The rate of formation of `C_(6)H_(6)+3H_(2) underset(k_(b))overset(k_(f))hArr C_(6)H_(12)` for the forward reaction is first order with respect to `C_(6)H_(6)` and `H_(12)` each. Which one of the options is/are correct?

Options

A. `k_(eq) = (k_(f))/(k_(b))`
B. `k_(eq) = ([C_(6)H_(12)])/([C_(6)H_(6)][H_(2)]^(3))`
C. `r_(f) =k_(f)[C_(6)H_(6)][H_(2)]`
D. `r_(b) = k_(b)[C_(6)H_(12)][H_(2)]^(-2)`

Explanation

To solve the problem regarding the reaction:

\[ C_6H_6 + 3H_2 \underset{k_b}{\overset{k_f}{\rightleftharpoons}} C_6H_{12} \]

where the forward reaction is first order with respect to both \( C_6H_6 \) and \( H_2 \), we will follow these steps:

### Step 1: Write the Rate Expressions
For the forward reaction, the rate can be expressed as:

\[ R_f = k_f [C_6H_6]^1 [H_2]^1 \]

This indicates that the reaction is first order with respect to both \( C_6H_6 \) and \( H_2 \).

### Step 2: Write the Rate Expression for the Backward Reaction
For the backward reaction, we can write the rate as:

\[ R_b = k_b [C_6H_{12}] \]

### Step 3: Establish the Equilibrium Condition
At equilibrium, the rate of the forward reaction equals the rate of the backward reaction:

\[ R_f = R_b \]

Substituting the expressions from Steps 1 and 2:

\[ k_f [C_6H_6]^1 [H_2]^1 = k_b [C_6H_{12}] \]

### Step 4: Write the Equilibrium Constant Expression
The equilibrium constant \( K_{eq} \) can be defined as:

\[ K_{eq} = \frac{[C_6H_{12}]}{[C_6H_6][H_2]^3} \]

From the rate expressions, we can also express \( K_{eq} \) as:

\[ K_{eq} = \frac{k_f}{k_b} \]

### Step 5: Relate the Two Expressions for \( K_{eq} \)
From the equilibrium condition, we can rearrange the equation:

\[ \frac{k_f}{k_b} = \frac{[C_6H_{12}]}{[C_6H_6][H_2]^3} \]

This gives us a relationship between the rate constants and the concentrations at equilibrium.

### Step 6: Analyze the Options
Now, based on the derived equations, we can analyze the provided options:

1. **Option 1**: \( K_{eq} = \frac{k_f}{k_b} \) - **Correct**
2. **Option 2**: \( K_{eq} = \frac{[C_6H_{12}]}{[C_6H_6][H_2]^3} \) - **Correct**
3. **Option 3**: \( R_f = k_f [C_6H_6][H_2] \) - **Correct**
4. **Option 4**: \( R_b = k_b [C_6H_{12}] \) - **Correct**

Thus, all options are correct.

### Final Answer:
All options (1, 2, 3, and 4) are correct.

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