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

Answer

To determine which equilibrium will shift forward upon the addition of an inert gas at constant pressure, we can analyze the reactions based on the principles of chemical equilibrium and the effect of volume changes on concentration.

### Step-by-Step Solution:

1. **Understanding the Reaction**:
For a general reaction:
\[ aA + bB \rightleftharpoons cC + dD \]
The equilibrium constant \( K_c \) is expressed in terms of concentrations, and \( K_p \) is expressed in terms of partial pressures when all species are gases.

2. **Effect of Adding Inert Gas**:
When an inert gas is added at constant pressure, the total pressure of the system increases, which causes the volume to increase (since pressure × volume = constant at constant temperature). This increase in volume leads to a decrease in the concentrations of all gaseous reactants and products.

3. **Le Chatelier's Principle**:
According to Le Chatelier's principle, if a system at equilibrium is disturbed, the system will shift in a direction that counteracts the disturbance. In this case, the addition of an inert gas decreases the concentration of all species, prompting the system to shift toward the side with more moles of gas to increase the concentration.

4. **Analyzing Each Reaction**:
- **Reaction 1**: \( N_2 + 3H_2 \rightleftharpoons 2NH_3 \)
- Moles of reactants = 4 (1 N2 + 3 H2)
- Moles of products = 2 (2 NH3)
- **Shift**: Backward (toward reactants) since there are more moles of reactants.

- **Reaction 2**: \( H_2 + I_2 \rightleftharpoons 2HI \)
- Moles of reactants = 2 (1 H2 + 1 I2)
- Moles of products = 2 (2 HI)
- **Shift**: No shift (equal moles on both sides).

- **Reaction 3**: \( PCl_3 + Cl_2 \rightleftharpoons PCl_5 \)
- Moles of reactants = 2 (1 PCl3 + 1 Cl2)
- Moles of products = 1 (1 PCl5)
- **Shift**: Backward (toward reactants) since there are more moles of reactants.

- **Reaction 4**: \( 2SO_3 \rightleftharpoons 2SO_2 + O_2 \)
- Moles of reactants = 2 (2 SO3)
- Moles of products = 3 (2 SO2 + 1 O2)
- **Shift**: Forward (toward products) since there are more moles of products.

5. **Conclusion**:
The only reaction that will shift forward upon the addition of an inert gas at constant pressure is:
\[ 2SO_3 \rightleftharpoons 2SO_2 + O_2 \]
Therefore, the answer is **Option D**.