One side of an iron plate one metre square is kept at 100°C and the other side is at 0°C. The thickness of the plate is 1 cm. If all the heat conducted across the plate in one minute is used in melting ice, calculate the amount of ice so melted. Thermal conductivity of iron `= 0.162` in C.G.S. unit L.H. of fusion of ice `= 80` cal/gm.

Explanation

To solve the problem step by step, we will calculate the amount of heat conducted through the iron plate and then determine how much ice can be melted with that heat.

### Step 1: Identify Given Values
- Area of the plate (A) = 1 m² = \(10^4 \, \text{cm}^2\)
- Temperature difference (\(\Delta T\)) = \(100°C - 0°C = 100°C\)
- Thickness of the plate (d) = 1 cm
- Thermal conductivity of iron (k) = 0.162 cal/cm·s·°C
- Latent heat of fusion of ice (L) = 80 cal/g

### Step 2: Calculate Heat Conducted in One Second
The formula for heat conducted per second (Q) through a material is given by:
\[ Q = \frac{k \cdot A \cdot \Delta T}{d} \]
Substituting the values:
\[ Q = \frac{0.162 \, \text{cal/cm·s·°C} \cdot 10^4 \, \text{cm}^2 \cdot 100 \, °C}{1 \, \text{cm}} \]

Calculating:
\[ Q = \frac{0.162 \cdot 10^4 \cdot 100}{1} = 162000 \, \text{cal/s} \]

### Step 3: Calculate Total Heat Conducted in One Minute
Since we need the total heat conducted in one minute (60 seconds), we multiply by 60:
\[ Q_{\text{total}} = Q \cdot 60 = 162000 \, \text{cal/s} \cdot 60 \, \text{s} = 9720000 \, \text{cal} \]

### Step 4: Calculate the Mass of Ice Melted
The heat used to melt ice can be calculated using the formula:
\[ Q_{\text{total}} = m \cdot L \]
Where \(m\) is the mass of ice melted. Rearranging gives:
\[ m = \frac{Q_{\text{total}}}{L} \]
Substituting the values:
\[ m = \frac{9720000 \, \text{cal}}{80 \, \text{cal/g}} = 121500 \, \text{g} \]

### Step 5: Convert Mass to Kilograms
To convert grams to kilograms:
\[ m_{\text{kg}} = \frac{121500 \, \text{g}}{1000} = 121.5 \, \text{kg} \]

### Final Answer
The amount of ice melted is approximately **121.5 kg**.

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