To solve the problem of determining the amount of ice remaining after mixing 500 g of ice at -5°C with 100 g of water at 20°C, we will follow these steps:
### Step 1: Calculate the heat required to raise the temperature of the ice to 0°C.
The formula to calculate the heat required (q) is:
\[ q = m \cdot s \cdot \Delta T \]
Where:
- \( m \) = mass of ice = 500 g
- \( s \) = specific heat of ice = 0.5 cal/g°C
- \( \Delta T \) = change in temperature = 0 - (-5) = 5°C
Calculating:
\[ q = 500 \, \text{g} \cdot 0.5 \, \text{cal/g°C} \cdot 5 \, \text{°C} = 1250 \, \text{cal} \]
### Step 2: Calculate the heat released by the water when it cools to 0°C.
Using the same formula:
\[ q = m \cdot s \cdot \Delta T \]
Where:
- \( m \) = mass of water = 100 g
- \( s \) = specific heat of water = 1 cal/g°C
- \( \Delta T \) = change in temperature = 20°C - 0°C = 20°C
Calculating:
\[ q = 100 \, \text{g} \cdot 1 \, \text{cal/g°C} \cdot 20 \, \text{°C} = 2000 \, \text{cal} \]
### Step 3: Determine the heat balance.
The ice requires 1250 cal to reach 0°C, and the water releases 2000 cal when it cools to 0°C.
Since the water releases more heat than the ice requires:
- Heat available after ice reaches 0°C:
\[ 2000 \, \text{cal} - 1250 \, \text{cal} = 750 \, \text{cal} \]
### Step 4: Calculate how much ice can be converted to water using the remaining heat.
The latent heat of fusion of ice is 80 cal/g. The amount of ice (m) that can be converted to water using the remaining heat is given by:
\[ q = m \cdot L_f \]
Where:
- \( L_f \) = latent heat of fusion = 80 cal/g
Rearranging for m:
\[ m = \frac{q}{L_f} = \frac{750 \, \text{cal}}{80 \, \text{cal/g}} = 9.375 \, \text{g} \]
### Step 5: Calculate the remaining amount of ice.
Initial mass of ice = 500 g
Mass of ice converted to water = 9.375 g
Remaining mass of ice:
\[ \text{Remaining ice} = 500 \, \text{g} - 9.375 \, \text{g} = 490.625 \, \text{g} \]
### Final Answer:
The amount of ice remaining in the mixture at equilibrium is approximately **490.63 g**.
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