# Thermal conductivity and resistivity The **thermal conductivity** of a material, written k or κ, is the rate at which [[Heat|heat]] flows through it per unit area for a unit [[Temperature|temperature]] gradient; **thermal resistivity** is its reciprocal. Both appear in Fourier's law of [[Thermal_conduction|conduction]], `q = −k·dT/dx`: the heat flux q flows down the temperature gradient in proportion to it.[^openstax-16] The range of k across ordinary matter spans five orders of magnitude, from [[Diamond|diamond]] near 2,000 W/(m·K) through [[Copper|copper]] near 400, [[Steel|steel]] near 50 and [[Glass|glass]] near 1 to [[Polyethylene|polyethylene]] near 0.4 and silica [[Aerogel|aerogel]] near 0.02, because heat is carried by different things in different materials: by [[Electron|electrons]] in metals, by lattice vibrations ([[Phonon|phonons]]) in insulators, and by molecules in flight in gases.[^spec-m17][^openstax-16] In the microsim below the reader picks a material from that ladder, or slides the temperature, and watches a bar held between a hot and a cold face: the flux it passes is read out from `q = k·ΔT/d`, while a second panel computes k two ways. For a metal it uses the Wiedemann–Franz law, `κ = L·σ·T`, with the Lorenz number L = 2.44×10⁻⁸ W·Ω/K², so that the [[Electrical_conductor|electrical]] conductivity σ of the sibling sim fixes the thermal one; for an insulator it uses the phonon-gas formula `κ = (1/3)·C·v·l`, heat capacity per unit volume times [[Sound|sound]] speed times phonon [[Mean_free_path|mean free path]].[^spec-m17] The equation answers one question: how much heat a slab of a given material lets through for a given temperature difference. On the [[Materials_science]] flagship this article serves the *Thermal conductivity* section of Part III — Fundamentals › Properties. Its sim is the sibling of [[Electrical_resistivity_and_conductivity]], which builds the same ladder for electric current; [[Debye_model]] supplies the phonon heat capacity C, and [[Thermal_expansion]] is the same lattice's third thermal property. ## Definition The definition comes in a one-dimensional form for a slab and a general form for any body. ### Simple definition For a slab of thickness d and face area A whose faces are held at T_hot and T_cold, the steady power through it is proportional to the area and the temperature difference and inversely proportional to the thickness: `P = k·A·(T_hot − T_cold)/d`.[^openstax-16] The constant k is the thermal conductivity, and 1/k the thermal resistivity. This is the sim's bar: with A = 1 cm², d = 10 cm and a 100 K difference, the power through diamond is 200 W, through copper 40 W, steel 5 W, glass 0.1 W, polyethylene 0.04 W and aerogel 2 mW.[^derived-tc] Doubling the thickness halves the power and doubling the area doubles it, while the temperature profile inside a uniform bar stays a straight line whatever the material. The quantity `R = d/(k·A)`, in kelvin per watt, is the thermal resistance of the slab, and it adds in series as electrical resistance does: 5 cm of [[Fiberglass|fibreglass]] (k = 0.042 W/(m·K)) resists ten times more than 10 cm of concrete (k = 0.84).[^openstax-16][^derived-tc] ### General definition For a body with a temperature field T(x, y, z) the heat flux is a vector, `q = −k·∇T`, pointing from hot to cold, Fourier's law as Fourier gave it in 1822.[^fourier1822] Combining it with [[Conservation_of_energy|conservation of energy]] gives the heat equation, `∂T/∂t = α·∇²T`, whose coefficient `α = k/(ρ·c_p)` is the [[Thermal_diffusivity|thermal diffusivity]]. In an anisotropic crystal k is a tensor and the flux need not be parallel to the gradient. The definition covers conduction only; [[Thermal_radiation|radiation]] through a transparent body and [[Convection|convection]] in a fluid are separate mechanisms that an effective k sometimes hides.[^openstax-16] ### Other quantities The diffusivity α sets how fast a temperature change spreads; it divides k by the volumetric heat capacity ρ·c_p, so copper responds in seconds where glass takes minutes. The R-value of [[Thermal_insulation|building insulation]] is the resistance per unit area, d/k, in m²·K/W, added across layers; conductance, the reciprocal of resistance, suits an interface, which has no thickness. ## Units In SI, thermal conductivity is measured in watts per metre-kelvin, W/(m·K), and thermal resistivity in metre-kelvins per watt; since a kelvin and a degree Celsius are the same size, W/(m·°C) is the same unit.[^openstax-16] Diffusivity is in m²/s and thermal resistance in K/W. ## Measurement Steady-state methods realize the simple definition directly: a guarded hot plate holds a slab between a heated and a cooled plate, guards the edges against side losses, and reads k from power, area, thickness and temperature difference; metals are measured on long bars with thermometers spaced along the heat flow. Transient methods measure the diffusivity instead: a laser flash heats one face of a thin disc, the temperature rise of the far face gives α, and k = α·ρ·c_p follows once the [[Density|density]] and [[Specific_heat_capacity|specific heat]] are known. Every method must separate conduction from radiation and convection, which makes gases and transparent solids the hardest cases. ## Experimental values The table lists the sim's material ladder beside the Portal Book's values where the book has the same or a comparable substance.[^spec-m17][^openstax-16] | Material | k, sim preset (W/(m·K)) | k, Portal Book Table 1.5 | Carrier | |---|---|---|---| | Diamond | 2,000 | — | phonons | | Silver | — | 420 | electrons | | Copper | 400 | 390 | electrons | | Steel | 50 | iron 80; stainless 14 | electrons, phonons | | Glass | 1 | 0.84 | phonons, short l | | Water | — | 0.6 | molecular collisions | | Polyethylene | 0.4 | — | phonons along chains | | Air | — | 0.023 | molecules in flight | | Silica aerogel | 0.02 | — | pore-trapped air | Two features repeat in every such table. The metals cluster in the order of their electrical conductivities, [[Silver|silver]] above copper above [[Aluminium|aluminium]], and the best solid conductor is not a metal but diamond, an [[Insulator_(electricity)|electrical insulator]] whose stiff, light lattice carries phonons far. At the bottom, building and clothing insulators all sit near still air, whose stillness is their job; only aerogel beats air itself. The polyethylene and aerogel presets have no book counterpart and are the sim's own values. ## Influencing factors Temperature, phase, crystal direction, purity and, in some cases, magnetic field all change k. ### Temperature The direction of the temperature dependence depends on the carrier. In a pure metal at and above room temperature k is nearly constant: σ falls as 1/T because phonons scatter the electrons more, and the Wiedemann–Franz product L·σ·T stays put, so copper's conductivity changes by only a few percent between 20 °C and 100 °C.[^derived-tc] At low temperature impurities fix the electron mean free path, σ becomes constant, and k falls in proportion to T. In a crystalline insulator k rises from zero as T³ at the lowest temperatures, where the phonon mean free path is the size of the sample and only the [[Debye_model|Debye]] heat capacity grows, peaks at a few tens of kelvin, and falls roughly as 1/T at high temperature, where phonons scatter off one another. In a glass k is small and simply rises with T; in a gas it rises as roughly the square root of T, since the molecules fly faster.[^likharev-ch6] The sim's slider applies the metallic and gas laws to their presets and marks the insulator curve ILLUSTRATIVE, since the position of the peak depends on the sample.[^spec-m17] ### Chemical phase The same substance conducts differently in each phase, and the change at a transition is a jump. [[Water|Water]] is the example in every table: [[Ice|ice]] 2.2 W/(m·K), liquid water 0.6, and steam near the value for air.[^openstax-16] The ordering, solid above liquid above gas, is general: a lattice passes vibrations coherently, a liquid from molecule to neighbour, and a gas only as fast as molecules fly between collisions. ### Thermal anisotropy In a crystal without cubic symmetry the conductivity depends on direction, and heat can flow at an angle to the gradient. [[Graphite|Graphite]] is the extreme case: within a layer the stiff covalent sheet carries phonons like diamond, while across the layers the weak bonds carry them hundreds of times worse. Wood conducts better along the grain than across it, which is why the Portal Book's table gives it as a range.[^openstax-16] ### Electrical conductivity In metals the two conductivities are tied by the Wiedemann–Franz law, `κ/σ = L·T`, because the same electrons carry both charge and heat.[^wf1853] The Lorenz number `L = (π²/3)·(k_B/e)² = 2.44×10⁻⁸ W·Ω/K²` follows from the free-electron theory with no adjustable constant, using only the [[Boltzmann_constant|Boltzmann constant]] and the electron charge.[^likharev-ch6][^nist-codata] For copper at 20 °C, with the resistivity 1.68×10⁻⁸ Ω·m of the sibling sim's preset, the law predicts κ = L·T/ρ = 426 W/(m·K), within seven percent of the tabulated 390 to 400.[^derived-tc] The law fails at intermediate temperatures, where small-angle phonon scattering impedes heat more than charge, and it says nothing about insulators, which is why diamond, twenty orders of magnitude more resistive than copper, still out-conducts it thermally. ### Magnetic field A [[Magnetic_field|magnetic field]] bends the paths of the electrons that carry heat in a metal, so k falls in a field as σ does, and a transverse temperature gradient appears, the thermal analogue of the Hall effect; both are small at ordinary fields. In a [[Superconductivity|superconductor]] the paired electrons carry no [[Entropy|entropy]], so the electronic share of k vanishes below the transition, and a field strong enough to destroy superconductivity restores it. ### Gaseous phases A gas conducts poorly, and nearly independently of pressure over the ordinary range, because halving the pressure halves the number of carriers but doubles the distance each flies between collisions. That independence ends when the container is smaller than the mean free path, about 70 nm for air at atmospheric pressure.[^derived-tc][^openstax-ch2] In a silica aerogel the pores are smaller than that, so a molecule hits a wall before it hits another molecule and the gas can no longer carry its share: the solid conducts worse than the air it is mostly made of, 0.02 against 0.023 W/(m·K).[^spec-m17][^openstax-16] ### Isotopic purity Phonons scatter off anything that breaks the regularity of the lattice, and an atom of the wrong mass at a lattice site is such a break even when it is chemically identical. Natural [[Carbon|carbon]] is about one percent carbon-13, natural [[Germanium|germanium]] and [[Silicon|silicon]] are mixtures of several [[Isotope|isotopes]], and crystals grown from a single isotope conduct measurably better, most strikingly diamond, above all near the conductivity maximum where impurity scattering is the only scattering left. ## Molecular origins Every mechanism of conduction is energy carried by something that moves and collides, and one kinetic formula, `k = (1/3)·C·v·l`, covers them all: C is the carriers' heat capacity per unit volume, v their speed and l the distance between collisions.[^likharev-ch6] ### Gases In a gas the carriers are the molecules: with n per unit volume, mean speed v̄ and mean free path λ, `k = (1/3)·n·c_v·v̄·λ`, c_v being the heat capacity per molecule.[^likharev-ch6] Since λ ∝ 1/n, the density cancels and k depends only on temperature and the molecule, k ∝ v̄ ∝ (T/m)^{1/2}. This is why [[Hydrogen|hydrogen]] and [[Helium|helium]] conduct several times better than air and why the sim's gas curve rises as the square root of T.[^openstax-ch2] The elementary estimate with the speeds of the [[Kinetic_theory_of_gases|kinetic theory]] gives the right order, 10⁻² W/(m·K), for air; the prefactor needs the full [[Ludwig_Boltzmann|Boltzmann]] treatment.[^derived-tc] ### Liquids A liquid is dense enough that a molecule cannot fly, so energy passes by collision from each molecule to its neighbours at roughly the speed of sound. Putting l equal to the intermolecular spacing n^{−1/3} and C equal to 3·n·k_B in the kinetic formula gives `k ≈ n^{2/3}·k_B·v_s`; for water, with 3.3×10²⁸ molecules per cubic metre and a sound speed of 1,480 m/s, this is 0.2 W/(m·K), the right order against the measured 0.6.[^derived-tc][^openstax-v1-17] Liquids therefore sit between gases and solids; water and liquid metals aside, most conduct a few times worse than water. ### Metals In a metal the conduction electrons dominate, because they are fast (the Fermi velocity, of order 10⁶ m/s) and travel tens of nanometres between collisions in a pure crystal at room temperature. Applying the kinetic formula to the [[Fermi_gas|electron gas]], with its heat capacity proportional to T and a mean free path set by the same collisions that limit the [[Electric_current|electric current]], gives `κ = (π²/3)·(k_B/e)²·σ·T`, the Wiedemann–Franz law with the Lorenz number already inside it.[^likharev-ch6] The phonons of a metal carry heat too, but the electrons short-circuit them; only in disordered [[Alloy|alloys]] such as [[Stainless_steel|stainless steel]], whose electrical resistivity is high, does the lattice share become comparable, which is why the sim gives steel its own preset.[^openstax-16] *Try:* read copper's σ at 300 K from the electrical sibling sim and check that L·σ·T lands within a few percent of the copper preset here; then pick steel and watch the law miss by the phonon share the alloy adds. ### Lattice waves, phonons, in dielectric solids In an electrical insulator the only carriers are the phonons of the [[Debye_model|Debye model]], and `κ = (1/3)·C·v·l` with C the lattice heat capacity per unit volume, v the speed of sound and l the phonon mean free path. C and v vary by factors of a few across solids; the ladder is a ladder of mean free paths. In a perfect crystal l would be infinite and so would k, so the finite values measure what scatters phonons: other phonons, through the anharmonicity of the bonds that also produces [[Thermal_expansion|thermal expansion]]; isotopes and impurities; [[Crystallographic_defect|defects]] and [[Grain_boundary|grain boundaries]]; and, at the lowest temperatures, the sample's own surfaces. In diamond the bonds are stiff, the sound fast and the anharmonic scattering weak, so l at room temperature is hundreds of spacings and diamond tops the ladder. In a glass the disorder scatters a phonon within a spacing or two, and k lands near 1 W/(m·K) for nearly every [[Amorphous_solid|amorphous solid]]. [[Polymer|Polymers]] sit just below, conducting along their chains and poorly between them. ## Prediction For gases the kinetic theory predicts k from first principles: with a model of the intermolecular force the Boltzmann equation gives the mean free path and the correction to the elementary estimate.[^likharev-ch6] For metals the Wiedemann–Franz law predicts k from σ, which is far easier to measure. For crystalline insulators the mean free path must be computed from the scattering rates of every mode, now done with first-principles lattice dynamics; the empirical recipe for high k is a stiff lattice, light atoms, a simple structure and a high Debye temperature, which describes diamond. ### In fluids For dense fluids no theory matches the kinetic theory of dilute gases. The liquid estimate above captures the order of magnitude and the trends (k rises with sound speed and density) but not the values, and it misses the hydrogen-bonded structure that makes water conduct three times better than the estimate. Engineering practice uses correlations fitted to measurements, and the sim's fluid values are tabulated, not computed.[^spec-m17] ## History The quantitative science began with Fourier, whose *Théorie analytique de la chaleur* of 1822 stated the law of conduction, set up the heat equation and solved it by the [[Fourier_analysis|series]] that carry his name.[^fourier1822] In 1853 Gustav Wiedemann and Rudolph Franz measured the thermal conductivities of a series of metals and found them in the same ratio as the electrical conductivities; Ludvig Lorenz later showed the ratio proportional to temperature.[^wf1853] The free-electron explanation, with the Lorenz number computed from k_B and e, came with the quantum theory of metals.[^likharev-ch6] ### Jan Ingenhousz and the thermal conductivity of different metals Before any of this, the ranking of metals as conductors of heat was demonstrated with an experiment that still runs in school laboratories: rods of different metals, coated with wax and set with one end in hot water, melt the wax to different lengths, and the lengths rank the conductivities. The experiment is associated with Jan Ingenhousz, the physician better known for discovering [[Photosynthesis|photosynthesis]], who described it in 1789.[citation needed][^ingenhousz-cn] It needs no thermometer, the wax front being the instrument, but the melted length depends on the square root of k and on the losses from the rod's surface, so it ranks the metals without measuring them; Wiedemann and Franz's apparatus was Ingenhousz's rods with the losses controlled and the temperatures read.[^wf1853] In the sim's terms each rod is the bar with one face hot and the rest open to the room, and the wax front marks where the falling temperature profile crosses the melting point. ## See also - [[Wiedemann–Franz_law]] — the metal branch of the sim's second panel - [[Thermal_insulation]] — the bottom of the ladder put to use - [[Phonon]] — the carrier in dielectric solids - [[Electrical_resistivity_and_conductivity]] — the sibling sim and its ladder - [[Thermal_conduction]] — Fourier's law and the transient rod, on the Energy flagship - [[Thermal_diffusivity]] - [[Debye_model]] ## References ### Notes Numerical values marked as computed for this article are derived from the equations on the page and the cited constants; they are not printed in a Portal Book. The sim's insulator temperature curve is ILLUSTRATIVE. ### Citations [^openstax-16]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 1, "Temperature and Heat" (pp. 17–74), §1.6 Mechanisms of Heat Transfer: conduction, `P = k·A·(T_h − T_c)/d`, Table 1.5 Thermal Conductivities of Common Substances, composite walls (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^openstax-ch2]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 2, "The Kinetic Theory of Gases" (pp. 75–114): molecular speeds `v_rms = (3·k·T/m)^{1/2}`, p. 89; mean free path (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^openstax-v1-17]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 1* (OpenStax). Chapter 17, "Sound" (pp. 819–874), Table 17.1 Speed of Sound in Various Media: fresh water (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 [^likharev-ch6]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 6, "Elements of kinetics" (pp. 187–225): the Boltzmann equation in the relaxation-time approximation, the Drude formula, the kinetic formula for transport coefficients and the Wiedemann–Franz law (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^wf1853]: Wiedemann, G.; Franz, R. (1853). "Ueber die Wärme-Leitungsfähigkeit der Metalle." *Annalen der Physik und Chemie* 165 (8): 497–531. https://doi.org/10.1002/andp.18531650802 [^fourier1822]: Fourier, Jean-Baptiste Joseph (1822). *Théorie analytique de la chaleur.* Paris: Firmin Didot. Pre-DOI work; no DOI exists. [^nist-codata]: National Institute of Standards and Technology. "CODATA Internationally Recommended Values of the Fundamental Physical Constants." *The NIST Reference on Constants, Units, and Uncertainty.* https://physics.nist.gov/cuu/Constants/ [^derived-tc]: Computed for this article: the bar powers and wall resistances from `P = k·A·ΔT/d` with the presets of [^spec-m17] and the concrete and fibreglass values of [^openstax-16]; diffusivities from α = k/(ρ·c_p) with handbook density and specific heat (page to pin against [^openstax-16] Table 1.3); the Lorenz number and the copper check from L = (π²/3)(k_B/e)² with the constants of [^nist-codata] and ρ = 1.68×10⁻⁸ Ω·m, α_ρ = 0.0039/K; the air mean free path λ = k·T/(√2·π·d²·p) with d = 0.37 nm at 300 K and 1 atm; the gas and liquid kinetic estimates as described in the text. [^spec-m17]: Matter & Energy Cluster contract, `_registry/plans/MATERIALS_SCIENCE_SECTIONS.md` row M17: sim concept (sibling of Electrical_resistivity_and_conductivity), the κ ladder (diamond 2,000, Cu 400, steel 50, glass 1, PE 0.4, aerogel 0.02 W/(m·K)), the Wiedemann–Franz branch with L = 2.44×10⁻⁸ W·Ω/K², the phonon branch `kappa = (1/3)·C·v·l`, and the hot-face/cold-face bar. [^ingenhousz-cn]: Citation needed. The wax-coated-rod experiment is universally attributed to Jan Ingenhousz and dated to 1789 in secondary accounts, but no Portal Book, primary paper with a DOI, or open text on this page's shelf carries it; the record that would settle it is Ingenhousz's *Nouvelles expériences et observations sur divers objets de physique* (Paris, 1785–1789), page to pin. ### Sources - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax), Chapters 1–2. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 1* (OpenStax), Chapter 17. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 - Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*, Chapter 6. https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics ### Further reading - Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM*, Chapter 6 (kinetics), for the Boltzmann-equation derivation of every transport coefficient on this page. - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*, Chapter 1 §1.6, for worked conduction problems with the Table 1.5 values. ## External links - The Wikipedia pair's *External links* section lists conductivity databases and calculators; the open texts under *Sources* are the Portal Book sources of this page. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Thermal_conductivity_and_resistivity.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Thermal conductivity and resistivity* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Thermal_conductivity_and_resistivity.html" data-title="Thermal conductivity and resistivity"></div> *Built from `MICROSIM_GUIDE/specs/sims/Thermal_conductivity_and_resistivity.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Thermal_conductivity_and_resistivity) : [Wikitube](https://en.wikitube.io/wiki/Thermal_conductivity_and_resistivity) · pinned revision [1374231229](https://en.wikipedia.org/w/index.php?oldid=1374231229) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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