If `log(K_(C))/(K_(P))-log[(1)/(RT)]=0`, then above is true for the following equilibrium reaction

Options

A. `NH_(2)(g)hArr(1)/(2)N_(2)(g)+(3)/(2)H_(2)(g)`
B. `CaCO_(3)(s)hArrCaO(s)+CO_(2)(g)`
C. `2NO_(2)(g)hArrN_(2)O_(4)(g)`
D. `H_(2)(g)+I_(2)(g)hArr2HI(g)`

Explanation

To solve the given equation \( \log\left(\frac{K_C}{K_P}\right) - \log\left(\frac{1}{RT}\right) = 0 \) and determine which equilibrium reaction it applies to, we can follow these steps:

### Step 1: Simplify the Equation
Start with the given equation:
\[ \log\left(\frac{K_C}{K_P}\right) - \log\left(\frac{1}{RT}\right) = 0 \]
This can be rewritten using the properties of logarithms:
\[ \log\left(\frac{K_C}{K_P}\right) = \log(RT) \]

### Step 2: Exponentiate Both Sides
To eliminate the logarithm, we can take the antilogarithm of both sides:
\[ \frac{K_C}{K_P} = RT \]

### Step 3: Rearrange the Equation
Rearranging the equation gives:
\[ \frac{K_P}{K_C} = \frac{1}{RT} \]
This means that:
\[ K_P = K_C \cdot RT \]

### Step 4: Relate to Δn
From the equilibrium constant expressions, we know:
\[ K_P = K_C \cdot (RT)^{\Delta n} \]
where \( \Delta n \) is the change in the number of moles of gas, calculated as:
\[ \Delta n = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)} \]

### Step 5: Set Up the Condition
From the equation \( K_P = K_C \cdot RT \), we can conclude:
\[ \Delta n = 1 \]
This means we are looking for reactions where the change in the number of moles of gas is equal to 1.

### Step 6: Analyze the Given Reactions
Now, we need to analyze the provided equilibrium reactions to find those that have \( \Delta n = 1 \).

1. **Reaction A**: Calculate the moles of products and reactants.
2. **Reaction B**: Repeat the calculation.
3. **Reaction C**: Check if \( \Delta n \) equals 1.
4. **Reaction D**: Check if \( \Delta n \) equals 1.

### Step 7: Identify Valid Reactions
After calculating \( \Delta n \) for each reaction:
- If \( \Delta n = 1 \), that reaction satisfies the condition.
- If \( \Delta n \neq 1 \), that reaction does not satisfy the condition.

### Conclusion
The reactions that satisfy the equation \( \log\left(\frac{K_C}{K_P}\right) - \log\left(\frac{1}{RT}\right) = 0 \) will be those where \( \Delta n = 1 \).