Question 11.5: Ice Water GOAL Solve a problem involving heat transfer and a...
Ice Water
GOAL Solve a problem involving heat transfer and a phase change from solid to liquid.
PROBLEM At a party, 6.00 kg of ice at -5.00°C is added to a cooler holding 30 liters of water at 20.0°C. What is the temperature of the water when it comes to equilibrium?
STRATEGY In this problem, it’s best to make a table. With the addition of thermal energy Q_{\text {ice }} the ice will warm to 0°C, then melt at 0°C with the addition of energy Q_{\text {melt }}. Next, the melted ice will warm to some final temperature T by absorbing energy Q_{\text {ice-water }}, obtained from the energy change of the original liquid water, Q_{\text {water }}. By conservation of energy, these quantities must sum to zero.
Calculate the mass of liquid water:
m_{\text {water }}=\rho_{\text {water }} V
=\left(1.00 \times 10^3 kg / m ^3\right)(30.0 L ) \frac{1.00 m ^3}{1.00 \times 10^3 L }
= 30.0 kg
Write the equation of thermal equilibrium:
\text { (1) } \quad Q_{\text {ice }}+Q_{\text {melt }}+Q_{\text {ice-water }}+Q_{\text {water }}=0
Construct a comprehensive table:
Substitute all quantities in the second through sixth columns into the last column and sum, which is the evaluation of Equation (1), and solve for T:
6.27 \times 10^4 J +2.00 \times 10^6 J
+\left(2.51 \times 10^4 J /{ }^{\circ} C \right)\left(T-0^{\circ} C \right)
+\left(1.26 \times 10^5 J /{ }^{\circ} C \right)\left(T-20.0^{\circ} C \right)=0
T = 3.03°C
REMARKS Making a table is optional. However, simple substitution errors are extremely common, and the table makes such errors less likely.
| Q | m (kg) | c (J/kg · °C) | L (J/kg) | T_f (°C) | T_i (°C) | Expression |
| Q_{\text {ice }} | 6.00 | 2 090 | 0 | -5.00 | m_{\text {ice }} c_{\text {ice }}\left(T_f-T_i\right) | |
| Q_{\text {melt }} | 6.00 | 3.33 × 10^5 | 0 | 0 | m_{\text {ice }} L_f | |
| Q_{\text {ice–water }} | 6.00 | 4 190 | T | 0 | m_{\text {ice }} c_{\text {water }}\left(T_f-T_i\right) | |
| Q_{\text {water }} | 30.0 | 4 190 | T | 20.0 | m_{\text {water }} c_{\text {water }}\left(T_f-T_i\right) |