Conisder the following reactions:
I. `A+B underset(k_(-1))overset(k_(1))hArr C`
II. `C + B overset(k_(2))rarr D`
Then `k_(1)[A][B]-k_(-1)[C]-k_(2)[C][B]` is equal to
(a) `(-d[A])/(dt)`
(b)`(-d[B])/(dt)`
(c)`(d[C])/(dt)`
(d)`(d[D])/(dt)`

Options

A. `(-d[A])/(dt)`
B. `(-d[B])/(dt)`
C. `(d[C])/(dt)`
D. `(d[D])/(dt)`

Explanation

To solve the problem, we need to analyze the given reactions and their rate expressions.

### Step 1: Write the rate expressions for the reactions
We have two reactions:
1. \( A + B \overset{k_1}{\rightleftharpoons} C \)
2. \( C + B \overset{k_2}{\rightarrow} D \)

For the first reaction, the rate of formation of \( C \) can be expressed as:
\[ \text{Rate of formation of } C = k_1 [A][B] - k_{-1} [C] \]

For the second reaction, the rate of disappearance of \( C \) can be expressed as:
\[ \text{Rate of disappearance of } C = k_2 [C][B] \]

### Step 2: Combine the rate expressions
The net rate of change of concentration of \( C \) can be expressed as:
\[ \frac{d[C]}{dt} = \text{Rate of formation of } C - \text{Rate of disappearance of } C \]
Substituting the expressions from above, we get:
\[ \frac{d[C]}{dt} = (k_1 [A][B] - k_{-1} [C]) - k_2 [C][B] \]

### Step 3: Rearranging the expression
Rearranging the equation gives:
\[ \frac{d[C]}{dt} = k_1 [A][B] - k_{-1} [C] - k_2 [C][B] \]

### Step 4: Identify the expression from the options
From the derived expression, we see that:
\[ k_1 [A][B] - k_{-1} [C] - k_2 [C][B] = \frac{d[C]}{dt} \]

Thus, the expression \( k_1 [A][B] - k_{-1} [C] - k_2 [C][B] \) is equal to \( \frac{d[C]}{dt} \).

### Conclusion
The correct answer is:
(c) \( \frac{d[C]}{dt} \)

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