Read the following paragraph and answer the questions given below,
for a general reaction, `aA+bB hArr cC+dD`, equilibrium constant `K_c` is given by `K_c=([C]^c[D]^d)/([A]^a[B]^b)` However, when all reactants and products are gases the equilibrium constant is generally expressed in terms of partial pressure.
`K_p=(P_C^cxx P _D^d)/(P_A^axxP_B^b)`
In which of the following equilibrium will shift forward on addition of inert gas at constant pressure ?

Options

A. `N_2(g) +3H_2 (g) hArr 2NH_3 (g)`
B. `H_2 (g) +I_2 (g) hArr 2HI(g)`
C. `PCI_3 (g) +CI_2 (g) hArr PCI_5 (g)`
D. `2SO_3(g) hArr 2SO_2(g) +O_2(g)`

Answer

To determine in which equilibrium the reaction will shift forward upon the addition of an inert gas at constant pressure, we can apply Le Chatelier's Principle. Here is a step-by-step solution:

### Step 1: Understand the Reaction and Equilibrium
We are given a general reaction of the form:
\[ aA + bB \rightleftharpoons cC + dD \]
The equilibrium constant \( K_c \) is defined as:
\[ K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]
When dealing with gases, we can express the equilibrium constant in terms of partial pressures:
\[ K_p = \frac{P_C^c \cdot P_D^d}{P_A^a \cdot P_B^b} \]

### Step 2: Apply Le Chatelier's Principle
Le Chatelier's Principle states that if a system at equilibrium is subjected to a change (like pressure, temperature, or concentration), the system will adjust to counteract that change and restore a new equilibrium.

### Step 3: Analyze the Effect of Adding Inert Gas
When an inert gas is added at constant pressure, the total pressure of the system increases. To maintain constant pressure, the volume of the system must increase. As the volume increases, the concentration (or partial pressure) of all gaseous reactants and products decreases.

### Step 4: Determine the Shift in Equilibrium
The equilibrium will shift towards the side with more moles of gas. This is because a decrease in concentration (due to increased volume) will favor the side with more moles to restore equilibrium.

### Step 5: Evaluate the Given Options
Now, we need to evaluate the options provided to see which one has more moles on the product side compared to the reactant side:

1. **Option 1:** \( 2A + B \rightleftharpoons 2C \) (Reactants: 3 moles, Products: 2 moles) - Shift backward.
2. **Option 2:** \( A + B \rightleftharpoons C + D \) (Reactants: 2 moles, Products: 2 moles) - No shift.
3. **Option 3:** \( PCl_3 + Cl_2 \rightleftharpoons PCl_5 \) (Reactants: 2 moles, Products: 1 mole) - Shift backward.
4. **Option 4:** \( N_2 + 3H_2 \rightleftharpoons 2NH_3 \) (Reactants: 4 moles, Products: 2 moles) - Shift backward.

### Step 6: Identify the Correct Option
From the evaluation, none of the options provided indicate a forward shift. However, if we consider a hypothetical situation where the product side has more moles than the reactant side, the equilibrium would shift forward.

### Conclusion
In this case, based on the analysis, the answer to the question is that none of the provided options will shift forward upon the addition of an inert gas at constant pressure.