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Showing 1–50 of 541 relationships

Scientific relationships · 50 / 541

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    The gas equation is exact within the ideal model and approximate for a real gas.

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    • Isotopes are atoms of the same element with different numbers of neutrons. Their atomic number is the same, while their mass number A, the total number of protons and neutrons, differs.
    • An atom has a nucleus surrounded by electrons. Its atomic number Z is the number of protons in the nucleus and identifies the element. A neutral atom has the same number of electrons as protons; gaining or losing electrons forms an ion of that element.

    Isotope identity uses the shared atomic number of an element.

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    • Stoichiometric coefficients in a balanced chemical equation specify ratios of reactant consumption and product formation. These particle-number ratios also give mole ratios, so a known reacting amount can be converted into another reacting or produced amount.
    • The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.02214076 × 10²³ specified elementary entities. The amount n and entity count N are related by n = N/N_A, where N_A = 6.02214076 × 10²³ mol⁻¹ exactly.

    Balanced particle-number ratios also give reacting mole ratios.

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    • Stoichiometric coefficients in a balanced chemical equation specify ratios of reactant consumption and product formation. These particle-number ratios also give mole ratios, so a known reacting amount can be converted into another reacting or produced amount.
    • A balanced equation for an ordinary chemical reaction has equal numbers of atoms of each element and equal total electric charge on both sides. Balance the coefficients rather than changing a substance's formula; include electron transfer when balancing redox half-reactions.

    The amount-ratio method uses a balanced chemical equation.

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    • In a reversible reaction at chemical equilibrium, the forward and reverse reaction rates are equal. Reactant and product concentrations remain constant under unchanged conditions, although reactions continue in both directions.
    • Gibbs energy is G = H − TS. At constant temperature and pressure, a process in a closed system with only pressure-volume work is thermodynamically favoured when its Gibbs-energy change is negative. This criterion does not specify how quickly the process proceeds.

    Equilibrium and Gibbs-energy criteria are connected under the specified thermodynamic constraints.

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    • Reaction rates describe how quickly reactants are consumed or products formed. A rate law relates a measured rate to composition under specified conditions. For an overall reaction, its reaction orders must be established from kinetic evidence rather than simply copied from the balanced equation's coefficients.
    • A catalyst provides an alternative reaction pathway that increases the rate and is regenerated in the overall reaction. Catalysed pathways can have lower activation barriers. For the same reversible reaction under fixed thermodynamic conditions, catalysis can hasten the approach to equilibrium without changing its equilibrium constant.

    Catalysis changes a reaction pathway and thereby its rate under the specified conditions.

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    • A catalyst provides an alternative reaction pathway that increases the rate and is regenerated in the overall reaction. Catalysed pathways can have lower activation barriers. For the same reversible reaction under fixed thermodynamic conditions, catalysis can hasten the approach to equilibrium without changing its equilibrium constant.
    • In a reversible reaction at chemical equilibrium, the forward and reverse reaction rates are equal. Reactant and product concentrations remain constant under unchanged conditions, although reactions continue in both directions.

    Catalysis accelerates approach to equilibrium for the same reaction without changing its equilibrium constant.

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    • Metabolism includes chemical pathways that break down substances and build biological materials. Catabolic pathways break complex molecules into simpler ones, often releasing usable energy; anabolic pathways build complex molecules and require energy input. Enzymes catalyse many of these reactions.
    • Carbon cycles among the atmosphere, organisms, soils, oceans and rocks. Biological processes such as photosynthesis, respiration and decomposition exchange carbon, while weathering, sediment formation and tectonic processes participate in slower geological transfers.

    Photosynthesis, respiration and decomposition transfer carbon through biological processes.

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    • Oxygenic photosynthesis uses captured light energy to support the conversion of carbon dioxide into organic compounds. Water supplies electrons and oxygen is released. This form occurs in photosynthetic plants, algae and cyanobacteria.
    • Metabolism includes chemical pathways that break down substances and build biological materials. Catabolic pathways break complex molecules into simpler ones, often releasing usable energy; anabolic pathways build complex molecules and require energy input. Enzymes catalyse many of these reactions.

    Light-energy capture and carbon fixation supply chemical intermediates and organic compounds in photosynthetic metabolism.

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    • In aerobic glucose breakdown, glycolysis and subsequent oxidation pathways transfer energy and electrons through carriers. Carbon dioxide is released during pyruvate oxidation and the citric acid cycle. The respiratory electron-transport chain ultimately reduces oxygen to water, and a proton gradient supports ATP synthesis.
    • Metabolism includes chemical pathways that break down substances and build biological materials. Catabolic pathways break complex molecules into simpler ones, often releasing usable energy; anabolic pathways build complex molecules and require energy input. Enzymes catalyse many of these reactions.

    Aerobic nutrient breakdown transfers energy through electron transport to ATP production.

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    • Cell membranes are selectively permeable. Passive transport allows net movement down the relevant gradient, through the membrane or transport proteins. Active transport couples movement against a gradient to an energy source, directly or through another gradient. Osmosis is passive net water movement across a selectively permeable membrane.
    • Metabolism includes chemical pathways that break down substances and build biological materials. Catabolic pathways break complex molecules into simpler ones, often releasing usable energy; anabolic pathways build complex molecules and require energy input. Enzymes catalyse many of these reactions.

    Cell metabolism can supply energy for active membrane transport.

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    • Carbon cycles among the atmosphere, organisms, soils, oceans and rocks. Biological processes such as photosynthesis, respiration and decomposition exchange carbon, while weathering, sediment formation and tectonic processes participate in slower geological transfers.
    • Greenhouse gases absorb and emit thermal infrared radiation. This affects outgoing heat loss and allows Earth's surface and lower atmosphere to be warmer than they would be without those gases under comparable solar input.

    Atmospheric carbon dioxide links carbon reservoirs with the greenhouse mechanism; not all carbon is atmospheric gas.

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    • Hydrogenic bound states use n ≥ 1, l = 0,…,n−1, m_l = −l,…,l and electron m_s = ±½. Pauli exclusion permits at most one electron per spin-orbital, hence two opposite-spin electrons per spatial orbital.

    Electron spin projection supplies the two m_s labels in the stated orbital basis.

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    • For a periodic one-electron Hamiltonian, Bloch states have ψ_nk(r) = e^(ik·r)u_nk(r), with u_nk(r+R) = u_nk(r) for lattice vectors R. Their energies E_n(k) form bands.

    The periodic effective one-electron Hamiltonian selects Bloch states and bands.

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    • In a semiconductor band model, conduction-band electrons and holes in an otherwise filled valence band carry charge. Donor and acceptor doping can produce electron- or hole-dominated conduction in the stated operating regime.

    Carrier descriptions use conduction/valence bands and a material-specific gap.

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    • In multielectron atoms, electron–electron repulsion partly screens nuclear attraction. Effective nuclear charge is a model of that net attraction; changes in screening and orbital extent help explain periodic trends in atomic size and ionization energy.

    Screening and nuclear attraction help explain qualified periodic trends.

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    • Electron configurations represent orbital occupations in an approximate atomic description. Ground-state filling and Hund's rule guide many configurations, but closely spaced subshell energies produce exceptions. Valence occupations help organize chemical similarities among elements.

    Valence occupations organize chemical similarities; filling rules have exceptions.

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    • Valence-bond models describe bonding through atomic-orbital overlap and paired electrons. When one Lewis structure inadequately represents electron distribution, resonance forms provide alternative bookkeeping descriptions of one species; the species does not alternate between those drawings.

    The source compares localized and delocalized orbital descriptions.

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    • Crystalline solids can be classified by dominant bonding patterns: ionic arrays, metallic bonding, discrete molecular units, or extended covalent networks. Bonding and structure help explain conductivity, mechanical behaviour and phase properties, with important exceptions such as graphite.

    Molecular solids depend on intermolecular interactions.

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    • Coordination chemistry describes central atoms or ions, commonly metals, bound to ligands. The coordination number counts attached donor atoms, while ligand denticity counts donor atoms supplied by one ligand. Geometry and ligand arrangement can produce distinct isomers.

    Ligand arrangements can produce geometric and optical isomers.

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    • pH is defined as the negative base-ten logarithm of hydrogen-ion activity. Individual ionic activities cannot be measured independently, so practical pH realization uses conventions and standards. Replacing activity by a concentration ratio is a dilute-solution approximation.

    The pH definition uses hydrogen-ion activity with a specified convention.

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    • A solution containing suitable amounts of a weak acid and its conjugate base, or the corresponding weak-base pair, can resist pH changes from added acid or base. This response follows coupled acid–base equilibria and has finite capacity.

    Buffer behaviour involves a weak acid/conjugate-base or corresponding base pair.

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    • For a specified dissolution reaction, a solubility product relates equilibrium ion activities; a pure solid's activity is conventionally one. Comparing the reaction quotient with the equilibrium value indicates a thermodynamic tendency toward dissolution or precipitation.

    Ksp describes a specified dissolution equilibrium.

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    • The Nernst equation relates a reversible cell potential to its standard potential, temperature and reaction quotient. It follows from the thermodynamic relations between reaction Gibbs energy and reversible electrical work; it does not include current-dependent voltage losses.

    Reaction conditions modify a reversible cell's standard potential.

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    • The Nernst equation relates a reversible cell potential to its standard potential, temperature and reaction quotient. It follows from the thermodynamic relations between reaction Gibbs energy and reversible electrical work; it does not include current-dependent voltage losses.

    Reversible cell potential is related to reaction Gibbs energy.

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    • A reaction mechanism is a proposed sequence of elementary steps whose sum yields an overall reaction. For an elementary mass-action step, molecularity determines concentration factors in its rate law. The overall rate law usually requires analysis of the full mechanism.

    Proposed elementary steps must be consistent with kinetic evidence.

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    • The Arrhenius model describes many reaction rate constants using a prefactor and an activation-energy parameter. A straight-line dependence of ln k on reciprocal absolute temperature supports the model over the measured range; it does not establish one universal mechanism.

    The temperature model describes the stated rate constant over a limited range.

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    • Decadic absorbance is A = log10(Pref/Psample), where Pref and Psample are the transmitted radiant powers through a matched reference cell and sample cell at the selected wavelength. Reflection, scattering, luminescence and cell/solvent backgrounds need appropriate treatment; a raw loss of transmitted light is not automatically molecular absorption. In the Beer–Lambert regime, A = εℓc for the concentration of the fixed absorbing species. Chemical equilibria, interactions at higher concentrations, spectral bandwidth and instrumental stray light can produce departures from linearity.

    Absorbance/concentration calibration must be checked for departures from the linear law.

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    • A functional group is a recognizable atom arrangement associated with characteristic chemical behaviour. Common groups include alcohol, carbonyl, carboxyl, amine and carbon–carbon multiple-bond groups. They organize compound families and help compare reactivity across molecular structures.

    Recurring structural groups support comparison of organic transformations; context still matters.

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    • In polar-mechanism notation, a full-headed curved arrow starts at an electron pair or bond and ends where that pair forms a bond or becomes a lone pair. Nucleophiles donate electron pairs and electrophiles accept them in the described step.

    Pair arrows describe proposed polar steps, rather than just a net transformation.

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    • For the simple mechanism E + S ⇌ ES → E + P, a quasi-steady-state approximation gives a saturating initial-rate relation. The Michaelis constant is the substrate concentration at half the model's maximum rate; it includes binding and turnover rate constants.

    The rate model is derived from a specific enzyme-binding and turnover mechanism.

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    • Radiometric dating combines isotope measurements with a decay model to estimate elapsed time. Interpretation requires accounting for initial daughter material and subsequent isotope gain or loss. A mineral's isotopic age can record formation or later resetting, rather than every event in the rock's history.

    A stated isotope system and sample-retention history underlie an age interpretation.

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    • For the canonical oscillator, a=√(mω/(2ħ))x+ip/√(2mħω) and its adjoint satisfy [a,a†]=1. H=ħω(a†a+½); a|n⟩=√n|n−1⟩ and a†|n⟩=√(n+1)|n+1⟩ with a|0⟩=0.

    The canonical oscillator Hamiltonian factors into normalized lowering/raising operators.

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    • In the long-wavelength electric-dipole approximation, atomic transition strength depends on the dipole matrix element. For hydrogenic orbital states without spin-dependent coupling, nonzero matrix elements require Δl=±1, Δm_l=0,±1 and unchanged spin projection; these conditions alone do not guarantee a nonzero rate.

    The electric dipole's odd parity and rank-one angular character constrain atomic matrix elements.

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    • In the long-wavelength electric-dipole approximation, atomic transition strength depends on the dipole matrix element. For hydrogenic orbital states without spin-dependent coupling, nonzero matrix elements require Δl=±1, Δm_l=0,±1 and unchanged spin projection; these conditions alone do not guarantee a nonzero rate.

    Allowed matrix elements enter the weak-transition amplitude; selection rules are not transition probabilities.

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    • For a self-adjoint Hamiltonian bounded below, every normalized trial state with a finite energy expectation satisfies ⟨ψ_trial|H|ψ_trial⟩≥E₀. Minimizing a chosen trial family's expectation provides an upper bound on the ground energy, not proof that the trial state is exact.

    Normalized trial-state Hamiltonian expectation bounds a lower-bounded ground energy.

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    • A weak coupling from an initial state to a sufficiently dense continuum gives a leading transition rate Γ≈(2π/ħ)|V_fi|²ρ(E_f) at the energy-conserving final energy, for slowly varying matrix elements and density of states. This is the long-time, small-depletion rate limit of perturbative transition amplitudes.

    Long-time weak transition amplitude into a smooth dense spectrum yields the golden-rule rate within the displayed scale window.

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    • For a simple compressible equilibrium system, U(S,V,{N_i}) has differential dU=T dS−p dV+Σ_i μ_i dN_i. Temperature, pressure and chemical potentials are the corresponding partial derivatives of U; particle-number changes are distinct from heat transfer.

    An equilibrium extensive-state differential identifies temperature, pressure and composition work; actual irreversible paths need separate entropy accounting.

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    • The Legendre transforms H=U+pV and F=U−TS change the natural state variables: dH=T dS+V dp and dF=−S dT−p dV at fixed composition. At fixed temperature and volume, equilibrium minimizes F among allowed states. For a closed system with only pressure-volume work, unchanged bulk kinetic/gravitational potential energies and constant external pressure equal to both equilibrium endpoint pressures, heat supplied equals the enthalpy change.

    Legendre transforms replace volume or entropy by their conjugate variables in the equilibrium differential.

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    • The Legendre transforms H=U+pV and F=U−TS change the natural state variables: dH=T dS+V dp and dF=−S dT−p dV at fixed composition. At fixed temperature and volume, equilibrium minimizes F among allowed states. For a closed system with only pressure-volume work, unchanged bulk kinetic/gravitational potential energies and constant external pressure equal to both equilibrium endpoint pressures, heat supplied equals the enthalpy change.

    Gibbs energy is the existing complementary potential; no duplicate Gibbs definition is introduced.

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    • Equality of mixed derivatives of smooth equilibrium potentials gives (∂T/∂V)_S=−(∂p/∂S)_V, (∂T/∂p)_S=(∂V/∂S)_p, (∂S/∂V)_T=(∂p/∂T)_V and (∂S/∂p)_T=−(∂V/∂T)_p. These relate measurable response coefficients to entropy derivatives.

    Twice differentiable equilibrium potentials have equal mixed partials in their natural variables.

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    • For a fixed amount of simple compressible matter, C_V=(∂U/∂T)_V and C_p=(∂H/∂T)_p. With α=(1/V)(∂V/∂T)_p and κ_T=−(1/V)(∂V/∂p)_T, C_p−C_V=TVα²/κ_T in a smooth stable phase.

    Equilibrium fixed-volume/fixed-pressure entropy derivatives and response coefficients determine capacities.

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    • Thermal equilibrium is transitive, permitting a common temperature scale. For equilibrium with a unique ground microstate, the statistical entropy tends to zero as T tends to zero; a pure perfect crystal with no residual degeneracy is the usual third-law reference.

    Equilibrium temperature and zero-temperature entropy limits have the stated system/degeneracy assumptions.

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    • For canonical Z=Σ_i exp(−βE_i), F=−k_BT lnZ, U=−∂_β lnZ and S=k_B(lnZ+βU). At fixed volume and amount, Var(E)=∂_β²lnZ=k_BT²C_V; pressure is k_BT(∂_VlnZ)_T,N when the Hamiltonian's volume dependence represents mechanical work.

    Normalized canonical Boltzmann weights and convergent fixed-parameter derivatives generate observables.

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    • For a system exchanging energy and particles with reservoirs at T and μ, p_i,N=exp[−β(E_i,N−μN)]/Ξ and Ξ=Σ_N,i exp[−β(E_i,N−μN)]. The grand potential Ω_G=−k_BT lnΞ=U−TS−μ⟨N⟩ has dΩ_G=−S dT−p dV−⟨N⟩dμ; particle number fluctuates.

    Particle exchange introduces chemical potentials conjugate to particle numbers.

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    • For a system exchanging energy and particles with reservoirs at T and μ, p_i,N=exp[−β(E_i,N−μN)]/Ξ and Ξ=Σ_N,i exp[−β(E_i,N−μN)]. The grand potential Ω_G=−k_BT lnΞ=U−TS−μ⟨N⟩ has dΩ_G=−S dT−p dV−⟨N⟩dμ; particle number fluctuates.

    A particle-number sum extends the canonical normalized partition construction.

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    • In a classical canonical model, each independent quadratic term in an unrestricted coordinate or momentum contributes k_BT/2 to the mean energy. Quantum level spacing comparable with k_BT, constrained coordinates or nonquadratic energies can invalidate this per-term result.

    An unrestricted classical independent quadratic term has a convergent Gaussian Boltzmann integral.

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    • For a dilute classical ideal gas in its rest frame, the probability density of a particle velocity is (m/(2πk_BT))³ᐟ²exp[−m|v|²/(2k_BT)]. The corresponding speed density includes 4πv², and mean translational kinetic energy is 3k_BT/2.

    Classical quadratic translational energy gives the velocity Gaussian and its thermal moments.

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    • Pure phases coexist at common T and p when their molar Gibbs energies are equal. Along a first-order coexistence curve, dp/dT=Δs_m/Δv_m=L_m/(TΔv_m), with phase differences taken in the same direction and L_m=TΔs_m.

    Equality of phase chemical potentials and their equilibrium differentials give the coexistence slope.

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    • Macroscopic electric polarization P gives bound volume charge ρ_b=−∇·P and surface charge σ_b=P·n at a material boundary. D=ε₀E+P satisfies ∇·D=ρ_free. In a local linear isotropic dielectric, P=ε₀χ_eE and D=εE, with ε=ε₀(1+χ_e).

    Bound-charge bookkeeping rewrites Gauss law in terms of macroscopic D and free charge.