5g of water at `30^@C and 5 g of ice at -29^@C` are mixed together in a calorimeter. Find the final temperature of mixture. Water equivalent of calorimeter is negligible, specific heat of ice = `0.5 cal//g-^@C` and latent heat of ice `=80 cal//g`.

Explanation

To solve the problem of finding the final temperature when mixing 5g of water at 30°C with 5g of ice at -29°C, we will follow these steps:

### Step 1: Identify the given data
- Mass of water (m_w) = 5 g
- Initial temperature of water (T_w) = 30°C
- Mass of ice (m_i) = 5 g
- Initial temperature of ice (T_i) = -29°C
- Specific heat of water (C_w) = 1 cal/g°C
- Specific heat of ice (C_i) = 0.5 cal/g°C
- Latent heat of fusion of ice (L_f) = 80 cal/g

### Step 2: Calculate the heat lost by water as it cools down
The heat lost by the water when it cools from 30°C to the final temperature T can be calculated using the formula:
\[ Q_{lost} = m_w \cdot C_w \cdot (T_w - T) \]
Substituting the values:
\[ Q_{lost} = 5 \cdot 1 \cdot (30 - T) = 150 - 5T \]

### Step 3: Calculate the heat gained by ice to reach 0°C
First, the ice must warm up from -29°C to 0°C:
\[ Q_{gained1} = m_i \cdot C_i \cdot (0 - T_i) \]
Substituting the values:
\[ Q_{gained1} = 5 \cdot 0.5 \cdot (0 - (-29)) = 5 \cdot 0.5 \cdot 29 = 72.5 \text{ cal} \]

### Step 4: Calculate the heat gained by ice to melt into water
Next, the ice at 0°C will absorb heat to melt:
\[ Q_{gained2} = m_i \cdot L_f \]
Substituting the values:
\[ Q_{gained2} = 5 \cdot 80 = 400 \text{ cal} \]

### Step 5: Set up the heat balance equation
At thermal equilibrium, the heat lost by the water will equal the total heat gained by the ice:
\[ Q_{lost} = Q_{gained1} + Q_{gained2} \]
Substituting the expressions we derived:
\[ 150 - 5T = 72.5 + 400 \]
\[ 150 - 5T = 472.5 \]

### Step 6: Solve for T
Rearranging the equation:
\[ 150 - 472.5 = 5T \]
\[ -322.5 = 5T \]
\[ T = -\frac{322.5}{5} = -64.5°C \]

### Step 7: Analyze the result
Since the calculated temperature of -64.5°C is lower than the initial temperature of the ice (-29°C), it is not a feasible temperature for the system. Therefore, the final equilibrium temperature must be 0°C, where the ice melts completely.

### Conclusion
The final temperature of the mixture when 5g of water at 30°C is mixed with 5g of ice at -29°C is **0°C**.

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