron $\mathrm{N}{\mathrm{i}}^0(\mathrm{dppe})$ fragment raises its count to 16; the overall drawn $18\to 16$ change also includes loss of four solvent-donor electrons. A falling displayed count does not turn this oxidative addition into a reductive elimination.

Before D, the aryl–alkyne intermediate is Ni(I), $d^9$, with $9+4_{\mathrm{P},\mathrm{P}}+2_{\mathrm{aryl}}+2_{\mathrm{alkyne}}=17$ electrons. Syn migration couples aryl carbon to one alkyne carbon and leaves a Ni–alkenyl bond. Insertion alone removes the separate two-electron alkyne donor and would give a 15-electron alkenyl fragment. The ester carbonyl oxygen then binds, as drawn, restoring the count to $9+4+2_{\mathrm{alkenyl}}+2_{\mathrm{O}}=17$. Ni remains $+1$.

In E the Ni-bound alkenyl carbon attacks the coordinated ester carbonyl carbon, forming the new ring-closing C–C bond. The C=O pi electrons move to oxygen, and an Ni–O alkoxide bond replaces Ni–C. This is carbonyl migratory insertion, equivalently intramolecular addition to a coordinated carbonyl. A neutral O donor plus an alkenyl X ligand become one alkoxide X ligand, so the drawn count falls $17\to 15$, while Ni remains $+1$. A solvating molecule could fill the resulting vacancy, but it is not shown and should not be silently added to the stated count.

zinc reduction is required in A, C and F. Reading the drawn ligands with consistent ionic bookkeeping givesIn A the two chlorides of the precursor are removed and neutral solvent donors occupy the reduced centre. In C the bromide is removed while the aryl ligand remains and the neutral alkyne coordinates; the drawn neutral intermediate has Ni(I). In F an anionic alkoxide-derived product is released and the neutral solvated Ni(0) complex is regenerated. zinc supplies electrons by oxidation to Zn(II); halide and alkoxide ions can be taken up in zinc-containing products.

A is precatalyst activation. The two one-electron reductions C and F together require two electrons per catalytic turnover, corresponding to one Zn atom if its oxidation ends at $+2$. The diagram does not establish whether each formal reduction is a single elementary electron-transfer event or a sequence of surface reactions and ligand transfers.

Apply the Felkin-Anh model to the stereocentre alpha to the ketone. Its large group is the chain continuing through the phenyl-bearing carbon, its medium group is methyl, and its small group is hydrogen. In the preferred transition-state arrangement, put the large group approximately perpendicular to C=O and approach from the opposite face, passing the small hydrogen along a Bürgi-Dunitz trajectory. The remote ether cannot form the usual small alpha-alkoxy chelate, and the coordinating solvent further supports using an open-addition prediction.^[ref]

The predicted new stereocentre is $S$. In the question's zigzag projection, this corresponds to the new OH bond directed behind the page and the newly introduced methyl toward the reader, with the original stereocentres unchanged. The new-centre priority order is OH $>$ phenyl $>$ the substituted alpha-carbon chain $>$ methyl. This is a model-based major-product prediction, rather than a quantitative diastereomeric ratio.

The functional group interaction profile compares donor strength $\alpha$ and acceptor strength $\beta$. In water, strong–strong contacts compensate for desolvation, giving the upper-right black region. Strong–weak contacts lose favourable solvent interactions without adequate replacement, giving the two grey regions. Weak–weak association gives the lower-left black region through the hydrophobic effect.

Ethanol's ethyl groups preferentially solvate weakly polar sites, removing this incentive for their association: the lower-left region becomes approximately neutral. Its hydroxyl group retains competition for hydrogen bonds elsewhere in the profile.^[ref]

Carbon-13 NMR spectra: 1 = C, 2 = A, 3 = B. Approximate shifts, read from the printed axes, are shown in the annotated structures.

The two methyls in A are chemically distinct because one is cis and the other trans to each substituent at the other alkene carbon. Their two measured shifts are secure; assigning an individual methyl to a particular side of the drawn double bond requires an additional stereospecific shift argument or experiment, so their spatial interchange is left explicit.

The alkene pi bond attacks Br${}_2$ as the Br-Br bond breaks to bromide, forming a bridged, unsymmetrical bromonium ion. The carbon bearing the two methyl groups has greater carbocation-like character.

The tethered alcohol oxygen attacks that carbon from the side opposite the bromine bridge. Its lone pair forms the new O-C bond while the attacked C-Br bridge bond opens, giving a six-membered cyclic oxonium ion. Bromide or solvent removes the O-H proton to give the neutral bromoether.

This explains why O becomes attached at the more substituted alkene carbon while Br remains at the other carbon. Polar solvent assists charge separation, and intramolecular capture competes effectively with external bromide attack. Opening is anti; attack on the two faces of the achiral starting material gives a racemic product at the newly formed CHBr stereocentre.

The ring-forming step is bromoetherification.

A is 3-Chlorobutan-2-one, $\mathrm{C}{\mathrm{H}}_3\mathrm{C}(=\mathrm{O})\mathrm{CH}(\mathrm{Cl})\mathrm{C}{\mathrm{H}}_3$. Counting chlorine as one hydrogen in the index of index of hydrogen deficiency gives $(2\times 4+2-7-1)/2=1$. A single carbonyl accounts for that unsaturation; the resonance at $203.2$ ppm identifies a ketone, while the $59.0$ ppm methine bears chlorine.

The carbon-13 NMR assignments are ${\delta }_C=203.2$ for $\mathrm{C}=\mathrm{O}$, $25.7$ for $\mathrm{C}{\mathrm{H}}_3\mathrm{CO}$, $59.0$ for $\mathrm{CHCl}$ and $20.1$ ppm for $\mathrm{CH}(\mathrm{Cl})\mathrm{C}{\mathrm{H}}_3$. The four distinct carbon environments agree with the unsymmetrical structure. The small signal near 77 ppm belongs to the usual deuterated chloroform solvent, rather than to a fifth carbon of A.

The six carbon environments are assigned as follows, with chemical shifts at the centres of the fluorine-split multiplets:The supplied carbon-13 spectrum is interpreted as proton-decoupled NMR, as indicated by its unsplit ethoxy signals. Proton decoupling removes C–H splitting but leaves C–F scalar spin-spin coupling. Three equivalent spin-$\frac{1}{2}$ fluorine nuclei give each resolved quartet its $1:3:3:1$ intensity pattern. The carbonyl's much smaller four-bond C–F interaction is unresolved in the plotted spectrum; “singlet” describes what is observed, rather than proving an exactly zero interaction.

The wide quartet is centred at $(127.48+116.74)/2=122.11$ ppm. The other two centres are $(132.04+130.63)/2=131.335$ and $(129.09+128.85)/2=128.97$ ppm. The signal near $77$ ppm is the deuterated chloroform solvent, not another carbon of the analyte. The structure labels skip ${\mathrm{C}}_i$: the ethoxy methylene is ${\mathrm{C}}_j$.

D is $(\mathrm{MeO}{)}_2\mathrm{CHC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{C}{\mathrm{O}}_2\mathrm{Me}$, and E is $\mathrm{OHCC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{C}{\mathrm{O}}_2\mathrm{Et}$. D results from transesterification; E results from acid-catalysed acetal hydrolysis.

For D, methoxide attacks the ester carbonyl carbon and moves the carbonyl $\pi$ pair to oxygen. Collapse of the tetrahedral reaction intermediate reforms C=O and expels ethoxide. Proton transfer with the methanol solvent regenerates methoxide. Excess methanol favours replacement of the ethyl ester by a methyl ester; the dimethyl acetal survives these basic conditions.

For E, protonate one acetal oxygen, then break its C–O bond with the pair going to oxygen, releasing methanol and forming an oxocarbenium ion. Water attacks that carbon. Deprotonation gives a hemiacetal; protonation of its remaining methoxy oxygen permits loss of a second methanol. The hydroxyl oxygen's lone pair forms C=O during this loss, and deprotonation gives the aldehyde, regenerating the acid catalyst.

The spectroscopy is consistent with these products. D has one ester carbonyl near $1745\mathrm{c}{\mathrm{m}}^{-1}$ and six carbon-13 NMR environments: acetal CH, two distinct chain $\mathrm{C}{\mathrm{H}}_2$ sites, carbonyl, ester $\mathrm{OC}{\mathrm{H}}_3$, and the two equivalent acetal $\mathrm{OC}{\mathrm{H}}_3$ groups. Thus seven carbon atoms give six resonances. E has separate ester and aldehyde carbonyl bands, approximately $1745$ and $1730\mathrm{c}{\mathrm{m}}^{-1}$, and six distinct carbon atoms: two carbonyls, two chain methylenes and the two ethyl carbons.