Question #77078

A solid sample of mass $m$ and unknown specific heat $c$ is heated to an initial measured temperature $T_1'$. It is then dropped into a calorimeter containing a liquid, with the system having a combined heat capacity $S$. The initial measured temperature of the calorimeter and the surrounding environment is $T_2'$. The mixture reaches a maximum measured temperature $T_3'$ after a time $t$. During this mixing phase, heat is lost to the surroundings according to Newton's Law of Cooling, with a cooling constant $k$ (in $\text{J} \cdot \text{K}^{-1} \cdot \text{s}^{-1}$). All temperatures are measured using a thermometer whose scale is printed on a metal strip with a coefficient of linear expansion $\alpha$. The thermometer is perfectly calibrated at $0^\circ\text{C}$, and the true temperature $T$ is strictly proportional to the absolute length of the thermometric liquid column. Assuming $\alpha$ is very small such that terms of order $\alpha^2$ and higher can be neglected, determine the specific heat $c$ of the sample.