One end of a copper rod of length 1 m and area of cross - section `400 xx (10^-4) m^2` is maintained at `100^@C`. At the other end of the rod ice is kept at `0^@C`. Neglecting the loss of heat from the surrounding, find the mass of ice melted in 1h. Given, `K_(Cu) = 401 W//m-K and L_f = 3.35 xx (10^5) J//kg.`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the rate of heat transfer through the copper rod.
We use the formula for heat conduction:

\[ \frac{dq}{dt} = \frac{K \cdot A \cdot (T_1 - T_2)}{L} \]

Where:
- \( K \) is the thermal conductivity of copper (\( 401 \, \text{W/m-K} \)),
- \( A \) is the area of cross-section (\( 400 \times 10^{-4} \, \text{m}^2 \)),
- \( T_1 \) is the temperature at one end (\( 100^\circ C \)),
- \( T_2 \) is the temperature at the other end (\( 0^\circ C \)),
- \( L \) is the length of the rod (\( 1 \, \text{m} \)).

Substituting the values:

\[ \frac{dq}{dt} = \frac{401 \times (400 \times 10^{-4}) \times (100 - 0)}{1} \]

Calculating this gives:

\[ \frac{dq}{dt} = 401 \times 400 \times 10^{-4} \times 100 \]

\[ = 401 \times 400 \times 10^{-4} \times 100 = 1604 \, \text{J/s} \]

### Step 2: Calculate the total heat transferred in one hour.
Since we need the heat transferred in one hour, we convert one hour to seconds:

\[ 1 \, \text{hour} = 60 \times 60 = 3600 \, \text{s} \]

Now, we can find the total heat transferred:

\[ Q = \frac{dq}{dt} \times t = 1604 \, \text{J/s} \times 3600 \, \text{s} \]

Calculating this gives:

\[ Q = 1604 \times 3600 = 5774400 \, \text{J} \]

### Step 3: Calculate the mass of ice melted using the heat transferred.
We use the formula relating the heat absorbed by the ice to the mass melted:

\[ Q = m \cdot L_f \]

Where:
- \( m \) is the mass of ice melted,
- \( L_f \) is the latent heat of fusion of ice (\( 3.35 \times 10^5 \, \text{J/kg} \)).

Rearranging the formula to find \( m \):

\[ m = \frac{Q}{L_f} = \frac{5774400}{3.35 \times 10^5} \]

Calculating this gives:

\[ m \approx \frac{5774400}{335000} \approx 17.24 \, \text{kg} \]

### Final Answer:
The mass of ice melted in one hour is approximately \( 17.24 \, \text{kg} \).

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