For the reaction, `H_(2)(g)+CO_(2)(g) hArr CO(g)+H_(2)O(g)`, if the initial concentration of `[H_(2)]=[CO_(2)]` and x mol `L^(1)` of `H_(2)` is consumed at equilibrium, the correct expression of `K_(p)` is:

Options

A. `x^(2)/((1-x)^(2))`
B. `((1+x)^(2))/((1-x)^(2))`
C. `(1+x^(2))/((2+x)^(2))`
D. `x^(2)/(1+x^(2))`

Explanation

To solve the problem, we need to analyze the given chemical reaction and determine the expression for the equilibrium constant \( K_p \).

### Step-by-Step Solution:

1. **Write the balanced chemical equation**:
\[ H_2(g) + CO_2(g) \rightleftharpoons CO(g) + H_2O(g) \]

2. **Set up initial concentrations**:
Let the initial concentration of both \( H_2 \) and \( CO_2 \) be \( 1 \, \text{mol/L} \) (as they are equal). Therefore:
\[ [H_2]_{initial} = [CO_2]_{initial} = 1 \, \text{mol/L} \]

3. **Define the change in concentration**:
If \( x \, \text{mol/L} \) of \( H_2 \) is consumed at equilibrium, then:
\[ [H_2]_{equilibrium} = 1 - x \]
\[ [CO_2]_{equilibrium} = 1 - x \]
\[ [CO]_{equilibrium} = x \]
\[ [H_2O]_{equilibrium} = x \]

4. **Write the expression for \( K_p \)**:
The equilibrium constant \( K_p \) is given by the expression:
\[ K_p = \frac{P_{CO} \cdot P_{H_2O}}{P_{H_2} \cdot P_{CO_2}} \]

5. **Relate partial pressures to concentrations**:
The partial pressures can be expressed in terms of concentrations and total pressure \( P \):
\[ P_{CO} = \frac{x}{2} P, \quad P_{H_2O} = \frac{x}{2} P, \quad P_{H_2} = \frac{1 - x}{2} P, \quad P_{CO_2} = \frac{1 - x}{2} P \]

6. **Substitute into the \( K_p \) expression**:
Substitute the expressions for partial pressures into the \( K_p \) formula:
\[ K_p = \frac{\left(\frac{x}{2} P\right) \cdot \left(\frac{x}{2} P\right)}{\left(\frac{1 - x}{2} P\right) \cdot \left(\frac{1 - x}{2} P\right)} \]

7. **Simplify the expression**:
\[ K_p = \frac{\frac{x^2}{4} P^2}{\frac{(1 - x)^2}{4} P^2} \]
The \( P^2 \) and \( 4 \) cancel out:
\[ K_p = \frac{x^2}{(1 - x)^2} \]

### Final Expression:
Thus, the correct expression for \( K_p \) is:
\[ K_p = \frac{x^2}{(1 - x)^2} \]