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Using the numerical rate constants in $\mathrm{d}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{s}}^{-1}$, the logarithms areA straight-line least-squares fit isEvery logarithmic residual has magnitude below $7.2\times 10^{-4}$, so the rates follow the fitted exponential to within $0.08\%$; $R^2\simeq 0.999997$. Thus a constant activation volume describes these data well.

Convert the slope to pascals, using $1\mathrm{atm}=101325$ Pa and $T=293.15$ K:Multiplying all rates by a common unit-conversion factor changes the intercept but leaves this slope and volume unchanged.

At fixed temperature, differentiating the Eyring equation with a pressure-independent prefactor and fixed concentration convention givesSince $\mathrm{d}G=V\mathrm{d}p-S\mathrm{d}T$, the derivative is the difference between the activated complex's partial molar volume and the sum for the reactants:If activation volume is approximately constant over the pressure range, integration givesThus plot $\ln k_2$ against pressure: its slope is $-\Delta V^{‡\circ }/RT$. Use pressure in pascals for a volume in ${\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$. The model neglects pressure dependence of the transmission coefficient, substantial changes in mechanism, and strong curvature from a pressure-dependent activation volume. Fixed concentration-unit conventions and compression corrections matter for non-first-order rates.^[ref]

The negative activation volume means the transition state occupies less partial molar volume than the two separated reactants. Pressure therefore favours reaching it and increases the forward rate, consistent with a compact associative cycloaddition in which new bonds begin to form.

The magnitude, about $25\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, is molecular in scale. It includes packing and solvent reorganization as well as intrinsic bond-length changes, so it should not be interpreted as a simple geometric volume carved out of two rigid molecular surfaces.

The negative activation volume is consistent with an associative substitution pathway: an incoming donor of the tetradentate ligand approaches square-planar Ni, giving a five-coordinate transition structure/intermediate before a donor of an original chelate leaves. Successive donor substitution and chelate closure give the tetradentate N${}_2$O${}_2$ complex. Proton transfers convert the incoming phenolic groups to coordinated phenoxides while releasing the original ligands.

With acid/base forms written explicitly, the net ligand exchange is $\mathrm{Ni}({\mathrm{L}}_{\mathrm{a}}{)}_2+{\mathrm{H}}_2{\mathrm{L}}_{\mathrm{d}}\to \mathrm{Ni}({\mathrm{L}}_{\mathrm{d}})+2H{\mathrm{L}}_{\mathrm{a}}$. The connected four-donor ligand benefits from the chelate effect.

The bulky N-tert-butyl groups in $\mathrm{Ni}({\mathrm{L}}_{\mathrm{c}}{)}_2$ hinder precisely the close donor approach needed for that associative route, and the starting tetrahedral geometry must reorganize to accept the planar tetradentate ligand. Consequently exchange can be too slow to observe under the stated conditions. Failure to observe a new product is a kinetic observation; it does not establish that all tetrahedral nickel complexes are inert or that the tetradentate product is thermodynamically impossible.

Homologous temperature and deformation. Homologous temperature is $\Theta =T/T_m$, with both temperatures in kelvin and $T_m$ the appropriate pressure- and composition-dependent melting temperature. It measures proximity to melting more meaningfully than an absolute temperature alone. Pressure, composition and water can alter both $T_m$ and the flow law, so equal $\Theta$ does not imply identical rheology.

A useful thermally activated material creep law iswith stress $\sigma$, grain size $d$, activation energy $E^{\ast }$ and activation volume $V^{\ast }$. Water activity can add another factor. Increasing temperature raises the rate at fixed stress when the activation enthalpy is positive, greatly lowering effective viscosity. Solid material can therefore flow on geological timescales well below melting.

Diffusion creep is approximately linear in stress: bulk-diffusion Nabarro-Herring creep has grain-size scaling $d^{-2}$, while grain-boundary Coble creep has $d^{-3}$ in their ideal regimes. Dislocation creep usually has a stress exponent greater than one and weaker grain-size dependence. Its effective viscosity scales as ${\eta }_{\mathrm{eff}}\propto {\sigma }^{1-n}$ at fixed temperature, so stronger forcing changes the apparent viscosity. At lower homologous temperature, brittle/frictional deformation or high-stress dislocation mechanisms can become important. The dominant process also depends on pressure and confinement; low temperature is not by itself a guarantee of brittle failure deep in a planet.

A deformation mechanism map compares the competing rates or strengths as functions of stress and homologous temperature at specified grain size, pressure and composition. Fine grains favour diffusion creep; larger stress favours nonlinear dislocation creep. There is no universal temperature dividing all these regimes. This sensitivity helps explain a strong outer lithosphere above a slowly flowing solid mantle and why modest thermal or water variations can reorganize deformation.

Determining the actual planform of mantle convection. The task is to infer the geometry of upwellings, downwellings and connecting flow in the present Earth, rather than to draw a preferred laboratory convection cell. Plate velocities and the distribution of subduction zones directly constrain surface motion and likely cold downwellings. Ridges, volcanism and hotspots provide further clues, but a ridge need not mark the centre of a deep plume, nor is every hotspot uniquely explained by one plume geometry.

Seismic tomography supplies three-dimensional velocity structure from body-wave travel times, waveforms and surface-wave dispersion. Fast slab-like structures can often be traced beneath subduction zones into the lower mantle. The 1997 global P-wave study reported long narrow lower-mantle structures connected with convergence and interpreted some slab segments as reaching the deepest mantle; that is evidence for deep circulation, not proof that every slab follows one route.^[ref] Temperature affects seismic speed, but composition, phase changes, water and anisotropy also matter, so velocity is not itself a flow map.

Gravity and the geoid constrain the spatially integrated density field and its induced boundary deformation. Dynamic topography supplies information about vertical stresses, after separating crustal structure, isostatic loads and thermal subsidence. Seismic anisotropy can constrain deformation and fabric orientations, subject to a mineral-physics relation between fabric and flow. Discontinuity topography and the mantle's heat budget provide further tests. Each observation has a different sensitivity to density, temperature and viscosity, making their combination more useful than any one image.

Compare these observations with three-dimensional geodynamic models that solve creeping-flow force balance, mass conservation and thermal transport, including buoyancy from temperature and composition, realistic plate motions, viscosity structure and phase boundaries. Predict seismic observables, gravity, geoid and dynamic topography from the same model, and assess data coverage and resolution before interpreting a feature. Whole-mantle exchange, impeded or deflected slabs near transition zones, and broad deep anomalies can coexist; the evidence does not require identical stacked or single-layer cells everywhere. The actual planform is constrained by joint seismic, gravitational, surface-kinematic and thermal evidence, with substantial nonuniqueness in its detailed flow and history.

Official primary terminology defining activation volume from the logarithmic pressure derivative of a rate constant. The concentration-unit convention must be held fixed when interpreting that derivative.