A piece of iron of mass `100g` is kept inside a furnace for a long time put in a calorimeter of water equivalent `10g` containing `240g` of water at `20^(@)C` The mixture attains an equilibrium temperature of `60^(@)C` Find the temperature of the furnace specific heat capacity of iron `= 470J kg^(-1)C^(-1)`

Explanation

To find the temperature of the furnace, we can use the principle of conservation of energy, which states that the heat lost by the iron will be equal to the heat gained by the water and the calorimeter.

### Step 1: Identify the given data
- Mass of iron, \( M_{\text{iron}} = 100 \, \text{g} = 0.1 \, \text{kg} \)
- Water equivalent of calorimeter, \( m_{\text{cal}} = 10 \, \text{g} = 0.01 \, \text{kg} \)
- Mass of water, \( M_{\text{water}} = 240 \, \text{g} = 0.24 \, \text{kg} \)
- Initial temperature of water and calorimeter, \( T_i = 20^\circ C \)
- Final equilibrium temperature, \( T_f = 60^\circ C \)
- Specific heat capacity of iron, \( C_{\text{iron}} = 470 \, \text{J/kg} \cdot \text{°C} \)

### Step 2: Write the heat lost by iron
The heat lost by the iron can be expressed as:
\[ Q_{\text{lost}} = M_{\text{iron}} \cdot C_{\text{iron}} \cdot (T_i - T_f) \]
Substituting the values:
\[ Q_{\text{lost}} = 0.1 \cdot 470 \cdot (T - 60) \]

### Step 3: Write the heat gained by water and calorimeter
The heat gained by the water and calorimeter can be expressed as:
\[ Q_{\text{gained}} = (M_{\text{water}} + m_{\text{cal}}) \cdot C_{\text{water}} \cdot (T_f - T_i) \]
Where \( C_{\text{water}} = 4200 \, \text{J/kg} \cdot \text{°C} \). Substituting the values:
\[ Q_{\text{gained}} = (0.24 + 0.01) \cdot 4200 \cdot (60 - 20) \]
Calculating this:
\[ Q_{\text{gained}} = 0.25 \cdot 4200 \cdot 40 \]

### Step 4: Set heat lost equal to heat gained
Since heat lost by iron equals heat gained by water and calorimeter:
\[ 0.1 \cdot 470 \cdot (T - 60) = 0.25 \cdot 4200 \cdot 40 \]

### Step 5: Solve for \( T \)
Calculating the right side:
\[ 0.25 \cdot 4200 \cdot 40 = 42000 \]
Now we have:
\[ 47 \cdot (T - 60) = 42000 \]
Dividing both sides by 47:
\[ T - 60 = \frac{42000}{47} \]
Calculating \( \frac{42000}{47} \):
\[ T - 60 \approx 893.62 \]
Adding 60 to both sides:
\[ T \approx 893.62 + 60 = 953.62 \]

### Final Answer
The temperature of the furnace is approximately \( 953.62^\circ C \).

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