Data-driven modeling in the introductory physics laboratory: Scaling analysis and data collapse in the specific heat of water experiment
In introductory physics laboratories, a central instructional goal is to help students construct and evaluate mathematical models from empirical data rather than applying given formulas. We present a data-driven redesign of the classic specific heat of water experiment that emphasizes scaling analysis and data collapse as tools for model construction. The activity combines structured experimental and analytical guidance with instructor-mediated questioning, while thermodynamic theory is deliberately postponed. Students collect temperature-time data under various experimental conditions, producing multiple data sets that initially appear unrelated. Through successive rescaling, students reduce the dimensionality of the variable space and achieve data collapse onto a single master curve, from which they formulate an empirical model relating energy input, mass, and temperature change. The analysis highlights a limitation of multiplicative scaling: the additive contribution of the calorimeter cannot be eliminated, leading to a structural non-identifiability of the subsystem contributions. To clarify the domain of validity of the model, a thermodynamic description is introduced a posteriori as a boundary-setting framework for interpreting the empirical model. In this sense, the central contribution of this work is to use data-driven modeling both to construct models and to reveal their intrinsic limitations. The experiment provides an accessible example of how scaling, data collapse, and theoretical reasoning can be integrated in an introductory laboratory.