All Problems Phase Transformations
Problem 2.216
A chunk of ice of mass \(m_{1}=100 \mathrm{~g}\) at a temperature \(t_{1}=\) \(=0^{\circ} \mathrm{C}\) was placed in a calorimeter in which water of mass \(m_{2}\) \(=100 \mathrm{~g}\) was at a temperature \(t_{2}\). Assuming the heat capacity of the calorimeter to be negligible, find the entropy increment of the system by the moment the thermal equilibrium is reached. Consider two cases: (a) \(t_{2}=60^{\circ} \mathrm{C}\) (b) \(t_{2}=94^{\circ} \mathrm{C}\).
\begin{aligned} &\text { (a) When } m_{2} c_{2} t_{2}<m_{1} q, \text { not all the ice will melt and }\\ &\Delta S=m_{2} c_{2}\left(\frac{T_{2}}{T_{1}}-1-\ln \frac{T_{2}}{T_{1}}\right)=9.2 \mathrm{~J} / \mathrm{K} \end{aligned} (b) When \(m_{2} c_{2} t_{2}>m_{1} q,\) the ice will melt completely and \[ \begin{array}{r} \qquad \Delta S=\frac{m_{1} q}{T_{1}}+c_{2}\left(m_{1} \ln \frac{T}{T_{1}}-m_{2} \ln \frac{T_{2}}{T}\right)=18 \mathrm{~J} / \mathrm{K} \\ \text { where } T=\frac{m_{1} T_{1}+m_{2} T_{\mathrm{a}}-m_{1} q / c_{2}}{m_{1}+m_{3}} \end{array} \]