= Hess's law calorimetry experiment
{c}
A practical test compares three transformations involving <sodium hydroxide>: <dissolution> of the <solid>, <neutralization> of its <chemical solution> by <hydrochloric acid>, and reaction of the <solid> directly with <hydrochloric acid>. In the ideal compatible-state notation, the <hydrogen ion> and <hydroxide> give
$$
\begin{aligned}
D:&\quad\mathrm{NaOH(s)\longrightarrow Na^+(aq)+OH^-(aq)},\\
N:&\quad\mathrm{H^+(aq)+OH^-(aq)\longrightarrow H_2O(l)},\\
D+N:&\quad\mathrm{NaOH(s)+H^+(aq)\longrightarrow Na^+(aq)+H_2O(l)}.
\end{aligned}
$$
The <hydroxide> cancels, whereas the newly formed <water> remains. <Sodium> and <chloride> <spectator ions> must still be included in the full material inventory when matching actual <chemical solutions>.
The teaching protocol https://www.pc.maricopa.edu/chemistry/151LLWeb/caloimetry.htm[Calorimetry: Heat of Neutralization Reaction & Hess's Law], Procedure A–C, uses nested insulated cups, a cardboard lid and a <thermometer>. It mixes $75\,\mathrm{mL}$ each of $2.0\,\mathrm{mol\,L^{-1}}$ <hydrochloric acid> and <sodium hydroxide> <chemical solutions>; separately it adds $6.00\,\mathrm g$ <solid> <sodium hydroxide> to diluted <hydrochloric acid>, or to $150\,\mathrm{mL}$ <water>. It records initial <temperatures>, stirs, reads <temperature> every $30\,\mathrm s$, repeats runs, and extrapolates cooling trends to mixing time. Rapid weighing limits moisture uptake. The Introduction neglects vessel <heat capacity> and room losses; these are approximations, not a measured calibration. The readout tests whether the inferred <enthalpies> add, rather than directly measuring a <state function>.<Calorimetry: Heat of Neutralization Reaction & Hess's Law, Chemistry 151LL>[Introduction; Procedure A.1–8, B.1–5 and C.1–6; Additional Data Treatment: cup approximation, preparation, repeat runs and mixing-time extrapolation]{ref}
Matching the <chemical equations> alone is insufficient. The direct-acid recipe uses $0.160\,\mathrm{mol}$ <hydrochloric acid>, whereas the solution-mixing recipe uses $0.150\,\mathrm{mol}$. The <dissolution> run also does not automatically reproduce the <water> inventory of the mixed-solution run. Exact closure requires matching initial and final <chemical solution> compositions and adding any necessary mixing or <dilution> <enthalpies>. Agreement after approximate corrections is a limited test, rather than proof of exact cancellation for those recipes.
To improve the <calorimetry>, use <calorimeter calibration> for the participating hardware, and estimate <calorimeter heat-loss correction> under comparable conditions. With an approximately constant total receiving <heat capacity> $C_{\rm tot}=mc_p+C_{\rm cal}$, common initial <temperature>, constant external <pressure>, only expansion <work>, negligible escaping material and negligible bulk <kinetic energy> and <potential energy> changes, the balance is
$$
q_{\rm rxn}+C_{\rm tot}\Delta T+Q_{\rm loss,out}=0,\qquad
\Delta H_{\rm rxn}=q_{\rm rxn}.
$$
Here $m$ is the <chemical solution> <mass>, $c_p$ its <specific heat capacity at constant pressure>, $C_{\rm cal}$ the hardware <heat capacity>, and $Q_{\rm loss,out}$ the signed outward <heat> loss. Different initial <temperatures> require separate warming terms. <Calorimeter calibration> and matching <chemical solution> compositions address different errors.
For an original constructed model, choose $C_{\rm tot}=120\,\mathrm{J\,K^{-1}}$ and $\Delta T=3\,\mathrm K$. Neglect outward <heat> loss and other corrections. The receiving <chemical solution> and hardware gain $q_{\rm receiver}=360\,\mathrm J$, so $q_{\rm rxn}=-360\,\mathrm J$. If the completed <extent of reaction> is $\xi=0.020\,\mathrm{mol}$ for a unit-coefficient <chemical equation>, the inferred reaction <enthalpy> per <mole> of that reaction is
$$
\frac{\Delta H_{\rm rxn}}{\xi}
=\frac{-360\,\mathrm J}{0.020\,\mathrm{mol}}
=-18\,\mathrm{kJ\,mol^{-1}}.
$$
These chosen numbers illustrate the signs and normalization; they are not measured <sodium hydroxide> data and are independent of the abstract $A,B,C$ example.
\Image[/hess-law-cancellation-and-calorimetry.png]
{title=<Stoichiometry> cancels intermediate states; <calorimetry> converts warming into signed reaction <enthalpy>}
{id=image-hess-law-cancellation-and-calorimetry}
{description=Left: original abstract <chemical equations> use $\tfrac12R_1-R_2$ to give $A\to C$ and $-7\,\mathrm{kJ}$ for one <mole>. Right: an independent chosen <heat capacity> model gives receiver gain $+360\,\mathrm J$ and reaction change $-360\,\mathrm J$. Neither panel reports experimental measurements.}
A loss of <heat> to the room makes the uncorrected warming underestimate the magnitude of a heat-releasing reaction. Incomplete <dissolution>, incomplete reaction, uncertain <amount of substance>, or a slow <thermometer> response can also spoil closure. A useful experimental residual is $r=\Delta H_{\rm direct}-(\Delta H_D+\Delta H_N)$ after all changes have been placed on the same amount and state basis. Compare $r$ with the <measurement uncertainty>, accounting for shared calibration errors; averaging repeated runs reduces scatter without removing a common bias.
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