To solve the problem of mixing 4 kg of ice at -20°C with 5 kg of water at 40°C, we need to calculate the heat exchange between the ice and the water until they reach thermal equilibrium. Here’s a step-by-step breakdown of the solution:
### Step 1: Calculate the heat gained by the ice to reach 0°C
The heat gained by the ice as it warms up from -20°C to 0°C can be calculated using the formula:
\[ Q_{\text{ice warming}} = m_{\text{ice}} \cdot S_{\text{ice}} \cdot \Delta T \]
Where:
- \( m_{\text{ice}} = 4 \, \text{kg} \)
- \( S_{\text{ice}} = 0.5 \, \text{kcal/kg°C} \)
- \( \Delta T = 0 - (-20) = 20°C \)
Calculating:
\[ Q_{\text{ice warming}} = 4 \, \text{kg} \cdot 0.5 \, \text{kcal/kg°C} \cdot 20°C = 40 \, \text{kcal} \]
### Step 2: Calculate the heat lost by the water as it cools down to 0°C
The heat lost by the water as it cools down from 40°C to 0°C can be calculated using the formula:
\[ Q_{\text{water cooling}} = m_{\text{water}} \cdot S_{\text{water}} \cdot \Delta T \]
Where:
- \( m_{\text{water}} = 5 \, \text{kg} \)
- \( S_{\text{water}} = 1 \, \text{kcal/kg°C} \)
- \( \Delta T = 40°C - 0 = 40°C \)
Calculating:
\[ Q_{\text{water cooling}} = 5 \, \text{kg} \cdot 1 \, \text{kcal/kg°C} \cdot 40°C = 200 \, \text{kcal} \]
### Step 3: Determine the net heat exchange
Now, we determine how much heat is available after the ice has warmed up to 0°C:
\[ Q_{\text{remaining}} = Q_{\text{water cooling}} - Q_{\text{ice warming}} \]
\[ Q_{\text{remaining}} = 200 \, \text{kcal} - 40 \, \text{kcal} = 160 \, \text{kcal} \]
### Step 4: Calculate the heat required to convert ice at 0°C to water at 0°C
The heat required to convert the ice at 0°C to water at 0°C is given by:
\[ Q_{\text{fusion}} = m_{\text{ice}} \cdot L_f \]
Where:
- \( L_f = 80 \, \text{kcal/kg} \)
Let \( m \) be the mass of ice that melts:
\[ Q_{\text{fusion}} = m \cdot 80 \, \text{kcal/kg} \]
Setting the heat gained from the water equal to the heat required for melting:
\[ 160 \, \text{kcal} = m \cdot 80 \, \text{kcal/kg} \]
Solving for \( m \):
\[ m = \frac{160 \, \text{kcal}}{80 \, \text{kcal/kg}} = 2 \, \text{kg} \]
### Step 5: Calculate the total mass of water in the equilibrium mixture
The total mass of water after the ice has melted is:
\[ \text{Total mass of water} = m_{\text{water}} + m_{\text{melted ice}} \]
\[ \text{Total mass of water} = 5 \, \text{kg} + 2 \, \text{kg} = 7 \, \text{kg} \]
### Conclusion
The final equilibrium mixture contains 7 kg of water.
