The final temperature of the system is 32.4 °C.
heat lost by Pb + heat gained by water = 0
[tex]q_{1} +q_{2} = 0[/tex]
[tex]m_{1}C_{1}\Delta T_{1} + m_{2}C_{2}\Delta T_{2} = 0[/tex]
[tex]q_{1}= m_{1}C_{1}\Delta T_{1} = 21.2 \:\text{g}\times 0.128\: \text{J}^{\circ}\text{C}^{-1}\text{g}\times (T_{f} -113\: ^{\circ}\text{C}) = 2.713T_{f}\: \text{J}^{\circ}\text{C}^{-1}\: -\: 306.6\:\text{J}\cdot\text{g}^{-1}\\[/tex]
[tex]q_{2}= m_{2}C_{2}\Delta T_{2} = 14.8 \:\text{g}\times 4.184\: \text{J}\cdot^{\circ}\text{C}^{-1}\text{g}^{-1}\times (T_{f} -28.9\: ^{\circ}\text{C}) = 61.9\: \Delta T_{f} \: \text{J}^{\circ}\text{C} \:- 1790 \: \text{J}\\[/tex]
Adding [tex]q_{1}[/tex] and [tex]q_{2}[/tex] and combinibg like terms, we get
[tex]168.0\text{T}_{f}\: ^{\circ}\text{C} ^{-1}\:- 2096 = 0]\\[/tex]
[tex]T_{f} = \frac{2096}{68.0 \: ^{\circ}\text{C}^{-1}} = 32.4 ^{\circ}\text{C}\\[/tex]
