For the reaction,
`3A + 2B to C+D`,
the differential rate law can be written as:

Options

A. `(1)/(3)(d[A])/(dt)=(d[C])/(dt)=k[A]^(n)[B]^(m)`
B. `-(d[A])/(dt)=(d[C])/(dt)=k[A]^(n)[B]^(m)`
C. `+(1)/(3)(d[A])/(dt)=-(d[C])/(dt)=k[A]^(n)[B]^(m)`
D. `-(1)/(3)(d[A])/(dt)=(d[C])/(dt)=k[A]^(n)[B]^(m)`

Explanation

To derive the differential rate law for the reaction \(3A + 2B \rightarrow C + D\), we will follow these steps:

### Step 1: Write the general form of the rate law
The rate law can be expressed in the form:
\[ \text{Rate} = k [A]^n [B]^m \]
where \(k\) is the rate constant, and \(n\) and \(m\) are the orders of the reaction with respect to reactants A and B, respectively.

### Step 2: Relate the rate to the change in concentration of reactants
For the reaction, we can express the rate in terms of the change in concentration of reactants. The rate of the reaction can be defined as:
\[ \text{Rate} = -\frac{1}{3} \frac{d[A]}{dt} = -\frac{1}{2} \frac{d[B]}{dt} \]
Here, the coefficients (3 for A and 2 for B) are used to relate the rate of change of concentration to the stoichiometry of the reaction.

### Step 3: Express the rate in terms of the products
Similarly, for the products formed, we can express the rate as:
\[ \text{Rate} = \frac{1}{1} \frac{d[C]}{dt} = \frac{1}{1} \frac{d[D]}{dt} \]
This means that the rate of formation of C and D is directly related to the rate of the reaction.

### Step 4: Combine the expressions
From the above expressions, we can equate the rates:
\[ -\frac{1}{3} \frac{d[A]}{dt} = -\frac{1}{2} \frac{d[B]}{dt} = \frac{d[C]}{dt} = \frac{d[D]}{dt} \]

### Step 5: Write the final differential rate law
Thus, the final form of the differential rate law can be expressed as:
\[ -\frac{1}{3} \frac{d[A]}{dt} = k [A]^n [B]^m \]
This means that the rate of the reaction is proportional to the concentrations of A and B raised to their respective powers.

### Conclusion
The differential rate law for the reaction \(3A + 2B \rightarrow C + D\) can be summarized as:
\[ \text{Rate} = k [A]^n [B]^m \]
where \(n\) and \(m\) are determined experimentally.

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