An icebox almost completely filled with ice at `0^(@)C` is dipped into a large volume of water at `20^(@)C` . The box has walls of surface area `2400cm^(2)` thickness `2.0mm` and thermal conductivity `0.06Wm^(-1)C^(-1)` . Calculate the rate at which the ice melts in the box. Latent heat of fusion of ice `=3.4xx10^(5)Jkg^(-1)` .

Explanation

To solve the problem, we need to calculate the rate at which ice melts in the icebox when it is dipped into water at a higher temperature. We will use the formula for heat conduction and the concept of latent heat of fusion.

### Step 1: Convert given values to SI units
- Surface area \( A = 2400 \, \text{cm}^2 = 2400 \times 10^{-4} \, \text{m}^2 = 0.24 \, \text{m}^2 \)
- Thickness \( B = 2.0 \, \text{mm} = 2.0 \times 10^{-3} \, \text{m} \)
- Thermal conductivity \( k = 0.06 \, \text{W/m°C} \)
- Temperature difference \( \Delta T = 20°C - 0°C = 20°C \)
- Latent heat of fusion of ice \( L = 3.4 \times 10^5 \, \text{J/kg} \)

### Step 2: Calculate the heat conducted through the icebox
Using the formula for heat conduction:
\[ Q = \frac{k \cdot A \cdot \Delta T}{B} \]
Substituting the values:
\[ Q = \frac{0.06 \, \text{W/m°C} \cdot 0.24 \, \text{m}^2 \cdot 20 \, \text{°C}}{2.0 \times 10^{-3} \, \text{m}} \]

### Step 3: Calculate the heat transfer rate
Calculating the above expression:
\[ Q = \frac{0.06 \cdot 0.24 \cdot 20}{2.0 \times 10^{-3}} = \frac{0.288}{0.002} = 144 \, \text{W} \]

### Step 4: Relate heat transfer to the melting of ice
The heat required to melt ice can be expressed as:
\[ Q = \dot{m} \cdot L \]
Where \( \dot{m} \) is the rate of mass melting (kg/s). Rearranging gives:
\[ \dot{m} = \frac{Q}{L} \]

### Step 5: Calculate the rate of melting
Substituting the values:
\[ \dot{m} = \frac{144 \, \text{W}}{3.4 \times 10^5 \, \text{J/kg}} = \frac{144}{340000} \approx 0.0004235 \, \text{kg/s} \]

### Step 6: Convert to kg/hour
To convert the rate from kg/s to kg/hour:
\[ \dot{m} \approx 0.0004235 \, \text{kg/s} \times 3600 \, \text{s/hour} \approx 1.525 \, \text{kg/hour} \]

### Final Result
The rate at which the ice melts in the box is approximately **1.53 kg/hour**.

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