In the reaction `C(s)+CO_(2)(g) hArr 2CO(g)`, the equilibrium pressure is `12` atm. If `50%` of `CO_(2)` reacts, calculate `K_(p)`.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Reaction and Initial Conditions
The reaction given is:
\[ C(s) + CO_2(g) \rightleftharpoons 2CO(g) \]

We know that the equilibrium pressure is 12 atm and that 50% of the \( CO_2 \) reacts.

### Step 2: Define Initial and Change in Concentration
Assume that initially, we have 1 mole of \( CO_2 \) (since the exact amount isn't specified, we can use 1 mole for simplicity).

- Initial moles of \( CO_2 \) = 1 mole
- Initial moles of \( CO \) = 0 moles

Since 50% of \( CO_2 \) reacts:
- Moles of \( CO_2 \) that react = \( 0.5 \) moles
- Remaining \( CO_2 \) = \( 1 - 0.5 = 0.5 \) moles
- Moles of \( CO \) produced = \( 2 \times 0.5 = 1 \) mole (from the stoichiometry of the reaction)

### Step 3: Calculate Total Moles at Equilibrium
At equilibrium, the total moles are:
- Moles of \( CO_2 \) = 0.5
- Moles of \( CO \) = 1
- Total moles = \( 0.5 + 1 = 1.5 \) moles

### Step 4: Calculate Mole Fractions
The mole fractions can be calculated as follows:
- Mole fraction of \( CO_2 \) (\( x_{CO_2} \)) = \( \frac{0.5}{1.5} = \frac{1}{3} \)
- Mole fraction of \( CO \) (\( x_{CO} \)) = \( \frac{1}{1.5} = \frac{2}{3} \)

### Step 5: Calculate Partial Pressures
Using the total pressure (12 atm):
- Partial pressure of \( CO_2 \) (\( P_{CO_2} \)) = \( x_{CO_2} \times P_{total} = \frac{1}{3} \times 12 \, \text{atm} = 4 \, \text{atm} \)
- Partial pressure of \( CO \) (\( P_{CO} \)) = \( x_{CO} \times P_{total} = \frac{2}{3} \times 12 \, \text{atm} = 8 \, \text{atm} \)

### Step 6: Write the Expression for \( K_p \)
The expression for the equilibrium constant \( K_p \) for the reaction is given by:
\[ K_p = \frac{(P_{CO})^2}{P_{CO_2}} \]

### Step 7: Substitute the Values into the Expression
Substituting the partial pressures we calculated:
\[ K_p = \frac{(8 \, \text{atm})^2}{4 \, \text{atm}} = \frac{64 \, \text{atm}^2}{4 \, \text{atm}} = 16 \, \text{atm} \]

### Step 8: Final Calculation
Thus, the value of \( K_p \) is:
\[ K_p = 16 \]

### Summary of Steps
1. Define initial conditions and the amount of \( CO_2 \) that reacts.
2. Calculate the total moles at equilibrium.
3. Determine the mole fractions of \( CO_2 \) and \( CO \).
4. Calculate the partial pressures using the total pressure.
5. Write the expression for \( K_p \) and substitute the values.
6. Calculate the final value of \( K_p \).