One mole of A reacts with one mole of B to form 0.1 mole of C and 0.1 mole of D . What will be the value of the equilibrium constant ?
`A+BrArrC+D`

Options

A. 0.1
B. 100
C. `(1)/(81)`
D. 81

Answer

To find the equilibrium constant \( K \) for the reaction \( A + B \rightleftharpoons C + D \), we can follow these steps:

### Step 1: Write the balanced chemical equation
The balanced equation for the reaction is:
\[ A + B \rightleftharpoons C + D \]

### Step 2: Identify the initial moles of reactants and products
According to the problem:
- Initial moles of \( A \) = 1 mole
- Initial moles of \( B \) = 1 mole
- Moles of \( C \) formed = 0.1 mole
- Moles of \( D \) formed = 0.1 mole

### Step 3: Determine the change in moles
Since 0.1 moles of \( C \) and \( D \) are formed, the change in moles of \( A \) and \( B \) will be:
- Change in moles of \( A \) = -0.1
- Change in moles of \( B \) = -0.1

### Step 4: Calculate the equilibrium moles
Now, we can calculate the moles at equilibrium:
- Moles of \( A \) at equilibrium = \( 1 - 0.1 = 0.9 \) moles
- Moles of \( B \) at equilibrium = \( 1 - 0.1 = 0.9 \) moles
- Moles of \( C \) at equilibrium = \( 0 + 0.1 = 0.1 \) moles
- Moles of \( D \) at equilibrium = \( 0 + 0.1 = 0.1 \) moles

### Step 5: Calculate the equilibrium concentrations
Assuming the reaction occurs in a 1 L container (for simplicity):
- Concentration of \( A \) = \( [A] = \frac{0.9 \text{ moles}}{1 \text{ L}} = 0.9 \, \text{M} \)
- Concentration of \( B \) = \( [B] = \frac{0.9 \text{ moles}}{1 \text{ L}} = 0.9 \, \text{M} \)
- Concentration of \( C \) = \( [C] = \frac{0.1 \text{ moles}}{1 \text{ L}} = 0.1 \, \text{M} \)
- Concentration of \( D \) = \( [D] = \frac{0.1 \text{ moles}}{1 \text{ L}} = 0.1 \, \text{M} \)

### Step 6: Write the expression for the equilibrium constant \( K \)
The equilibrium constant \( K \) is given by the formula:
\[ K = \frac{[C][D]}{[A][B]} \]

### Step 7: Substitute the equilibrium concentrations into the expression
Substituting the values we found:
\[ K = \frac{(0.1)(0.1)}{(0.9)(0.9)} \]

### Step 8: Calculate the value of \( K \)
Calculating the above expression:
\[ K = \frac{0.01}{0.81} \approx 0.01235 \]

### Final Answer
Thus, the value of the equilibrium constant \( K \) is approximately \( 0.01235 \).

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