Question #42015

A solid of mass $M$ is heated to a temperature measured as $\theta_H$ by a thermometer and dropped into a calorimeter of combined heat capacity $C$ (including water), initially at a measured temperature of $0^\circ\text{C}$. The final steady temperature of the mixture, measured by the same thermometer, is $\theta$. The thermometer's scale is calibrated correctly at $0^\circ\text{C}$ but has a coefficient of linear expansion $\alpha$. During the mixing, the system loses heat to the surroundings (at $0^\circ\text{C}$) given by $Q_{\text{loss}} = \lambda T_f$, where $T_f$ is the true final temperature and $\lambda$ is a constant. Assuming $\alpha$ is very small, which expression gives the specific heat capacity $S$ of the solid, correct to first order in $\alpha$?