The commercial production of water gas utilizes the reaction under standard conditions `:C+H_(2)O_((g)) rarr H_(2)+CO`. The heat required for this endothermic reaction may be supplied by adding a limited amount of air and burning some carbon to `CO_(20`. How many gram of carbon must be burnt to `CO_(2)` to provide enough heat for the water gas conversion of `100g` carbon ? Neglect all heat losses to the environment. Also `DeltaH_(f)^(@)` of `CO,H_(2)O_((g))` and `CO_(2)` are `-110.53, -241.81` and `-393.51kJ //mol` respectively.

Options

A. 30.5 g
B. 39.7 g
C. 33.36 g
D. 42.5 g

Explanation

To solve the problem, we need to determine how many grams of carbon must be burned to CO₂ to provide enough heat for the conversion of 100 grams of carbon into water gas (H₂ and CO).

### Step-by-Step Solution:

1. **Identify the Reaction and Enthalpy Change**:
The reaction for the production of water gas is:
\[ C + H_2O(g) \rightarrow H_2 + CO \]
The enthalpy change (ΔH) for this reaction can be calculated using the standard enthalpies of formation (ΔH_f^°) provided:
\[ \Delta H = \Delta H_f^{\text{CO}} + \Delta H_f^{\text{H}_2} - \Delta H_f^{\text{H}_2O} \]
Given values:
- ΔH_f^°(CO) = -110.53 kJ/mol
- ΔH_f^°(H₂O) = -241.81 kJ/mol
- ΔH_f^°(H₂) = 0 kJ/mol (as it is in its standard state)

Substituting the values:
\[ \Delta H = (-110.53) + 0 - (-241.81) = 131.28 \text{ kJ/mol} \]

2. **Calculate the Energy Required for 100 g of Carbon**:
First, we need to calculate the number of moles of carbon in 100 grams:
\[ \text{Moles of C} = \frac{100 \text{ g}}{12 \text{ g/mol}} = 8.33 \text{ mol} \]
The total energy required for the conversion of 100 g of carbon is:
\[ \text{Energy required} = \text{Moles of C} \times \Delta H = 8.33 \text{ mol} \times 131.28 \text{ kJ/mol} = 1094.1 \text{ kJ} \]

3. **Determine the Mass of Carbon Needed to be Burned**:
The combustion of carbon to CO₂ is represented by the reaction:
\[ C + O_2 \rightarrow CO_2 \]
The enthalpy change for this reaction is:
\[ \Delta H = -\Delta H_f^{\text{CO}_2} = -(-393.51 \text{ kJ/mol}) = 393.51 \text{ kJ/mol} \]
To find out how many grams of carbon need to be burned to produce enough energy (1094.1 kJ), we can set up the equation:
\[ \text{Energy from burning carbon} = \text{Moles of C burned} \times 393.51 \text{ kJ/mol} \]
Rearranging gives:
\[ \text{Moles of C burned} = \frac{1094.1 \text{ kJ}}{393.51 \text{ kJ/mol}} \approx 2.78 \text{ mol} \]
Now, convert moles of carbon to grams:
\[ \text{Mass of C burned} = \text{Moles of C burned} \times 12 \text{ g/mol} = 2.78 \text{ mol} \times 12 \text{ g/mol} \approx 33.36 \text{ g} \]

### Final Answer:
Approximately **33.36 grams** of carbon must be burned to CO₂ to provide enough heat for the water gas conversion of 100 grams of carbon.

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