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 derive the differential rate equation for the reaction \(3A + B \rightarrow C + D\), we can follow these steps:

### Step 1: Write the general form of the rate of reaction
The rate of a reaction can be expressed in terms of the change in concentration of reactants and products over time. For the given reaction, we can express the rate as:
\[ \text{Rate} = -\frac{1}{3} \frac{d[A]}{dt} = -\frac{d[B]}{dt} = \frac{d[C]}{dt} = \frac{d[D]}{dt} \]
Here, the negative sign indicates that the concentration of reactants decreases over time, while the positive sign indicates that the concentration of products increases.

### Step 2: Apply stoichiometric coefficients
In the reaction \(3A + B \rightarrow C + D\), the stoichiometric coefficients are 3 for A and 1 for B. This means that for every 3 moles of A that react, 1 mole of B reacts. Therefore, we need to account for these coefficients in our rate expressions:
- For A: The rate of disappearance is given by \(-\frac{1}{3} \frac{d[A]}{dt}\).
- For B: The rate of disappearance is given by \(-\frac{d[B]}{dt}\).
- For C: The rate of appearance is given by \(\frac{d[C]}{dt}\).
- For D: The rate of appearance is given by \(\frac{d[D]}{dt}\).

### Step 3: Set up the differential rate equation
From the above expressions, we can set up the differential rate equation as:
\[ -\frac{1}{3} \frac{d[A]}{dt} = -\frac{d[B]}{dt} = \frac{d[C]}{dt} = \frac{d[D]}{dt} \]

### Step 4: Finalize the differential rate equation
Thus, the complete differential rate equation for the reaction \(3A + B \rightarrow C + D\) can be summarized as:
\[ -\frac{1}{3} \frac{d[A]}{dt} = -\frac{d[B]}{dt} = \frac{d[C]}{dt} = \frac{d[D]}{dt} \]

### Conclusion
The differential rate equation captures the relationship between the rates of change of concentrations of reactants and products in the reaction.

---