Question 11.3: Finding a Specific Heat Goal Solve a calorimetry problem inv...
Finding a Specific Heat
Goal Solve a calorimetry problem involving only two substances.
Problem A 125-g block of an unknown substance with a temperature of 90.0^{\circ} \mathrm{C} is placed in a Styrofoam cup containing 0.326 \mathrm{~kg} of water at 20.0^{\circ} \mathrm{C}. The system reaches an equilibrium temperature of 22.4^{\circ} \mathrm{C}. What is the specific heat, c_{x}, of the unknown substance if the heat capacity of the cup is neglected?
Strategy The water gains thermal energy Q_{\text {cold }}, while the block loses thermal energy Q_{\text {hot. }} Using Equation 11.3,
Q=m c\Delta T (11.3)
substitute expressions into Equation 11.4
Q_{\mathrm{cold}}=-Q_{\mathrm{hol}} (11.4)
and solve for the unknown specific heat, c_{x}.
Let T be the final temperature, and let T_{w} and T_{x} be the initial temperatures of the water and block, respectively. Apply Equations 11.3 and 11.4:
\begin{aligned} Q_{\text {cold }} & =-Q_{\text {hot }} \\ m_{w} c_{w}\left(T-T_{w}\right) & =-m_{x} c_{x}\left(T-T_{x}\right) \end{aligned}
Solve for c_{x} and substitute numerical values:
\begin{aligned} c_{x} & =\frac{m_{w} c_{w}\left(T-T_{w}\right)}{m_{x}\left(T_{x}-T\right)} \\ & =\frac{(0.326 \mathrm{~kg})\left(4190 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\right)\left(22.4^{\circ} \mathrm{C}-20.0^{\circ} \mathrm{C}\right)}{(0.125 \mathrm{~kg})\left(90.0^{\circ} \mathrm{C}-22.4^{\circ} \mathrm{C}\right)} \\ c_{x} & =388 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C} \end{aligned}
Remarks Comparing our results to values given in Table 11.1, the unknown substance is probably copper.
TABLE 11.1
Specific Heats of Some Materials at Atmospheric Pressure
| Substance | J/kg · °C | cal/g · °C |
| Aluminum | 900 | 0.215 |
| Beryllium | 1,820 | 0.436 |
| Cadmium | 230 | 0.055 |
| Copper | 387 | 0.0924 |
| Germanium | 322 | 0.077 |
| Glass | 837 | 0.200 |
| Gold | 129 | 0.0308 |
| Ice | 2,090 | 0.500 |
| Iron | 448 | 0.107 |
| Lead | 128 | 0.0305 |
| Mercury | 138 | 0.033 |
| Silicon | 703 | 0.168 |
| Silver | 234 | 0.056 |
| Steam | 2,010 | 0.480 |
| Water | 4,186 | 1.00 |