Heat of Solution Calorimetry theory

The integral heat of solution (ΔH_int or ΔH_soln) is the total enthalpy change that occurs when 1 mole of solute is dissolved in a specified finite amount of solvent (e.g., n₂ moles of solute in n₁ moles of water to form a solution of definite molality m). It represents the macroscopic heat evolved or absorbed: ΔH_int = q_p / n₂. In contrast, the differential heat of solution (ΔH_diff = ∂H/∂n₂) represents the infinitesimal enthalpy change when 1 mole of solute is added to an infinitely large volume of solution at a fixed concentration, such that the concentration remains unchanged. As concentration approaches infinite dilution (m ⟶ 0), the integral and differential heats of solution converge to the identical limiting value: lim(m ⟶ 0) ΔH_int = lim(m ⟶ 0) ΔH_diff = ΔH_soln°, known as the standard heat of solution at infinite dilution.

The enthalpy of solution is governed by the thermodynamic balance: ΔH_soln = ΔH_lattice + ΔH_hyd. For KNO₃, the crystal lattice energy required to dissociate the ions is ΔH_lattice = +687.0 kJ/mol, whereas the exothermic hydration enthalpy released upon solvating K⁺ and NO₃⁻ ions is ΔH_hyd = −652.1 kJ/mol. Because the lattice energy exceeds the hydration energy by +34.89 kJ/mol, dissolution is endothermic and absorbs heat from the solvent, lowering its temperature. Despite this positive enthalpy change, KNO₃ dissolves readily and spontaneously at 25 °C because the process is driven by a massive increase in entropy (ΔS_soln > 0). The breakdown of the rigid, highly ordered 3D crystal lattice into freely translating, randomly distributed aqueous ions creates substantial positional and conformational disorder (ΔS_soln ≈ +115 J/(mol·K)). At room temperature (T = 298.15 K), the entropic term T ΔS_soln ≈ (298.15)(0.115) = +34.3 kJ/mol compensates for the enthalpy, yielding a negative Gibbs free energy of dissolution: ΔG_soln = ΔH_soln − T ΔS_soln < 0.

Because the dissolving salt drops the calorimeter temperature below the surrounding room temperature (T(t) < T_room), heat spontaneously conducts and radiates inward from the room into the calorimeter according to Newton's law: dq/dt = +k · (T_room − T(t)). This inward heat leak warms the solution before complete dissolution is achieved, preventing the observed minimum temperature T_min from reaching the true adiabatic lower bound T_ad. By tracking the pre-period drift for 300 s and the post-period re-warming slope for 600 s, the post-dissolution cooling/warming line is extrapolated backward to the dissolution midpoint time t_mid (where 63.2% of the temperature drop occurred). The vertical distance between the extrapolated pre-period and post-period lines at t_mid yields the true corrected adiabatic temperature depression ΔT_drop, restoring 0.05 to 0.15 °C of lost temperature drop.

The total system heat capacity is the sum of the thermal capacities of all components that undergo the temperature drop: C_eff = m_water · c_p + C_cal. For 100.00 mL of pure water with density ρ = 1.000 g/mL and specific heat capacity c_p = 4.184 J/(g·K), the water component is m_water · c_p = 100.0 g × 4.184 J/(g·K) = 418.4 J/K. The calorimeter hardware contribution C_cal (the 'water equivalent' of the inner silvered glass wall, immersed thermistor sheath, and PTFE stirrer paddle) is 27.5 ± 0.5 J/K, determined by electrical calibration. Adding both contributions yields C_eff = 418.4 + 27.5 = 445.90 J/K = 0.44590 kJ/K. While dissolved salt slightly modifies the specific heat capacity of the resulting solution, this concentration-dependent variation is linear and is smoothly accommodated by multi-point linear regression.

While both salts dissolve endothermically, potassium nitrate (KNO₃) is far superior as a precision educational standard: (1) Non-Hygroscopic Nature: KNO₃ does not absorb moisture from ambient air during weighing, whereas ammonium nitrate (NH₄NO₃) is extremely deliquescent and rapidly liquefies if exposed to humidity. (2) Safety & Stability: KNO₃ is chemically stable, non-volatile, and safe to handle in solid form, whereas pure NH₄NO₃ is subject to strict regulatory oversight due to its explosive hazard and sensitivity to organic contamination. (3) Clean Physical Chemistry: KNO₃ forms simple spherical K⁺ and planar NO₃⁻ ions without secondary hydrolysis, providing a clean benchmark with literature ΔH_soln° = +34.89 kJ/mol.

In an individual dissolution trial, any slight difference between the temperature of the dry solid salt and the water solvent prior to addition introduces a systematic thermal bias: q_bias = m_salt · c_salt · (T_salt − T_water). Furthermore, mechanical paddle rotation continuously dissipates stirring energy into the liquid (q_stir ≈ 0.02 W). In our protocol across six salt masses (2.000 g to 7.000 g), the governing relation is ΔT_drop = [ ΔH_soln / (M_KNO3 · C_eff) ] · m + ΔT₀. In ordinary least-squares regression, constant thermal offsets are absorbed entirely into the baseline intercept ΔT₀, leaving the empirical slope m_slope completely unaffected. Inverting this slope via ΔH_soln = m_slope · M_KNO3 · C_eff isolates the true thermodynamic enthalpy with zero baseline distortion.

According to the van 't Hoff equation: d(ln K_sp)/dT = ΔH_soln° / (R T²). For salts with an endothermic heat of solution (ΔH_soln° > 0, such as KNO₃ where ΔH_soln° = +34.89 kJ/mol), the right-hand side is positive: d(ln K_sp)/dT > 0. Consequently, the solubility product K_sp increases steeply with increasing temperature. In the case of potassium nitrate, its solubility increases dramatically from 13.3 g per 100 g water at 0 °C to 31.6 g at 20 °C, and surges to 246 g at 100 °C. This steep solubility gradient is the direct thermodynamic consequence of its large endothermic heat of solution, explaining why KNO₃ is readily purified by recrystallization from hot water.