Calorimetry

Result

Calorimetry

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Statement

Heat exchanged during a reaction is measured calorimetrically via the temperature change of a surrounding medium of known heat capacity, \(q=mc\Delta T\). At constant volume (a sealed, rigid bomb calorimeter), no expansion work is possible, so the measured heat equals the internal energy change exactly: \(q_v=\Delta U\). At constant pressure (an open, coffee-cup-style calorimeter), the measured heat equals the enthalpy change exactly: \(q_p=\Delta H\) — the very definition \(H\equiv U+PV\) exists specifically because this combination is what a constant-pressure calorimeter measures directly.

Why it matters

Every tabulated formation enthalpy used in enthalpy-of-formation, and every combustion enthalpy referenced throughout this unit and the network more broadly, ultimately traces back to a calorimetric measurement of exactly this kind. This result also finally makes explicit a distinction used implicitly throughout the unit: enthalpy \(\Delta H\) and internal energy \(\Delta U\) are not the same quantity, and which one a given calorimeter measures depends entirely on whether pressure or volume is held constant during the measurement.

Hypotheses

The first law of thermodynamics holds: \(\Delta U=q+w\), where \(w\) is work done on the system.This is energy conservation applied to a thermodynamic system: any change in internal energy must be accounted for entirely by heat exchanged with the surroundings and work done on or by the system, with nothing left over. At constant external pressure, the only work exchanged is expansion/compression work, \(w=-P\Delta V\).Other forms of work (electrical, for instance) are assumed absent or separately accounted for; for an ordinary chemical reaction in an open vessel exposed to constant atmospheric pressure, expansion or compression against that constant external pressure is the only work term present.

Proof

\text{Constant volume: } w=-P\Delta V=0 \ (\Delta V=0) \ \Longrightarrow\ \Delta U = q_v

In a sealed, rigid bomb calorimeter, the reaction vessel's volume cannot change, so no expansion work is possible regardless of how much gas is produced or consumed; the first law (Hypotheses) then reduces directly to \(\Delta U=q_v\). A

\text{Constant pressure: } \Delta U = q_p - P\Delta V \ \Longrightarrow\ q_p = \Delta U + P\Delta V

Substituting the constant-pressure work term (Hypotheses) into the first law and solving for \(q_p\). A

\text{At constant } P: \ P\Delta V = \Delta(PV) \ \Longrightarrow\ q_p = \Delta U + \Delta(PV) = \Delta(U+PV) \equiv \Delta H

Since pressure is constant throughout, \(P\Delta V\) equals \(\Delta(PV)\) exactly; substituting into Step 2 and recognising \(U+PV\) as precisely the defined quantity enthalpy, \(H\), shows \(q_p=\Delta H\) exactly. This is, in fact, why enthalpy is defined the way it is: \(H\equiv U+PV\) specifically because this combination is what an ordinary, constant-pressure (open-to-the-atmosphere) calorimetric measurement directly reports. A

\Delta H = \Delta U + \Delta n_{\text{gas}}\,RT

Relating a bomb calorimeter's measured \(\Delta U\) to the more commonly tabulated \(\Delta H\): using the ideal gas law (ideal-gas-law) for the gas-phase contribution to \(\Delta(PV)\), and treating solids' and liquids' volume changes as negligible by comparison, \(\Delta(PV)\approx\Delta(n_{\text{gas}}RT)=RT\,\Delta n_{\text{gas}}\) at fixed temperature, where \(\Delta n_{\text{gas}}\) is the change in moles of gas-phase species between products and reactants. A

Result

q_v = \Delta U\ (\text{bomb, constant }V), \qquad q_p = \Delta H\ (\text{open, constant }P), \qquad \Delta H = \Delta U + \Delta n_{\text{gas}}RT

Reading. Which thermodynamic quantity a calorimeter measures directly depends entirely on whether the measurement is made at constant volume or constant pressure; a bomb calorimeter's raw result requires one further, precisely calculable correction to obtain the more commonly tabulated enthalpy change.

Scope. The \(\Delta n_{\text{gas}}RT\) correction can be small (even exactly zero, when gas moles are unchanged by the reaction) or a meaningful fraction of the total reaction enthalpy, depending on how much the moles of gas-phase species change during the specific reaction (Worked examples, Problems).

Corollaries & converses

For a reaction where gas-phase moles are unchanged (\(\Delta n_{\text{gas}}=0\), e.g. glucose combustion, \(\text{C}_6\text{H}_{12}\text{O}_6(s)+6\text{O}_2(g)\to6\text{CO}_2(g)+6\text{H}_2\text{O}(l)\), where \(6\) mol \(\text{O}_2\) consumed exactly balances \(6\) mol \(\text{CO}_2\) produced), the bomb-calorimeter correction vanishes entirely and \(\Delta H=\Delta U\) exactly, with no correction needed at all.
Reactions in solution, where volume changes are typically very small compared with gas-phase reactions, generally have \(\Delta H\approx\Delta U\) to good approximation even without an explicit \(\Delta n_{\text{gas}}\) correction — the distinction between the two matters most specifically for reactions that produce or consume significant amounts of gas.
Converse: given an independently known, tabulated \(\Delta H\) for a reaction and its \(\Delta n_{\text{gas}}\), the corresponding bomb-calorimeter \(\Delta U\) (and hence the expected \(q_v\)) can be predicted in advance, useful for calibrating or checking a calorimetric measurement against known reference data (Worked examples).

Fails without

Treat a bomb calorimeter's measured \(q_v\) as though it were directly \(\Delta H\), without applying the Step 4 correction: for any reaction with \(\Delta n_{\text{gas}}\ne0\), this introduces a real, quantifiable error (Worked examples' benzoic acid case shows a genuine, if modest, discrepancy of about \(1.2\,\text{kJ/mol}\) for a typical combustion reaction) — the two quantities are only guaranteed identical when gas-phase moles happen not to change (Corollaries).
Assume constant pressure without verifying the calorimeter is actually open to a constant external pressure (rather than sealed and rigid): Step 2's derivation of \(q_p=\Delta H\) depends specifically on the constant-pressure work term \(w=-P\Delta V\); a sealed, constant-volume vessel instead satisfies Step 1's \(q_v=\Delta U\), a genuinely different quantity.

Common errors

Reporting a bomb calorimeter's raw measured heat as \(\Delta H\) without the necessary \(\Delta n_{\text{gas}}RT\) correction (Fails without, first bullet).
Getting the sign of the correction wrong: \(\Delta H=\Delta U+\Delta n_{\text{gas}}RT\), so a reaction that consumes more gas moles than it produces (\(\Delta n_{\text{gas}}<0\)) has \(\Delta H\) slightly less negative (less exothermic) than \(\Delta U\), not more.
Forgetting the sign convention linking the calorimeter's own temperature change to the reaction's heat: heat released by an exothermic reaction is absorbed by the surrounding calorimeter medium (raising its temperature), so \(q_{\text{reaction}}=-q_{\text{calorimeter medium}}\), not the same sign.
Using the calorimeter's total heat capacity for the wrong mass (e.g. omitting the calorimeter vessel's own heat capacity in addition to the solution/water it contains, for setups where the vessel itself absorbs a non-negligible amount of heat).

Discussion

Calorimetry as a quantitative technique developed alongside the broader 18th- and 19th-century emergence of thermochemistry, with Antoine Lavoisier and Pierre-Simon Laplace's ice calorimeter (1780s) among the earliest systematic instruments, well before the first law of thermodynamics was formally established in the 1840s (the same historical gap already noted in hess-law). The modern bomb calorimeter, a sealed, rigid, high-pressure vessel allowing complete, controlled combustion of a sample, was developed in the late 19th century and remains the standard instrument for precisely measuring combustion enthalpies; benzoic acid, used as the Worked example here, remains the standard calibration reference substance for bomb calorimeters in real laboratory practice today, precisely because its combustion enthalpy is known to very high precision.

The distinction between \(\Delta H\) and \(\Delta U\) is often small in absolute terms (a few kilojoules per mole out of several thousand, as the benzoic acid example shows) but is not merely a pedantic technicality: precise thermochemical reference tables (the same tables underlying enthalpy-of-formation's tabulated values) are built specifically from carefully corrected bomb-calorimeter measurements, and omitting the correction systematically, even by a small percentage, would compound into meaningful inaccuracy across an entire reference table used by countless subsequent calculations.

Common misconception: that "heat measured by a calorimeter" is a single, unambiguous quantity always equal to enthalpy change. Which thermodynamic state function a calorimeter's raw measurement corresponds to depends entirely on the specific experimental conditions (constant volume versus constant pressure) under which that measurement was made, exactly as the Result's two distinct formulas make explicit.

Worked examples

\text{Coffee-cup: } 50.0\,\text{mL of }1.00\,\text{M HCl} + 50.0\,\text{mL of }1.00\,\text{M NaOH},\ \Delta T=+6.9^\circ\text{C},\ m=100.0\,\text{g},\ c=4.18\,\text{J/(g}\cdot^\circ\text{C)}

\(q_{\text{solution}}=mc\Delta T=100.0\times4.18\times6.9\approx2884\,\text{J}\); since the reaction released this heat into the solution, \(q_{\text{rxn}}=-2884\,\text{J}=-2.884\,\text{kJ}\). At constant (atmospheric) pressure, \(q_p=\Delta H\) directly (Step 3), with no correction needed. A

n = (0.0500\,\text{L})(1.00\,\text{mol/L}) = 0.0500\,\text{mol}\ (1{:}1\text{ stoichiometry}); \quad \Delta H_{\text{rxn}} = \frac{-2.884\,\text{kJ}}{0.0500\,\text{mol}} \approx -57.7\,\text{kJ/mol}

This closely matches the well-established standard enthalpy of neutralisation for a strong acid and strong base (\(\approx-57.3\,\text{kJ/mol}\)), a value that is essentially the same for any strong-acid/strong-base pair, since the net reaction is simply \(\text{H}^+(aq)+\text{OH}^-(aq)\to\text{H}_2\text{O}(l)\) regardless of the specific spectator ions involved. A

\Delta H_{\text{neutralisation}}\approx-57.7\,\text{kJ/mol}\ (\text{matches literature }\approx-57.3\,\text{kJ/mol})

Reading. A constant-pressure calorimetric measurement gives \(\Delta H\) directly, with no correction step needed, reproducing a well-known standard thermochemical value from simple temperature-change data.

Scope. This direct \(q_p=\Delta H\) relationship applies to any open, constant-atmospheric-pressure calorimetric measurement.

Problems

Benzoic acid combustion, \(\text{C}_6\text{H}_5\text{COOH}(s)+\tfrac{15}{2}\text{O}_2(g)\to7\text{CO}_2(g)+3\text{H}_2\text{O}(l)\), has a known standard \(\Delta H^\circ=-3227\,\text{kJ/mol}\) (the real, standard bomb-calorimeter calibration value). Compute \(\Delta n_{\text{gas}}\) and the corresponding \(\Delta U\) a bomb calorimeter should measure at \(298\,\text{K}\).
Solution
\(\Delta n_{\text{gas}}=7-\tfrac{15}{2}=-0.5\,\text{mol}\). \(\Delta U=\Delta H-\Delta n_{\text{gas}}RT=-3227-(-0.5)(8.314)(298)/1000=-3227+1.24=-3225.8\,\text{kJ/mol}\) — a small but genuine, precisely calculable \(1.2\,\text{kJ/mol}\) difference from \(\Delta H\), exactly the correction size referenced in Fails without.
A bomb calorimeter combustion of naphthalene, \(\text{C}_{10}\text{H}_8(s)+12\text{O}_2(g)\to10\text{CO}_2(g)+4\text{H}_2\text{O}(l)\), measures \(q_v=-5157\,\text{kJ/mol}\). Find the corresponding \(\Delta H\) at \(298\,\text{K}\).
Solution
\(\Delta n_{\text{gas}}=10-12=-2\,\text{mol}\). \(\Delta H=\Delta U+\Delta n_{\text{gas}}RT=-5157+(-2)(8.314)(298)/1000=-5157-4.96\approx-5162.0\,\text{kJ/mol}\) — here the correction (\(\approx5\,\text{kJ/mol}\)) is larger than in Problem 1, since naphthalene combustion has a larger \(|\Delta n_{\text{gas}}|\).
Explain, using the first law of thermodynamics directly, why a bomb calorimeter's sealed, rigid design guarantees it measures \(\Delta U\) rather than \(\Delta H\), regardless of how much gas is produced or consumed by the reaction inside it.
Solution
A sealed, rigid vessel cannot change volume no matter how much gas is generated internally (the rigid walls physically prevent expansion), so \(\Delta V=0\) always, and hence the expansion-work term \(w=-P\Delta V=0\) always as well (Step 1). By the first law, \(\Delta U=q+w=q_v+0=q_v\) exactly, regardless of \(\Delta n_{\text{gas}}\) — the rigidity of the container, not the specific reaction, is what fixes the calorimeter to measure \(\Delta U\).
Explain why glucose combustion (Corollaries, \(\Delta n_{\text{gas}}=0\)) is a case where a bomb calorimeter's raw \(q_v\) measurement can be reported directly as \(\Delta H\) without any correction, while benzoic acid combustion (Problem 1) cannot.
Solution
The correction term \(\Delta n_{\text{gas}}RT\) is exactly zero whenever gas-phase moles are unchanged by the reaction (\(\Delta n_{\text{gas}}=0\)), as for glucose combustion (\(6\) mol \(\text{O}_2\) consumed exactly balances \(6\) mol \(\text{CO}_2\) produced); in that specific case \(\Delta H=\Delta U+0=\Delta U\) exactly, and the raw bomb-calorimeter measurement already equals \(\Delta H\) with no further adjustment needed. Benzoic acid's combustion instead has \(\Delta n_{\text{gas}}=-0.5\ne0\) (Problem 1), so the correction term is genuinely nonzero and cannot be skipped without introducing the error quantified there.