`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 equation \( \log\left(\frac{K_P}{K_C}\right) + \log(RT) = 0 \) and find out for which reaction this relation holds true, we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ \log\left(\frac{K_P}{K_C}\right) + \log(RT) = 0 \]
We can rearrange this to isolate \( \log\left(\frac{K_P}{K_C}\right) \):
\[ \log\left(\frac{K_P}{K_C}\right) = -\log(RT) \]

### Step 2: Applying Logarithmic Properties
Using the property of logarithms, we can rewrite the right-hand side:
\[ \log\left(\frac{K_P}{K_C}\right) = \log\left(\frac{1}{RT}\right) \]
This implies:
\[ \frac{K_P}{K_C} = \frac{1}{RT} \]

### Step 3: Cross-Multiplying
Cross-multiplying gives us:
\[ K_P = K_C \cdot \frac{1}{RT} \]
or equivalently:
\[ K_P = \frac{K_C}{RT} \]

### Step 4: Relating \( K_P \) and \( K_C \)
From the relationship between \( K_P \) and \( K_C \), we know that:
\[ K_P = K_C \cdot RT^{\Delta n_g} \]
where \( \Delta n_g \) is the change in the number of moles of gas (moles of products - moles of reactants).

### Step 5: Setting Up the Equation
Comparing the two expressions for \( K_P \):
\[ K_C \cdot \frac{1}{RT} = K_C \cdot RT^{\Delta n_g} \]
We can simplify this to:
\[ \frac{1}{RT} = RT^{\Delta n_g} \]

### Step 6: Solving for \( \Delta n_g \)
This implies:
\[ RT^{\Delta n_g + 1} = 1 \]
Taking logarithm on both sides, we find that:
\[ \Delta n_g + 1 = 0 \implies \Delta n_g = -1 \]

### Step 7: Finding the Reaction
Now we need to find a reaction where \( \Delta n_g = -1 \). This means that the number of moles of gaseous products minus the number of moles of gaseous reactants equals -1.

### Step 8: Analyzing the Options
We analyze the given reactions (not provided in the question but typically would be):
1. \( A(g) + B(g) \rightarrow C(g) \) (Δn_g = 1)
2. \( 2A(g) \rightarrow B(g) \) (Δn_g = -1)
3. \( A(g) + 2B(g) \rightarrow C(g) \) (Δn_g = 0)
4. \( 2A(g) + 3B(g) \rightarrow 2C(g) \) (Δn_g = -2)

From the above analysis, the reaction that satisfies \( \Delta n_g = -1 \) is the second option.

### Conclusion
Thus, the reaction for which the relation \( \log\left(\frac{K_P}{K_C}\right) + \log(RT) = 0 \) is true is the one with \( \Delta n_g = -1 \).

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