For a given reaction 3A + B `to` C + D the rate of reaction can be represented by:

Options

A. `-1/3(d[A])/(dt) = -(d[B])/(dt) = (+d[C])/(dt) = (+d[D])/(dt)`
B. `-1/3(d[A])/(dt) = (+d[A])/(dt) = k[A]^m[B]^n`
C. `+1/3(d[A])/(dt) =(-d[C])/(dt) = k[A]^n[B]^m`
D. None of these

Answer

To solve the question regarding the rate of the reaction \(3A + B \rightarrow C + D\), we can follow these steps:

### Step 1: Understand the Reaction
The reaction involves the consumption of reactants \(A\) and \(B\) to produce products \(C\) and \(D\). The stoichiometric coefficients indicate how many moles of each substance are involved in the reaction.

### Step 2: Define the Rate of Reaction
The rate of a reaction can be defined in terms of the change in concentration of reactants and products over time. The general expression for the rate of a reaction can be represented as:

\[ \text{Rate} = -\frac{1}{\nu_A} \frac{d[A]}{dt} = -\frac{1}{\nu_B} \frac{d[B]}{dt} = \frac{1}{\nu_C} \frac{d[C]}{dt} = \frac{1}{\nu_D} \frac{d[D]}{dt} \]

where \(\nu\) represents the stoichiometric coefficients of the respective substances.

### Step 3: Apply Stoichiometric Coefficients
For the reaction \(3A + B \rightarrow C + D\), the stoichiometric coefficients are:
- For \(A\): 3
- For \(B\): 1
- For \(C\): 1
- For \(D\): 1

Thus, the rate expressions become:

\[ \text{Rate} = -\frac{1}{3} \frac{d[A]}{dt} = -\frac{1}{1} \frac{d[B]}{dt} = \frac{1}{1} \frac{d[C]}{dt} = \frac{1}{1} \frac{d[D]}{dt} \]

### Step 4: Write the Rate Expressions
From the above, we can express the rate of the reaction as:

\[ -\frac{1}{3} \frac{d[A]}{dt} = -\frac{d[B]}{dt} = \frac{d[C]}{dt} = \frac{d[D]}{dt} \]

### Step 5: Analyze the Options
Now, we need to check the provided options against our derived expressions. The first option states:

\[ -\frac{1}{3} \frac{d[A]}{dt} = -\frac{d[B]}{dt} + \frac{d[C]}{dt} + \frac{d[D]}{dt} \]

This option is consistent with our derived expressions, as it correctly represents the rates of disappearance of reactants and appearance of products.

### Conclusion
Thus, the correct representation of the rate of the reaction is given by the first option.