Consider the following reactions:
I. `C(s) + 1/2 O_(2)(g) rarr CO(g)," "Delta H_(1) = x_(1)`
II. `CO(g) + 1/2 O_(2)(g) rarr CO_(2)(g), " "Delta H_(2) = x_(2)`
III. `C(s) + CO_(2)(g) rarr 2CO(g)," "Delta H_(3) = x_(3)`
Select the correct statements:

Options

A. Heat of formation of `CO_(2)` is `(x_1 + x_2)`
B. Heat of combusion of C is `(x_1 + x_2)`
C. `Delta H_(3) = Delta H_(1) - Delta H_(2)`
D. All of the above are correct statement(s)

Explanation

To solve the question regarding the given reactions and their enthalpy changes, we will analyze each reaction step by step.

### Given Reactions:
1. **Reaction I**: \( C(s) + \frac{1}{2} O_2(g) \rightarrow CO(g) \) with \( \Delta H_1 = x_1 \)
2. **Reaction II**: \( CO(g) + \frac{1}{2} O_2(g) \rightarrow CO_2(g) \) with \( \Delta H_2 = x_2 \)
3. **Reaction III**: \( C(s) + CO_2(g) \rightarrow 2CO(g) \) with \( \Delta H_3 = x_3 \)

### Step 1: Analyze the first statement
**Statement 1**: The heat of formation of \( CO_2 \) is \( x_1 + x_2 \).

To check this, we can combine Reaction I and Reaction II:
- From Reaction I, we have \( C + \frac{1}{2} O_2 \rightarrow CO \) (enthalpy change = \( x_1 \))
- From Reaction II, we have \( CO + \frac{1}{2} O_2 \rightarrow CO_2 \) (enthalpy change = \( x_2 \))

When we add these two reactions:
\[ C + \frac{1}{2} O_2 + CO + \frac{1}{2} O_2 \rightarrow CO + CO_2 \]
The \( CO \) on the left cancels with \( CO \) on the right, resulting in:
\[ C + O_2 \rightarrow CO_2 \]
Thus, the overall enthalpy change for the formation of \( CO_2 \) is:
\[ \Delta H = x_1 + x_2 \]
This confirms that **Statement 1 is correct**.

### Step 2: Analyze the second statement
**Statement 2**: The heat of combustion of carbon is \( x_1 + x_2 \).

The heat of combustion of carbon refers to the reaction:
\[ C(s) + O_2(g) \rightarrow CO_2(g) \]
This reaction can also be represented as the sum of Reaction I and Reaction II:
- From Reaction I: \( C + \frac{1}{2} O_2 \rightarrow CO \) (enthalpy change = \( x_1 \))
- From Reaction II: \( CO + \frac{1}{2} O_2 \rightarrow CO_2 \) (enthalpy change = \( x_2 \))

Adding these gives:
\[ C + \frac{1}{2} O_2 + CO + \frac{1}{2} O_2 \rightarrow CO + CO_2 \]
Again, the \( CO \) cancels out, leading to:
\[ C + O_2 \rightarrow CO_2 \]
Thus, the heat of combustion of carbon is indeed:
\[ \Delta H = x_1 + x_2 \]
This confirms that **Statement 2 is also correct**.

### Step 3: Analyze the third statement
**Statement 3**: \( \Delta H_3 = \Delta H_1 - \Delta H_2 \).

From Reaction III:
\[ C + CO_2 \rightarrow 2CO \]
We need to express this in terms of \( \Delta H_1 \) and \( \Delta H_2 \).

Using Hess's law, we can manipulate the reactions:
1. Start with Reaction II in reverse:
\[ CO_2 \rightarrow CO + \frac{1}{2} O_2 \quad (\Delta H = -x_2) \]
2. Then add Reaction I:
\[ C + \frac{1}{2} O_2 \rightarrow CO \quad (\Delta H = x_1) \]

Adding these two gives:
\[ C + CO_2 \rightarrow 2CO \]
Thus, the enthalpy change for this reaction is:
\[ \Delta H_3 = x_1 - x_2 \]
This confirms that **Statement 3 is correct**.

### Conclusion
Since all three statements are correct, the final answer is:
**All the above options are correct.**