The rate of reaction is expressed as :
`(1)/(2)(+d)/(d t)[C] = (1)/(3)(-d)/(d t)[D] = (1)/(4)(+d)/(d t)[A] = -(d)/(d t)[B]`
The reaction is:

Options

A. `(1)/(4)A + (1)/(2)(C ) rarr B+(1)/(3)D`
B. `4A + 2C rarr B + 3D`
C. `B + 3D rarr 4A + 2C`
D. (d) `B + (1)/(3)D rarr (1)/(4)A + (1)/(2)C`

Explanation

To determine the reaction based on the given rate expression, we can follow these steps:

### Step 1: Analyze the Rate Expression
The rate of reaction is given as:
\[ \frac{1}{2} \left( \frac{d[C]}{dt} \right) = \frac{1}{3} \left( -\frac{d[D]}{dt} \right) = \frac{1}{4} \left( \frac{d[A]}{dt} \right) = -\left( \frac{d[B]}{dt} \right) \]

### Step 2: Identify the Stoichiometric Coefficients
From the rate expression, we can identify the stoichiometric coefficients for each species involved in the reaction:
- For \([C]\), the coefficient is \(2\) (since it is in the denominator).
- For \([D]\), the coefficient is \(3\) (since it is in the denominator).
- For \([A]\), the coefficient is \(4\) (since it is in the denominator).
- For \([B]\), the coefficient is \(1\) (since it is in the denominator).

### Step 3: Write the Balanced Chemical Equation
Based on the stoichiometric coefficients, we can write the balanced chemical equation:
\[ 3D + 4A \rightarrow 2C + B \]

### Step 4: Confirm the Reaction
We can confirm that the rates of disappearance of reactants and formation of products are consistent with the coefficients:
- The rate of disappearance of \(D\) is \(-\frac{d[D]}{dt}\) and has a coefficient of \(3\).
- The rate of disappearance of \(A\) is \(-\frac{d[A]}{dt}\) and has a coefficient of \(4\).
- The rate of formation of \(C\) is \(\frac{d[C]}{dt}\) and has a coefficient of \(2\).
- The rate of formation of \(B\) is \(\frac{d[B]}{dt}\) and has a coefficient of \(1\).

### Final Reaction
Thus, the final balanced reaction can be expressed as:
\[ 3D + 4A \rightarrow 2C + B \]