Equal masses of ice `(at 0^@C)` and water are in contact. Find the temperature of water needed to just melt the complete ice.

Explanation

To solve the problem of finding the temperature of water needed to just melt the complete ice, we can follow these steps:

### Step 1: Understand the Problem
We have equal masses of ice at 0°C and water at an unknown temperature \( T \). The goal is to find the temperature \( T \) of the water required to completely melt the ice.

### Step 2: Identify Relevant Concepts
To melt ice, we need to provide it with heat, specifically the latent heat of fusion. The heat lost by the water will equal the heat gained by the ice.

### Step 3: Write the Heat Transfer Equation
The heat required to melt the ice can be expressed as:
\[ Q_{\text{ice}} = m \cdot L_f \]
where:
- \( m \) is the mass of the ice,
- \( L_f \) is the latent heat of fusion of ice (approximately 80 cal/g).

The heat lost by the water as it cools down can be expressed as:
\[ Q_{\text{water}} = m \cdot s \cdot (T - 0) \]
where:
- \( s \) is the specific heat of water (approximately 1 cal/g°C),
- \( T \) is the initial temperature of the water.

### Step 4: Set Up the Equation
Since the heat gained by the ice equals the heat lost by the water, we can set the two equations equal to each other:
\[ m \cdot L_f = m \cdot s \cdot T \]

### Step 5: Cancel Out the Mass
Since the masses are equal, we can cancel \( m \) from both sides:
\[ L_f = s \cdot T \]

### Step 6: Substitute Known Values
Substituting the known values:
\[ 80 \text{ cal/g} = 1 \text{ cal/g°C} \cdot T \]

### Step 7: Solve for \( T \)
Now, we can solve for \( T \):
\[ T = 80 \text{ °C} \]

### Final Answer
The temperature of the water needed to just melt the complete ice is \( 80 \text{ °C} \).

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