# Calorimetry **Calorimetry** is the measurement of the [[Heat|heat]] exchanged when a body changes its [[Temperature|temperature]], its [[Phase_(matter)|phase]] or its chemical composition. The instrument is a [[Calorimeter|calorimeter]]: an insulated vessel whose contents have a known [[Heat_capacity|heat capacity]], so that the heat released or absorbed by whatever happens inside it can be read from the temperature change of the whole. The method dates from the eighteenth century, when Joseph Black separated heat from temperature and [[Antoine_Lavoisier|Lavoisier]] and Laplace built the first instrument to weigh heat by the ice it melted.[^black1803][^lavoisier-laplace] In [[Chemistry|chemistry]] calorimetry is how the [[Enthalpy|enthalpy]] of a [[Chemical_reaction|reaction]] is measured, and every thermochemical table rests on it. In the microsim below the reader runs a coffee-cup calorimeter: the reaction is chosen from three presets — the neutralisation of a strong acid by a strong base, the dissolving of ammonium nitrate, and the [[Combustion|combustion]] of a fuel in a sealed bomb — and the amount of reactant is set with a slider. The thermometer moves by ΔT = q/C_cal, where C_cal is the heat capacity of the calorimeter and its contents, and the readout converts the heat to a molar enthalpy of reaction through ΔH_rxn = −q/n. An [[Exothermic_reaction|exothermic]] reaction warms the water and an [[Endothermic_process|endothermic]] one cools it, which is the pair's exothermic–endothermic distinction turned into a number on a thermometer. On the [[Chemistry]] flagship this article is the child of Part V — Energy, section *Heat of reaction: calorimetry* (row K25), the experimental row that feeds the ΔH values used by the spontaneity dial of [[Chemical_thermodynamics|chemical thermodynamics]]. ## History Joseph Black, lecturing in Glasgow and Edinburgh in the 1760s, showed that ice absorbs a large quantity of heat while melting without any rise in temperature, and named it [[Latent_heat|latent heat]]; he also showed that equal masses of different substances need different amounts of heat for the same temperature rise, the [[Specific_heat_capacity|specific heat]].[^black1803] Lavoisier and Laplace turned the first observation into an instrument: their ice calorimeter of 1783 surrounded the sample with ice and weighed the meltwater, and with it they measured the heat of combustion of carbon and the heat given off by a guinea pig, showing that respiration is a slow combustion.[^lavoisier-laplace] Rumford's cannon-boring experiments of 1798 showed that friction produces heat without limit, and [[James_Prescott_Joule|James Prescott Joule]]'s paddle-wheel measurements, reported in full in 1850, fixed the [[Mechanical_equivalent_of_heat|mechanical equivalent of heat]] to within about one percent of the modern 4.184 J per [[Calorie|calorie]] (derived from his 772 foot-pounds per British thermal unit).[^rumford1798][^joule1850] Hess's 1840 finding that the heat of a reaction is independent of the route by which it runs made calorimetric data additive and gave [[Thermochemistry|thermochemistry]] its bookkeeping rule.[^hess1840] Marcellin Berthelot introduced the bomb calorimeter, a sealed steel vessel in which the sample burns in oxygen under pressure, and his *Thermochimie* of 1897 collected the heats of combustion measured with it.[^berthelot1897] The twentieth century added electrical calibration, adiabatic shields and the differential scanning and titration instruments used today on [[Polymer|polymers]] and proteins; the physics of the measurement has not changed since Black. ## Classical calorimetric calculation of heat The rule behind every calorimeter is that the heat absorbed by a body of heat capacity C when its temperature rises by ΔT is q = C·ΔT, and that within an insulated vessel the heat lost by one part is the heat gained by the rest. For the coffee-cup preset the vessel is two nested foam cups holding a known mass of aqueous solution; its heat capacity is very nearly that of the water, 4.184 J per gram per kelvin, and the cup itself is neglected.[^af2e-9-2] The reaction's heat is then q_rxn = −q_solution = −m·c·ΔT, and dividing by the number of moles that reacted gives the molar enthalpy, ΔH_rxn = −q/n, which is an enthalpy rather than an internal energy because the cup is open to the atmosphere and the process runs at constant [[Pressure|pressure]].[^af2e-9-3] The worked case in the Portal Book is the sim's neutralisation preset: 50.0 mL of 1.00 M hydrochloric acid mixed with 50.0 mL of 1.00 M [[Sodium_hydroxide|sodium hydroxide]] in a coffee-cup calorimeter warms from 22.0 °C to 28.9 °C, so the 100 g of solution gains q = 100 g × 4.184 J/(g·K) × 6.9 K ≈ 2.9 kJ, and the reaction of 0.0500 mol of H⁺ with 0.0500 mol of OH⁻ released it: ΔH ≈ −2.9 kJ/0.0500 mol = −58 kJ/mol (derived), against −55.8 kJ/mol from the tabulated enthalpies of formation of liquid water and the hydroxide ion (derived).[^af2e-9-2][^af2e-appg] The gap is the heat that leaked into the cups and the air. In the sim the slider scales the amount; doubling the reactants doubles q but leaves ΔH per mole unchanged. The ammonium nitrate preset runs the other way: dissolving the salt absorbs about 26 kJ/mol (derived from tabulated formation enthalpies), so 10 g in 100 g of water pulls roughly 3.2 kJ out of the solution and cools it by about 7 K (ILLUSTRATIVE: the sim gives the solution water's heat capacity), which is how a chemical cold pack works.[^nbs-tables] ### Cases with differentiable equation of state for a one-component body For a simple body with a smooth [[Ideal_gas_law|equation of state]] p(V, T), the heat needed for a small change is δq = C_V·dT + Λ_V·dV in the variables (T, V), or δq = C_p·dT + Λ_p·dp in the variables (T, p). C_V and C_p are the heat capacities at constant volume and at constant pressure; Λ_V and Λ_p are the latent heats with respect to volume and to pressure, the heat the body absorbs when it is made to expand at fixed temperature or to change pressure at fixed temperature. This is the classical scheme of the nineteenth-century treatises.[^bryan1907] ### Calorimetry through phase change, equation of state shows one jump discontinuity Where the equation of state jumps — at melting, boiling or a solid-state transition — the heat capacity is not defined at the transition and the heat is a latent heat per unit mass, q = m·L, absorbed at constant temperature. Melting ice takes 6.01 kJ/mol, or 334 kJ/kg (derived); boiling water takes about 2.2×10⁶ J/kg, roughly 0.4 eV per molecule; and a body of ice, water and steam heated through both transitions spends more heat on the two plateaus than on all the warming in between.[^nist-water][^likharev-latent] The Portal Book's gallium example runs the measurement backwards: hot air melting 202 g of a 300 g gallium sample in a calorimeter gives a heat of fusion of 80.2 kJ/kg.[^up2-gallium] ### Cumulation of heating Heat is not a property a body possesses but a quantity transferred along a path, so the total heat absorbed in a process is the integral of δq along that path and depends on the route. Warming a gas at constant volume and then expanding it isothermally takes a different total heat from expanding first and warming afterwards, though the end state is the same. Only the sum q + w, the change in [[Internal_energy|internal energy]], is the same along every route; that is the [[First_law_of_thermodynamics|first law]], and it is why a calorimeter measures a state function only when the path is fixed, at constant volume or at constant pressure.[^bryan1907] ### Mathematical aspects of the above rules In the language of differential forms, δq is an inexact differential: there is no function Q(T, V) whose partial derivatives are C_V and Λ_V, because ∂C_V/∂V is not equal to ∂Λ_V/∂T. The second law supplies an integrating factor: dividing by the absolute temperature turns δq into the exact differential of the [[Entropy|entropy]], dS = δq_rev/T, which is a [[State_function|state function]]. The calorimetric coefficients are therefore not independent; the integrability of δq/T ties Λ_V to the equation of state, as worked out below.[^likharev-thermo] ### Physical scope of the above rules of calorimetry The rules apply to a closed body of fixed composition passing through equilibrium states, so that its temperature and pressure are defined at every instant. They exclude [[Friction|friction]] and other irreversible work inside the calorimeter, which appear as heat with no change of state, and they exclude chemical reaction unless composition is treated as a further variable. The chemist's calorimeter sidesteps the second exclusion by measuring only the initial and final states of a completed reaction and letting Hess's law do the rest.[^af2e-9-3] ## Experimentally conveniently measured coefficients The three response coefficients that a laboratory can measure most directly are the pressure coefficient at constant volume, the [[Thermal_expansion|thermal expansion]] at constant pressure and the compressibility at constant temperature. For a body of volume V they are defined as (∂p/∂T)_V, α = (1/V)(∂V/∂T)_p and κ_T = −(1/V)(∂V/∂p)_T, and only two are independent, because the cyclic rule of partial derivatives gives (∂p/∂T)_V = α/κ_T.[^bryan1907] ### Pressure increase at constant volume Sealed in a rigid vessel and warmed, a gas raises its pressure by (∂p/∂T)_V per kelvin; for an ideal gas the coefficient is nR/V, so the pressure rises by 1/273 of its value at 0 °C for each kelvin. It is the coefficient a constant-volume gas thermometer reads, and it reappears below as the latent heat with respect to volume. ### Expansion at constant pressure Warmed under a constant load a body expands by the fraction α per kelvin; for an ideal gas α = 1/T exactly, about 3.4×10⁻³ K⁻¹ at 20 °C (derived), while for liquids and solids it is one to three orders of magnitude smaller. The expansion does work against the surroundings, which is why heating at constant pressure costs more heat than heating at constant volume. ### Compressibility at constant temperature Squeezed slowly enough to stay at the temperature of its surroundings, a body shrinks by the fraction κ_T per unit of added pressure; for an ideal gas κ_T = 1/p. Liquids and solids are nearly incompressible on this scale, and the smallness of their α and κ_T together is what makes C_p and C_V almost equal for them. ## Relation between classical calorimetric quantities The two forms of δq describe the same body, so the coefficients in (T, V) and in (T, p) are connected through the equation of state. Substituting dV = (∂V/∂T)_p dT + (∂V/∂p)_T dp into C_V·dT + Λ_V·dV gives C_p = C_V + Λ_V·(∂V/∂T)_p and Λ_p = Λ_V·(∂V/∂p)_T. The difference of the heat capacities is thus the latent heat of expansion times the expansion per kelvin — the heat that goes into pushing back the atmosphere rather than into raising the temperature — and both relations follow from calorimetry and the equation of state alone, before any appeal to the second law.[^bryan1907] ## Connection between calorimetry and thermodynamics The first law makes the calorimeter an instrument for measuring changes of internal energy and enthalpy: at constant volume no work is done and q_V = ΔU, while at constant pressure q_p = ΔH, with H = U + pV.[^af2e-9-3] This is why the coffee-cup preset reports an enthalpy and the bomb preset an internal energy, and why the two differ by the pressure–volume work of any gas produced or consumed, Δ(pV) ≈ Δn_gas·R·T for an ideal gas. The second law then makes the calorimeter an instrument for measuring entropy: integrating C_p/T from a temperature near 0 K, with each latent heat divided by its transition temperature, gives the absolute entropy S° of the [[Third_law_of_thermodynamics|third law]], and those integrals are the S° column of every thermodynamic table.[^af2e-ch12] Calorimetry supplies the data; [[Thermodynamics|thermodynamics]] supplies the relations that turn measured heat into ΔU, ΔH and S and, through G = H − TS, into the prediction of what will react and how far. ## Special interest of thermodynamics in calorimetry: the isothermal segments of a Carnot cycle Thermodynamics needs calorimetry for its definition of temperature. In a [[Carnot_cycle|Carnot cycle]] the working body absorbs heat Q_H on an isotherm at T_H and rejects Q_L on an isotherm at T_L, and because the entropy change on each isotherm is the same, Q_H = T_H·(S₂ − S₁) and Q_L = T_L·(S₂ − S₁), so Q_H/Q_L = T_H/T_L.[^likharev-carnot] The ratio of the two isothermal heats, which are calorimetric quantities, defines the ratio of the two absolute temperatures independently of any thermometric substance; that is Kelvin's thermodynamic scale, and the efficiency ceiling η = 1 − T_L/T_H follows from it.[^likharev-carnot][^yan-carnot] For an ideal gas the heat on the isotherm is the integral of the latent heat with respect to volume, Q = ∫Λ_V dV = nRT·ln(V₂/V₁), which is how the classical coefficients enter the cycle. ## Special interest of calorimetry in thermodynamics: relations between classical calorimetric quantities Calorimetry, in turn, needs thermodynamics to make its four coefficients into two. The second law's integrating factor turns two calorimetric quantities that cannot be measured directly — the latent heats Λ_V and Λ_p — into derivatives of the equation of state, which can. ### Relation of latent heat with respect to volume, and the equation of state Requiring δq/T to be an exact differential gives Λ_V = T·(∂p/∂T)_V: the heat absorbed on isothermal expansion equals the absolute temperature times the pressure coefficient at constant volume. For an ideal gas the right-hand side is p, so all the heat absorbed on an isotherm is spent as work and the internal energy does not change, which is Joule's result for the free expansion of air. Applied across a phase boundary, the same relation becomes the [[Clausius–Clapeyron_relation|Clausius–Clapeyron equation]], dp/dT = L/(T·Δv), which ties the latent heat measured in a calorimeter to the slope of the coexistence curve on the [[Phase_diagram|phase diagram]]; for water the slope passes through the [[Triple_point|triple point]] at 0.612 kPa and 273.16 K.[^likharev-latent] ### Difference of specific heats Combining Λ_V = T·(∂p/∂T)_V with C_p − C_V = Λ_V·(∂V/∂T)_p gives the general result C_p − C_V = T·(∂p/∂T)_V·(∂V/∂T)_p = T·V·α²/κ_T. For an ideal gas it reduces to C_p − C_V = nR, so that the measured C_V/R of 1.50 for [[Helium|helium]], [[Neon|neon]] and [[Argon|argon]] implies C_p/R = 2.50 (derived), and C_V = (d/2)·R counts the [[Degrees_of_freedom_(physics_and_chemistry)|degrees of freedom]] d of the molecule.[^likharev-cpcv][^up2-cv] Because α is small for condensed matter the difference is a small fraction of C_p for liquids and solids, which is the justification for the single 4.184 J/(g·K) that the coffee-cup preset uses. ## Practical constant-volume calorimetry (bomb calorimetry) for thermodynamic studies The sim's third preset is the bomb. A weighed sample is sealed in a steel vessel with oxygen under pressure, the bomb is submerged in a known mass of water inside an insulated jacket, and the sample is ignited electrically; the heat capacity of the assembly is found beforehand by burning a standard such as benzoic acid, or by electrical heating.[^af2e-9-2] Because the volume is fixed, no expansion work is done and the measured heat is ΔU of combustion; the enthalpy follows from ΔH = ΔU + Δn_gas·R·T, a correction of a few kJ/mol against the thousands released.[^af2e-9-3] The Portal Book's worked example is the sim's default: burning 3.12 g of glucose raises the temperature of a bomb containing 775 g of water from 23.8 °C to 35.6 °C, the bomb itself having a heat capacity of 893 J/K. The total heat capacity is 775 g × 4.184 J/(g·K) + 893 J/K = 4,136 J/K, so the combustion released q = 4,136 J/K × 11.8 K ≈ 48.8 kJ, or 15.6 kJ per gram (derived); per mole of glucose (180.2 g/mol) that is about 2,820 kJ, within one percent of the tabulated heat of combustion of about 2,800 kJ/mol.[^af2e-9-2][^nist-glucose] In the sim the slider sets the sample mass, the thermometer answers through the same q = C_cal·ΔT, and the temperature rise scales with mass while the kJ/g readout stays put. Bomb calorimetry is how the [[Heat_of_combustion|heats of combustion]] of [[Fuel|fuels]] and the [[Food_energy|energy content of foods]] are determined — the nutritional calorie is a kilocalorie of exactly this heat, corrected for the fraction the body cannot oxidise — and it is the source of most standard enthalpies of formation of organic compounds, which are obtained by combining measured combustion enthalpies through [[Hess's_law|Hess's law]].[^af2e-9-3] ## See also - [[Enthalpy]] - [[Standard_enthalpy_of_reaction]] - [[Exothermic_reaction]] - [[Endothermic_process]] - [[Specific_heat_capacity]] - [[Calorimeter]] - [[Hess's_law]] - [[Heat_capacity]] - [[Thermochemistry]] - [[Latent_heat]] ## References [^black1803]: Black, Joseph (1803). *Lectures on the Elements of Chemistry, Delivered in the University of Edinburgh*. Edited by John Robison. Edinburgh: Mundell and Son. (Posthumous publication of the lectures in which latent and specific heat were introduced.) [^lavoisier-laplace]: Lavoisier, A. L.; Laplace, P. S. (1783). "Mémoire sur la chaleur." *Mémoires de l'Académie royale des sciences*, année 1780 (Paris, 1784). [^rumford1798]: Thompson, Benjamin, Count Rumford (1798). "An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction." *Philosophical Transactions of the Royal Society of London* 88: 80–102. [^joule1850]: Joule, J. P. (1850). "On the Mechanical Equivalent of Heat." *Philosophical Transactions of the Royal Society of London* 140: 61–82. [^hess1840]: Hess, H. (1840). "Thermochemische Untersuchungen." *Annalen der Physik und Chemie* 126 (3): 385–404. [^berthelot1897]: Berthelot, M. (1897). *Thermochimie: données et lois numériques*. 2 vols. Paris: Gauthier-Villars. [^bryan1907]: Bryan, G. H. (1907). *Thermodynamics: An Introductory Treatise Dealing Mainly with First Principles and Their Direct Applications*. Leipzig: B. G. Teubner. (The classical calorimetric coefficients C_V, C_p, Λ_V, Λ_p and their relations.) [^af2e-9-2]: Flowers, Paul; Neth, Edward; Robinson, William; et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 9 "Thermochemistry", §9.2 "Calorimetry" — the coffee-cup HCl/NaOH example and the glucose bomb example (chapter pp. 421–474; page to pin). https://openstax.org/books/chemistry-atoms-first-2e/pages/9-2-calorimetry — Portal Book 051, https://open.umn.edu/opentextbooks/textbooks/chemistry-atoms-first [^af2e-9-3]: Flowers et al. (2019), *Chemistry: Atoms First 2e*, Chapter 9, §9.3 "Enthalpy" (q_p = ΔH; Hess's law; enthalpies of combustion and formation; chapter pp. 421–474; page to pin). Portal Book 051. [^af2e-ch12]: Flowers et al. (2019), *Chemistry: Atoms First 2e*, Chapter 12 "Thermodynamics", §12.2–12.3 (entropy and the third law; chapter pp. 597–622; page to pin). Portal Book 051. [^af2e-appg]: Flowers et al. (2019), *Chemistry: Atoms First 2e*, Appendix G "Standard Thermodynamic Properties for Selected Substances", pp. 1093–1110 (ΔH_f° of H₂O(l) and OH⁻(aq)). Portal Book 051. [^nbs-tables]: Wagman, D. D.; Evans, W. H.; Parker, V. B.; et al. (1982). "The NBS Tables of Chemical Thermodynamic Properties." *Journal of Physical and Chemical Reference Data* 11, Supplement 2 (ΔH_f° of NH₄NO₃(cr), NH₄⁺(aq) and NO₃⁻(aq); the +25.7 kJ/mol enthalpy of solution is derived from them). [^nist-water]: NIST Chemistry WebBook, SRD 69. Water (CAS 7732-18-5), phase-change data: enthalpy of fusion 6.01 kJ/mol at 273.15 K. https://webbook.nist.gov/chemistry/ [^nist-glucose]: NIST Chemistry WebBook, SRD 69. D-Glucose (CAS 50-99-7), condensed-phase thermochemistry data: standard enthalpy of combustion of the solid. https://webbook.nist.gov/chemistry/ [^likharev-latent]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 4 "Phase Transitions", pp. 107 and 113 (water triple point 0.612 kPa, 273.16 K; latent heat of vaporization 2.2×10⁶ J/kg ≈ 0.4 eV per molecule). Portal Book 075. [^likharev-thermo]: Likharev (2013), *Essential Graduate Physics, Part SM*, Chapter 1 "Review of Thermodynamics", pp. 10–12 (dE = T dS − P dV; entropy as a state function). Portal Book 075. [^likharev-carnot]: Likharev (2013), *Essential Graduate Physics, Part SM*, Chapter 1, pp. 22–24 (Carnot cycle: Q_H = T_H(S₂ − S₁), Q_L = T_L(S₂ − S₁), η = 1 − T_L/T_H). Portal Book 075. [^likharev-cpcv]: Likharev (2013), *Essential Graduate Physics, Part SM*, Chapter 1, p. 20 (C_P − C_V = N for the ideal gas in energy units). Portal Book 075. [^yan-carnot]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*. Chapter 6 "Entropy and the Second Law of Thermodynamics", p. 272 (Carnot efficiency and the thermodynamic temperature scale). Portal Book 115, https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics [^up2-cv]: Sanny, Jeff; Ling, Samuel J.; et al. (2016). *University Physics Volume 2*. OpenStax. Chapter 2 "The Kinetic Theory of Gases", pp. 97–98 (C_V = (d/2)R; measured C_V/R of He, Ne, Ar = 1.50). Portal Book 078. [^up2-gallium]: Sanny, Ling et al. (2016), *University Physics Volume 2*, Chapter 2, Example 2.9, pp. 99–100 (gallium calorimetry, L_f = 80.2 kJ/kg). Portal Book 078. ## Books - Flowers, Neth, Robinson et al., *Chemistry: Atoms First 2e* (OpenStax, 2019), Chapter 9 "Thermochemistry" — Portal Book 051. - Ball, *Introductory Chemistry* (2011), Chapter 7 "Energy and Chemistry", pp. 317–368 — Portal Book 056, https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry - Sanny and Ling, *University Physics Volume 2* (OpenStax, 2016), Chapter 2 — Portal Book 078. - Likharev, *Essential Graduate Physics, Part SM* (2013), Chapters 1 and 4 — Portal Book 075. - Yan, *Introduction to Engineering Thermodynamics* (2022), Chapters 4 and 6 — Portal Book 115. - Bryan, *Thermodynamics* (Teubner, 1907); Callen, *Thermodynamics and an Introduction to Thermostatistics*, 2nd ed. (Wiley, 1985); Truesdell, *The Tragicomical History of Thermodynamics 1822–1854* (Springer, 1980). ## External links - [NIST Chemistry WebBook](https://webbook.nist.gov/chemistry/), phase-change and combustion data by species - [Chemistry: Atoms First 2e](https://openstax.org/details/books/chemistry-atoms-first-2e) at OpenStax - The Wikipedia pair's external links list further open resources <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Calorimetry.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Calorimetry* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Calorimetry.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Calorimetry.html" data-title="Calorimetry"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Calorimetry) : [Wikitube](https://en.wikitube.io/wiki/Calorimetry) · pinned revision [1366292308](https://en.wikipedia.org/w/index.php?oldid=1366292308) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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