`log.(K_(P))/(K_(C))+logRT=0`. For which of the following reaction is this relation true?

Options

A. `PCl_(5)hArrPCl_(3)+Cl_(2)`
B. `2SO_(2)+O_(2)hArr 2SO_(3)`
C. `N_(2)+3H_(2)hArr 2NH_(3)`
D. Both (2) and (3)

Answer

To solve the problem, we need to analyze the relationship given in the question:

\[ \log\left(\frac{K_P}{K_C}\right) + \log(RT) = 0 \]

### Step 1: Understand the relationship between \( K_P \) and \( K_C \)

The relationship between the equilibrium constants \( K_P \) (in terms of pressure) and \( K_C \) (in terms of concentration) is given by:

\[ K_P = K_C (RT)^{\Delta n} \]

where \( \Delta n \) is the change in the number of moles of gas, defined as:

\[ \Delta n = n_{products} - n_{reactants} \]

### Step 2: Take the logarithm of both sides

Taking the logarithm of both sides of the equation gives us:

\[ \log(K_P) = \log(K_C) + \Delta n \log(RT) \]

### Step 3: Rearranging the equation

Rearranging this equation, we can express it as:

\[ \log(K_P) - \log(K_C) = \Delta n \log(RT) \]

Using the properties of logarithms, we can rewrite the left side:

\[ \log\left(\frac{K_P}{K_C}\right) = \Delta n \log(RT) \]

### Step 4: Setting up the equation from the question

From the question, we have:

\[ \log\left(\frac{K_P}{K_C}\right) + \log(RT) = 0 \]

This implies:

\[ \log\left(\frac{K_P}{K_C}\right) = -\log(RT) \]

### Step 5: Equating the two expressions

From our rearranged equation, we have:

\[ \Delta n \log(RT) = -\log(RT) \]

This leads us to:

\[ \Delta n = -1 \]

### Step 6: Finding the correct reaction

To find the reaction for which \( \Delta n = -1 \), we need to analyze the given reactions and calculate \( \Delta n \) for each:

1. **First Reaction**:
- Products: 2 moles (gaseous)
- Reactants: 1 mole (gaseous)
- \( \Delta n = 2 - 1 = 1 \) (not valid)

2. **Second Reaction**:
- Products: 2 moles (gaseous)
- Reactants: 3 moles (gaseous)
- \( \Delta n = 2 - 3 = -1 \) (valid)

3. **Third Reaction**:
- Products: 2 moles (gaseous)
- Reactants: 4 moles (gaseous)
- \( \Delta n = 2 - 4 = -2 \) (not valid)

4. **Fourth Reaction**:
- Products: 3 moles (gaseous)
- Reactants: 4 moles (gaseous)
- \( \Delta n = 3 - 4 = -1 \) (valid)

### Conclusion

The second reaction is the one where \( \Delta n = -1 \), making it the correct answer.