For the reaction `2A(g)hArrB(g)+3C(g),` at a given temperature ,`K_(c)=16.` What must be the volume of the flask , if a mixture of 2 mole rach of A,B and C exist in equilibrium ?

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. 1
D. none of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Write the balanced chemical equation
The reaction is given as:
\[ 2A(g) \rightleftharpoons B(g) + 3C(g) \]

### Step 2: Define the equilibrium constant expression
The equilibrium constant \( K_c \) for the reaction is defined as:
\[ K_c = \frac{[B]^{b} \cdot [C]^{c}}{[A]^{a}} \]
where \( a, b, c \) are the stoichiometric coefficients from the balanced equation. For our reaction:
\[ K_c = \frac{[B] \cdot [C]^3}{[A]^2} \]

### Step 3: Set up the concentrations
We are given that there are 2 moles of each species (A, B, and C) at equilibrium. Let the volume of the flask be \( V \) liters. Therefore, the concentrations of each species will be:
- Concentration of \( A \): \([A] = \frac{2}{V}\)
- Concentration of \( B \): \([B] = \frac{2}{V}\)
- Concentration of \( C \): \([C] = \frac{2}{V}\)

### Step 4: Substitute concentrations into the equilibrium expression
Now substituting the concentrations into the \( K_c \) expression:
\[ K_c = \frac{\left(\frac{2}{V}\right) \cdot \left(\frac{2}{V}\right)^3}{\left(\frac{2}{V}\right)^2} \]

### Step 5: Simplify the equation
Substituting and simplifying:
\[ K_c = \frac{\left(\frac{2}{V}\right) \cdot \left(\frac{8}{V^3}\right)}{\left(\frac{4}{V^2}\right)} = \frac{\frac{16}{V^4}}{\frac{4}{V^2}} = \frac{16}{4} \cdot \frac{1}{V^2} = \frac{4}{V^2} \]

### Step 6: Set \( K_c \) equal to the given value
We know that \( K_c = 16 \), so we set up the equation:
\[ \frac{4}{V^2} = 16 \]

### Step 7: Solve for \( V^2 \)
Cross-multiplying gives:
\[ 4 = 16V^2 \]
\[ V^2 = \frac{4}{16} = \frac{1}{4} \]

### Step 8: Solve for \( V \)
Taking the square root of both sides:
\[ V = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Final Answer
The volume of the flask must be:
\[ \boxed{\frac{1}{2} \text{ liters}} \]