The two opposite faces of a cubical piece of iron (thermal conductivity = 0.2 CGS units) are at `100^(@)C` and `0^(@)C` in ice. If the area of a surface is `4cm^(2)` , then the mass of ice melted in 10 minutes will b

Options

A. 30 gm
B. 300 gm
C. 5 gm
D. 50 gm

Answer

To solve the problem, we will use the formula for heat conduction and the properties of the materials involved. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have a cubical piece of iron with two opposite faces maintained at temperatures of 100°C and 0°C. We need to find the mass of ice melted in 10 minutes due to the heat conducted through the iron.

### Step 2: Use the Formula for Heat Conduction
The formula for heat transfer (Q) through conduction is given by:
\[ Q = \frac{k \cdot A \cdot (T_H - T_C)}{L} \cdot t \]
where:
- \( Q \) = heat transferred (in calories)
- \( k \) = thermal conductivity of the material (in CGS units)
- \( A \) = area of the surface (in cm²)
- \( T_H \) = hot temperature (in °C)
- \( T_C \) = cold temperature (in °C)
- \( L \) = length of the cube (in cm)
- \( t \) = time (in seconds)

### Step 3: Identify the Given Values
From the problem, we have:
- \( k = 0.2 \) (thermal conductivity in CGS units)
- \( A = 4 \, \text{cm}^2 \)
- \( T_H = 100 \, °C \)
- \( T_C = 0 \, °C \)
- \( t = 10 \, \text{minutes} = 10 \times 60 = 600 \, \text{seconds} \)

### Step 4: Calculate the Length of the Cube
Since the area \( A \) of one face of the cube is given as \( 4 \, \text{cm}^2 \), we can find the side length \( a \) of the cube:
\[ a^2 = 4 \implies a = 2 \, \text{cm} \]
Thus, the length \( L \) (which is the same as the side length of the cube) is:
\[ L = 2 \, \text{cm} \]

### Step 5: Substitute the Values into the Formula
Now we can substitute all the known values into the heat conduction formula:
\[ Q = \frac{0.2 \cdot 4 \cdot (100 - 0)}{2} \cdot 600 \]
Calculating the numerator:
\[ Q = \frac{0.2 \cdot 4 \cdot 100}{2} \cdot 600 \]
\[ = \frac{80}{2} \cdot 600 \]
\[ = 40 \cdot 600 = 24000 \, \text{calories} \]

### Step 6: Relate Heat to Mass of Ice Melted
The heat \( Q \) is also related to the mass of ice melted \( m \) by the equation:
\[ Q = m \cdot L_F \]
where \( L_F \) is the latent heat of fusion of ice, which is \( 80 \, \text{cal/g} \).

### Step 7: Calculate the Mass of Ice Melted
Rearranging the equation gives:
\[ m = \frac{Q}{L_F} = \frac{24000}{80} \]
Calculating the mass:
\[ m = 300 \, \text{grams} \]

### Final Answer
The mass of ice melted in 10 minutes is **300 grams**.

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