Calorimetry

Every substance requires a specific amount of energy to raise its temperature by 1 °C (or 1 K). This is called the specific heat capacity, $c$, measured in J g$^{-1}$ K$^{-1}$. For liquid water, $c = 4.184$ J g$^{-1}$ K$^{-1}$ — one of the highest values of any common substance, which is why water is such an effective heat absorber in calorimeters. The total heat absorbed or released by the calorimeter is $q = mc\Delta T$, where $m$ is mass and $\Delta T = T_{\text{final}} - T_{\text{initial}}$.

A simple coffee-cup calorimeter consists of two nested polystyrene cups (excellent insulators), a lid, a thermometer, and a stirrer. Reactants are dissolved or mixed in a measured mass of water. Because the system is open to atmospheric pressure, $q_p = \Delta H$. The heat gained by the solution equals the heat lost by the reaction: $q_{\text{rxn}} = -q_{\text{solution}} = -mc\Delta T$. For example, dissolving $\ce{NaOH}$ in water releases heat ($\Delta T > 0$, exothermic, $q_{\text{rxn}} < 0$).

A bomb calorimeter encloses the reaction in a thick-walled steel vessel (the 'bomb') submerged in a water bath. Because volume is constant, no expansion work occurs ($w = -P_{\text{ext}}\Delta V = 0$), so $q_V = \Delta U$. The entire apparatus — bomb, water, stirrer, thermometer — has a combined heat capacity called the calorimeter constant, $C_{\text{cal}}$, measured in kJ K$^{-1}$. The heat released is $q_{\text{rxn}} = -C_{\text{cal}}\Delta T$. Bomb calorimeters are used for combustion reactions such as burning glucose: $\ce{C6H12O6 + 6O2 -> 6CO2 + 6H2O}$.

The relationship between internal energy and enthalpy is $\Delta H = \Delta U + \Delta(PV)$. For ideal gases at constant temperature, $\Delta(PV) = \Delta n_{\text{gas}} RT$, where $\Delta n_{\text{gas}}$ is the change in moles of gas during the reaction and $R = 8.314$ J mol$^{-1}$ K$^{-1}$. Thus $\Delta H = \Delta U + \Delta n_{\text{gas}} RT$. For many reactions in solution, $\Delta n_{\text{gas}} = 0$ and $\Delta H \approx \Delta U$, but for combustion reactions with significant gas changes this correction is essential.

Before measuring an unknown reaction, the calorimeter constant $C_{\text{cal}}$ must be determined by burning a standard compound of known internal energy of combustion. Benzoic acid ($\ce{C6H5COOH}$) is the international standard, with $\Delta U_{\text{comb}} = -3226.7$ kJ mol$^{-1}$. A precisely weighed sample is ignited electrically; measuring $\Delta T$ allows calculation of $C_{\text{cal}} = |q_{\text{rxn}}| / \Delta T$. This calibration step is what makes bomb calorimetry a primary thermodynamic measurement technique.

• q — Heat transferred (J or kJ): Positive when the substance gains heat (endothermic process); negative when it loses heat (exothermic process).
• m — Mass (g): Mass of the substance absorbing or releasing heat, typically the water or solution in the calorimeter.
• c — Specific heat capacity (J g⁻¹ K⁻¹): Intrinsic property of the substance; for liquid water c = 4.184 J g⁻¹ K⁻¹.
• \Delta T — Change in temperature (K or °C): T_final minus T_initial. A positive value indicates the solution warmed (exothermic reaction); negative means the solution cooled (endothermic reaction).

Problem. A student dissolves 4.00 g of ammonium nitrate ($\ce{NH4NO3}$, molar mass = 80.04 g mol$^{-1}$) in 100.0 g of water at 22.0 °C in a coffee-cup calorimeter. The final temperature is 18.6 °C. Assuming the specific heat capacity of the solution is 4.184 J g$^{-1}$ K$^{-1}$ and the calorimeter absorbs negligible heat, calculate (a) $q_{\text{rxn}}$ in kJ and (b) $\Delta H_{\text{soln}}$ per mole of $\ce{NH4NO3}$ dissolved.

Answer. q_rxn = +1.48 kJ (endothermic); ΔH_soln = +29.6 kJ mol⁻¹

• First Law of Thermodynamics — Calorimetry is the experimental application of the First Law (ΔU = q + w); understanding that energy is conserved is the theoretical foundation of all calorimetric measurements.
• Enthalpy and Hess's Law — Coffee-cup calorimetry directly measures ΔH, and Hess's Law allows unmeasurable enthalpies to be calculated from calorimetric data of related reactions.
• Specific Heat Capacity and Thermal Energy — The equation q = mcΔT, which depends on specific heat capacity, is the core mathematical tool used in every calorimetric calculation.
• Bond Enthalpies — Calorimetric measurements of combustion reactions provide experimental bond enthalpy data, connecting thermodynamic measurement to molecular structure.