`0.75` gram of petroleum was burnt in a bomb calorimeter which contains 2 kg of water and has a water equivalent 500 gram. The rise in temperature was `3^(@)C` . The calorific value of petroleum is `10^(n)` cal/g. Determine the value of n.

Explanation

To solve the problem, we need to calculate the total heat absorbed by the water and the calorimeter, and then use that to find the calorific value of petroleum.

### Step-by-Step Solution:

1. **Calculate the Heat Absorbed by Water:**
- The formula for heat absorbed (Q) is given by:
\[ Q = m \cdot c \cdot \Delta T \]
- Where:
- \( m \) = mass of water = 2 kg = 2000 g (since 1 kg = 1000 g)
- \( c \) = specific heat of water = 1 cal/g°C
- \( \Delta T \) = rise in temperature = 3°C
- Substituting the values:
\[ Q_{\text{water}} = 2000 \, \text{g} \cdot 1 \, \text{cal/g°C} \cdot 3 \, \text{°C} = 6000 \, \text{cal} \]

2. **Calculate the Heat Absorbed by the Calorimeter:**
- The heat absorbed by the calorimeter can be calculated using the water equivalent:
\[ Q_{\text{calorimeter}} = W \cdot \Delta T \]
- Where:
- \( W \) = water equivalent of the calorimeter = 500 g
- \( \Delta T \) = rise in temperature = 3°C
- Substituting the values:
\[ Q_{\text{calorimeter}} = 500 \, \text{g} \cdot 3 \, \text{°C} = 1500 \, \text{cal} \]

3. **Calculate Total Heat Absorbed:**
- Total heat absorbed by the system is the sum of heat absorbed by water and the calorimeter:
\[ Q_{\text{total}} = Q_{\text{water}} + Q_{\text{calorimeter}} = 6000 \, \text{cal} + 1500 \, \text{cal} = 7500 \, \text{cal} \]

4. **Calculate the Calorific Value of Petroleum:**
- The calorific value (CV) is defined as the total heat absorbed divided by the mass of the petroleum burnt:
\[ \text{CV} = \frac{Q_{\text{total}}}{\text{mass of petroleum}} = \frac{7500 \, \text{cal}}{0.75 \, \text{g}} = 10000 \, \text{cal/g} \]

5. **Express the Calorific Value in the Form \(10^n\):**
- We have found that the calorific value is \(10000 \, \text{cal/g}\), which can be expressed as:
\[ 10000 = 10^4 \, \text{cal/g} \]
- Therefore, \( n = 4 \).

### Final Answer:
The value of \( n \) is \( 4 \).