For the reaction `H_(2)(g) + CO_(2)(g) rarr CO(g) + H_(2)O(g)` ,
If the initial concentration of `[H_(2)] = [CO_(2)] = 1 "and" x mol//L` of hydrogen is consumed at equilibrium, the correct expression of Kp is :

Options

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

Explanation

To solve the problem, we need to determine the expression for \( K_p \) for the reaction:

\[ H_2(g) + CO_2(g) \rightleftharpoons CO(g) + H_2O(g) \]

Given that the initial concentrations of \( [H_2] \) and \( [CO_2] \) are both 1 mol/L, and \( x \) mol/L of hydrogen is consumed at equilibrium, we can follow these steps:

### Step 1: Write the initial concentrations
At the start of the reaction (t = 0):
- \( [H_2] = 1 \, \text{mol/L} \)
- \( [CO_2] = 1 \, \text{mol/L} \)
- \( [CO] = 0 \, \text{mol/L} \)
- \( [H_2O] = 0 \, \text{mol/L} \)

### Step 2: Determine the changes in concentration
As the reaction proceeds to equilibrium, \( x \) mol/L of \( H_2 \) is consumed. Therefore, the changes in concentrations will be:
- \( [H_2] \) decreases by \( x \)
- \( [CO_2] \) decreases by \( x \)
- \( [CO] \) increases by \( x \)
- \( [H_2O] \) increases by \( x \)

### Step 3: Write the equilibrium concentrations
At equilibrium, the concentrations will be:
- \( [H_2] = 1 - x \)
- \( [CO_2] = 1 - x \)
- \( [CO] = x \)
- \( [H_2O] = x \)

### Step 4: Write the expression for \( K_c \)
The equilibrium constant \( K_c \) is defined as:

\[ K_c = \frac{[CO][H_2O]}{[H_2][CO_2]} \]

Substituting the equilibrium concentrations into the expression:

\[ K_c = \frac{x \cdot x}{(1 - x)(1 - x)} = \frac{x^2}{(1 - x)^2} \]

### Step 5: Relate \( K_p \) and \( K_c \)
Since all reactants and products are gases, we can relate \( K_p \) and \( K_c \) using the equation:

\[ K_p = K_c \cdot (RT)^{\Delta n} \]

Where \( \Delta n \) is the change in the number of moles of gas, calculated as:

\[ \Delta n = \text{moles of products} - \text{moles of reactants} = (1 + 1) - (1 + 1) = 0 \]

Since \( \Delta n = 0 \):

\[ K_p = K_c \cdot (RT)^0 = K_c \]

### Step 6: Final expression for \( K_p \)
Thus, we find that:

\[ K_p = K_c = \frac{x^2}{(1 - x)^2} \]

### Conclusion
The correct expression for \( K_p \) is:

\[ K_p = \frac{x^2}{(1 - x)^2} \]

---