Following three gaseous equilibrium reactions are occuring at `27^@C`
A, `2CO+O_2hArr 2CO_2`
B, `PCI_5hArr PCI_3+CI_2`
C, `2HIhArr H_2+I_2`
The correct order of `(K_p)/(K_c)` for the following reactions is

Options

A. `A lt C lt B`
B. `A lt B lt C`
C. `C lt B lt A`
D. `B lt C lt A`

Answer

To solve the problem of determining the correct order of \( \frac{K_p}{K_c} \) for the given reactions, we will follow these steps:

### Step 1: Identify the reactions and calculate \( \Delta n_g \)

1. **Reaction A**:
\[ 2CO + O_2 \rightleftharpoons 2CO_2 \]
- Moles of gaseous products = 2 (from \( 2CO_2 \))
- Moles of gaseous reactants = 2 (from \( 2CO \)) + 1 (from \( O_2 \)) = 3
- \( \Delta n_g = \text{moles of products} - \text{moles of reactants} = 2 - 3 = -1 \)

2. **Reaction B**:
\[ PCl_5 \rightleftharpoons PCl_3 + Cl_2 \]
- Moles of gaseous products = 1 (from \( PCl_3 \)) + 1 (from \( Cl_2 \)) = 2
- Moles of gaseous reactants = 1 (from \( PCl_5 \))
- \( \Delta n_g = 2 - 1 = 1 \)

3. **Reaction C**:
\[ 2HI \rightleftharpoons H_2 + I_2 \]
- Moles of gaseous products = 1 (from \( H_2 \)) + 1 (from \( I_2 \)) = 2
- Moles of gaseous reactants = 2 (from \( 2HI \))
- \( \Delta n_g = 2 - 2 = 0 \)

### Step 2: Calculate \( \frac{K_p}{K_c} \) for each reaction

Using the relation:
\[ K_p = K_c \cdot R T^{\Delta n_g} \]
we can find \( \frac{K_p}{K_c} \):
\[ \frac{K_p}{K_c} = R T^{\Delta n_g} \]

1. **For Reaction A**:
\[ \Delta n_g = -1 \Rightarrow \frac{K_p}{K_c} = R T^{-1} = \frac{1}{RT} \]

2. **For Reaction B**:
\[ \Delta n_g = 1 \Rightarrow \frac{K_p}{K_c} = R T^{1} = RT \]

3. **For Reaction C**:
\[ \Delta n_g = 0 \Rightarrow \frac{K_p}{K_c} = R T^{0} = 1 \]

### Step 3: Compare the values of \( \frac{K_p}{K_c} \)

- For Reaction A: \( \frac{K_p}{K_c} = \frac{1}{RT} \) (less than 1)
- For Reaction B: \( \frac{K_p}{K_c} = RT \) (greater than 1)
- For Reaction C: \( \frac{K_p}{K_c} = 1 \)

### Step 4: Order the values

Now, we can order the values:
- \( \frac{K_p}{K_c} \) for Reaction A < \( \frac{K_p}{K_c} \) for Reaction C < \( \frac{K_p}{K_c} \) for Reaction B

Thus, the correct order is:
\[ \text{A} < \text{C} < \text{B} \]

### Final Answer:
The correct order of \( \frac{K_p}{K_c} \) for the reactions is:
\[ \text{A, C, B} \]

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