In which one of the following equilibria, `K_(p) = K_(c)` ?

Options

A. `2NO(g) rarr N_(2)(g) + O_(2)(g)`
B. `2C(s) + O_(2)(g) rarr 2CO(g)`
C. `2HI(g) rarr H_(2)(g) + I_(2)(g)`
D. `NO_(2)(g) + SO_(2)(g) rarr NO(g) + SO_(2)(g)`

Explanation

To determine in which of the given equilibria \( K_p = K_c \), we need to understand the relationship between \( K_p \) and \( K_c \). The equation that relates these two constants is:

\[ K_p = K_c \times (R T)^{\Delta n} \]

where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( \Delta n \) is the change in the number of moles of gas, calculated as the moles of gaseous products minus the moles of gaseous reactants.

For \( K_p \) to equal \( K_c \), \( \Delta n \) must be zero. This means the number of moles of gas on the reactant side must equal the number of moles of gas on the product side.

Let's analyze each equilibrium given in the question:

1. **Equation 1: \( 2 \text{NO}_g \rightleftharpoons \text{N}_2_g + \text{O}_2_g \)**
- Reactants: 2 moles of NO (g)
- Products: 1 mole of N2 (g) + 1 mole of O2 (g) = 2 moles
- \( \Delta n = 2 - 2 = 0 \)
- Therefore, \( K_p = K_c \).

2. **Equation 2: \( 2 \text{C}_s + \text{O}_2_g \rightleftharpoons 2 \text{CO}_g \)**
- Reactants: 0 moles of C (s) + 1 mole of O2 (g) = 1 mole
- Products: 2 moles of CO (g)
- \( \Delta n = 2 - 1 = 1 \)
- Therefore, \( K_p \neq K_c \).

3. **Equation 3: \( 2 \text{HI}_g \rightleftharpoons \text{H}_2_g + \text{I}_2_g \)**
- Reactants: 2 moles of HI (g)
- Products: 1 mole of H2 (g) + 1 mole of I2 (g) = 2 moles
- \( \Delta n = 2 - 2 = 0 \)
- Therefore, \( K_p = K_c \).

4. **Equation 4: \( \text{NO}_2_g + \text{SO}_2_g \rightleftharpoons \text{NO}_g + \text{SO}_2_g \)**
- Reactants: 1 mole of NO2 (g) + 1 mole of SO2 (g) = 2 moles
- Products: 1 mole of NO (g) + 1 mole of SO2 (g) = 2 moles
- \( \Delta n = 2 - 2 = 0 \)
- Therefore, \( K_p = K_c \).

### Conclusion:
From the analysis, \( K_p = K_c \) for the following equations:
- Equation 1: \( 2 \text{NO}_g \rightleftharpoons \text{N}_2_g + \text{O}_2_g \)
- Equation 3: \( 2 \text{HI}_g \rightleftharpoons \text{H}_2_g + \text{I}_2_g \)
- Equation 4: \( \text{NO}_2_g + \text{SO}_2_g \rightleftharpoons \text{NO}_g + \text{SO}_2_g \)

Thus, the answer is **Equations 1, 3, and 4**.