If `C+O_(2) to CO_(2)+94.2kcal`
`H_(2)+(1)/(2)O_(2) to H_(2)O+68.3"kcal"`
`CH_(4)+2O_(2) to CO_(2)+2H_(2)O+210.8"kcal"`
Then, the heat of formation of methane will be

Options

A. 47.3kcal
B. 20.0kcal
C. 45.9kcal
D. `-47.3`kcal

Answer

To find the heat of formation of methane (CH₄), we can use the given reactions and their enthalpy changes. The heat of formation of a compound is defined as the change in enthalpy when one mole of the compound is formed from its elements in their standard states.

### Given Reactions:
1. \( C + O_2 \rightarrow CO_2 \) (ΔH = -94.2 kcal)
2. \( H_2 + \frac{1}{2}O_2 \rightarrow H_2O \) (ΔH = -68.3 kcal)
3. \( CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O \) (ΔH = -210.8 kcal)

### Step-by-Step Solution:

**Step 1: Write the formation reaction for methane.**
The formation reaction of methane from its elements is:
\[ C + 2H_2 \rightarrow CH_4 \]

**Step 2: Rearrange the given reactions.**
We need to manipulate the given reactions to derive the formation reaction of methane.

- From Reaction 1, we can use:
\[ C + O_2 \rightarrow CO_2 \] (ΔH = -94.2 kcal)

- From Reaction 2, we need to multiply it by 2 to get 2 moles of \( H_2O \):
\[ 2H_2 + O_2 \rightarrow 2H_2O \] (ΔH = 2 * -68.3 kcal = -136.6 kcal)

- Reaction 3 is already in the correct form but we will reverse it to get methane:
\[ CO_2 + 2H_2O \rightarrow CH_4 + 2O_2 \] (ΔH = +210.8 kcal)

**Step 3: Combine the reactions.**
Now, we combine these reactions:
1. \( C + O_2 \rightarrow CO_2 \) (ΔH = -94.2 kcal)
2. \( 2H_2 + O_2 \rightarrow 2H_2O \) (ΔH = -136.6 kcal)
3. \( CO_2 + 2H_2O \rightarrow CH_4 + 2O_2 \) (ΔH = +210.8 kcal)

Adding these reactions:
- The \( CO_2 \) from Reaction 1 cancels with \( CO_2 \) in Reaction 3.
- The \( O_2 \) from Reaction 2 cancels with the \( O_2 \) in Reaction 3.
- The \( 2H_2O \) from Reaction 2 cancels with \( 2H_2O \) in Reaction 3.

This gives us:
\[ C + 2H_2 \rightarrow CH_4 \]

**Step 4: Calculate the total enthalpy change.**
Now we can calculate the total enthalpy change for the formation of methane:
\[ \Delta H_{formation} = (-94.2) + (-136.6) + (+210.8) \]
\[ \Delta H_{formation} = -94.2 - 136.6 + 210.8 = -20.0 \text{ kcal} \]

### Conclusion:
The heat of formation of methane (CH₄) is **-20.0 kcal**.