Which is a good fuel `CH_(4)` or`C_(2)H_(6)` ?
The standard enthalpy of formation of `CH_(4), C_(2)H_(6), CO_(2)` and `H_(2)O` are `- 74.8, - 84.6 , - 393 . 5` and `-286kJ mol^(-1)` respectively.

Explanation

To determine which is a better fuel between \( CH_4 \) (methane) and \( C_2H_6 \) (ethane), we will calculate the heat of combustion for both compounds using the given standard enthalpy of formation values. The fuel with the higher calorific value (heat released per unit mass) will be considered the better fuel.

### Step-by-Step Solution:

1. **Write the Combustion Reactions:**
- For \( CH_4 \):
\[ CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O \]
- For \( C_2H_6 \):
\[ C_2H_6 + \frac{7}{2}O_2 \rightarrow 2CO_2 + 3H_2O \]

2. **Calculate the Heat of Combustion for \( CH_4 \):**
- Using the formula:
\[ \Delta H_{combustion} = \Sigma \Delta H_{products} - \Sigma \Delta H_{reactants} \]
- Products:
\[ \Delta H_{products} = \Delta H_{CO_2} + 2 \times \Delta H_{H_2O} = -393.5 + 2 \times (-286) = -393.5 - 572 = -965.5 \, kJ \]
- Reactants:
\[ \Delta H_{reactants} = \Delta H_{CH_4} + 2 \times \Delta H_{O_2} = -74.8 + 0 = -74.8 \, kJ \]
- Therefore:
\[ \Delta H_{combustion} = -965.5 - (-74.8) = -890.7 \, kJ \]

3. **Calculate the Molar Mass of \( CH_4 \):**
- Molar mass of \( CH_4 = 12 + 4 = 16 \, g/mol \)

4. **Calculate the Calorific Value of \( CH_4 \):**
\[ \text{Calorific Value} = \frac{\Delta H_{combustion}}{\text{Molar Mass}} = \frac{890.7 \, kJ}{16 \, g/mol} \approx 55.66 \, kJ/g \]

5. **Calculate the Heat of Combustion for \( C_2H_6 \):**
- Products:
\[ \Delta H_{products} = 2 \times \Delta H_{CO_2} + 3 \times \Delta H_{H_2O} = 2 \times (-393.5) + 3 \times (-286) = -787 - 858 = -1645 \, kJ \]
- Reactants:
\[ \Delta H_{reactants} = \Delta H_{C_2H_6} + \frac{7}{2} \times \Delta H_{O_2} = -84.6 + 0 = -84.6 \, kJ \]
- Therefore:
\[ \Delta H_{combustion} = -1645 - (-84.6) = -1560.4 \, kJ \]

6. **Calculate the Molar Mass of \( C_2H_6 \):**
- Molar mass of \( C_2H_6 = 2 \times 12 + 6 = 30 \, g/mol \)

7. **Calculate the Calorific Value of \( C_2H_6 \):**
\[ \text{Calorific Value} = \frac{\Delta H_{combustion}}{\text{Molar Mass}} = \frac{1560.4 \, kJ}{30 \, g/mol} \approx 52.01 \, kJ/g \]

8. **Comparison of Calorific Values:**
- Calorific value of \( CH_4 \) is approximately \( 55.66 \, kJ/g \)
- Calorific value of \( C_2H_6 \) is approximately \( 52.01 \, kJ/g \)

### Conclusion:
Since \( CH_4 \) has a higher calorific value than \( C_2H_6 \), \( CH_4 \) is a better fuel.

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