On the basic of thermochemical equations (I),(II)and (III), Find out which of the algebraic relationships given in options (a)to (d) is correct ?
(I)`C("graphite")+O_(2)(g)to CO_(2)(g),`
`Delta_(r)H=XKJmol^(-1)`
(II)`C("graphite")+(1)/(2)O_(2)(g)toCO(g),`
`Delta_(r)H=yKJ mol^(-1)`
`(III) CO(g)+(1)/(2)O_(2)(g)toCO_(2)(g), Delta_(r)H=zKJ mol^(-1)`

Options

A. `z = x + y`
B. `x = y - z`
C. `x = y + z`
D. `y = 2z - x`

Explanation

To solve the problem based on the given thermochemical equations, we need to analyze the relationships between the enthalpy changes (ΔH) for the reactions provided. Let's break it down step by step.

### Step 1: Write down the given reactions and their enthalpy changes

1. **Reaction I**:
\[ C(\text{graphite}) + O_2(g) \rightarrow CO_2(g), \quad \Delta_rH = X \text{ kJ/mol} \]

2. **Reaction II**:
\[ C(\text{graphite}) + \frac{1}{2}O_2(g) \rightarrow CO(g), \quad \Delta_rH = y \text{ kJ/mol} \]

3. **Reaction III**:
\[ CO(g) + \frac{1}{2}O_2(g) \rightarrow CO_2(g), \quad \Delta_rH = z \text{ kJ/mol} \]

### Step 2: Understand the relationships between the reactions

- **Reaction I** produces CO2 from graphite and O2.
- **Reaction II** produces CO from graphite and O2.
- **Reaction III** converts CO to CO2 using O2.

### Step 3: Apply Hess's Law

According to Hess's Law, the total enthalpy change for a reaction is the sum of the enthalpy changes for the individual steps. We can express the enthalpy change for Reaction III in terms of the enthalpy changes of Reactions I and II.

From Reaction I, we have:
\[ \Delta_rH_{I} = X \]

From Reaction II, we have:
\[ \Delta_rH_{II} = y \]

For Reaction III, we can express it as:
\[ \Delta_rH_{III} = \Delta_rH_{I} - \Delta_rH_{II} \]

This leads to:
\[ z = X - y \]

### Step 4: Rearranging the equation

From the equation \( z = X - y \), we can rearrange it to find a relationship:
\[ X = z + y \]

### Step 5: Identify the correct option

Now, we need to match this relationship with the provided options (a) to (d). The correct algebraic relationship based on our derivation is:
\[ X = z + y \]

### Conclusion

Thus, the correct answer among the options provided is:

**Option (c)**: \( X = z + y \)

---