Consider the general hypothetical reaction
`A(s)hArr2B(g)+3C(g)`
If the concentration of C at equlibrium is doubled then after the equlibrium is re-established the concentration of B will be

Options

A. Twice of its original value
B. Half of its original value
C. `2sqrt2` times of original value
D. `(1)/(2sqrt2)` time of original value

Answer

To solve the problem, we need to analyze the equilibrium of the reaction given:

**Reaction:**
\[ A(s) \rightleftharpoons 2B(g) + 3C(g) \]

### Step 1: Write the expression for the equilibrium constant (Kc)
The equilibrium constant expression for the reaction can be written as:
\[ K_c = \frac{[B]^2[C]^3}{1} \]
Since A is a solid, its concentration does not appear in the equilibrium expression.

### Step 2: Define initial concentrations
Let:
- \([B] = P_B\) (initial concentration of B at equilibrium)
- \([C] = P_C\) (initial concentration of C at equilibrium)

Thus, the equilibrium constant can be expressed as:
\[ K_c = P_B^2 P_C^3 \]

### Step 3: Change in concentration of C
According to the problem, the concentration of C is doubled:
\[ P_C' = 2P_C \]

### Step 4: Write the new equilibrium constant expression
When the concentration of C is doubled, we need to find the new equilibrium constant \(K_c'\):
\[ K_c' = \frac{[B']^2[C']^3}{1} \]
Substituting \(P_C' = 2P_C\):
\[ K_c' = [B']^2(2P_C)^3 = [B']^2 \cdot 8P_C^3 \]

### Step 5: Set the two equilibrium constants equal
Since the equilibrium constant does not change with concentration changes, we can set the two expressions equal:
\[ K_c = K_c' \]
Thus:
\[ P_B^2 P_C^3 = [B']^2 \cdot 8P_C^3 \]

### Step 6: Simplify the equation
We can cancel \(P_C^3\) from both sides (assuming \(P_C \neq 0\)):
\[ P_B^2 = [B']^2 \cdot 8 \]

### Step 7: Solve for the new concentration of B
Rearranging gives:
\[ [B']^2 = \frac{P_B^2}{8} \]
Taking the square root:
\[ [B'] = \frac{P_B}{\sqrt{8}} = \frac{P_B}{2\sqrt{2}} \]

### Conclusion
Thus, the concentration of B after the equilibrium is re-established will be:
\[ [B'] = \frac{P_B}{2\sqrt{2}} \]