Mathematical Interpretation of Heat Loss in Liquids Based on Newton's Law of Cooling

What happens when a hot liquid is left exposed to cooler surroundings, and can mathematics accurately predict the rate at which it surrenders its thermal energy? This research addresses that question by examining Newton's Law of Cooling through controlled laboratory experiments on six liquid samples: distilled water, olive oil, ethanol, glycerol, saline solution, and milk. Each liquid was heated to a defined initial temperature and allowed to cool at ambient conditions (22.0 °C) while temperature readings were recorded at 30-second intervals over a 10-minute observation window. The primary objective was to determine the cooling constant k for each liquid and to evaluate how well the classical exponential decay model fits the observed temperature profiles. Material properties such as specific heat capacity, thermal conductivity, and viscosity were measured alongside the cooling data to identify which physical parameters most strongly influence the cooling rate. Results showed that ethanol exhibited the highest cooling constant (k = 0.0512 s⁻¹), while glycerol displayed the lowest (k = 0.0163 s⁻¹). A strong inverse correlation (r = −0.91) between viscosity and cooling rate was observed across the six samples. The coefficient of determination (R²) exceeded 0.97 for all six liquids, confirming that Newton's model provides an excellent first-order approximation under the conditions tested. Numerical simulations using a fourth-order Runge-Kutta method were conducted in parallel, and the simulated curves matched the empirical data within a root-mean-square error of less than 0.38 °C across all trials. Residual analysis indicated minor systematic deviations at early time points (t < 60 s), suggesting that convective effects may not be fully captured by the simple exponential framework. These findings reaffirm the practical reliability of Newton's Law for moderate temperature differentials and highlight the role of liquid viscosity as a dominant factor governing heat dissipation behavior. The work offers a straightforward yet rigorous approach suitable for undergraduate thermal physics education and for engineering contexts where rapid cooling estimates are needed.