For the reaction `2A + 3B + 3/2C rarr 3P` the correct relation between rate of reaction of species A,B,C is

Options

A. `d(n_A)/dt = 2/3 d(n_B)/dt = 4/3 d(n_C/dt)`
B. `2d(n_A)/dt = 3 d(n_B)/dt = 3/2 d(n_C/dt)`
C. `3/2 d(n_A)/dt = d(n_B)/dt = 3/4 d(n_C/dt)`
D. `d(n_A)/dt = d(n_B)/dt = d(n_C/dt)`

Answer

To find the correct relation between the rates of reaction of species A, B, and C for the reaction:

\[ 2A + 3B + \frac{3}{2}C \rightarrow 3P \]

we will derive the relationship step by step.

### Step 1: Write the Rate of Reaction Expression

The rate of reaction can be expressed in terms of the change in concentration of the reactants. For the given reaction, we can express the rate of reaction (R) in terms of A, B, and C as follows:

\[ R = -\frac{1}{2} \frac{d[A]}{dt} = -\frac{1}{3} \frac{d[B]}{dt} = -\frac{2}{3} \frac{d[C]}{dt} \]

### Step 2: Relate the Rates of Change

From the rate expressions, we can relate the rates of change of concentrations of A, B, and C based on their stoichiometric coefficients:

1. For species A:
\[ R = -\frac{1}{2} \frac{d[A]}{dt} \implies \frac{d[A]}{dt} = -2R \]

2. For species B:
\[ R = -\frac{1}{3} \frac{d[B]}{dt} \implies \frac{d[B]}{dt} = -3R \]

3. For species C:
\[ R = -\frac{2}{3} \frac{d[C]}{dt} \implies \frac{d[C]}{dt} = -\frac{3}{2}R \]

### Step 3: Combine the Relations

Now we can express the rates of change of A, B, and C in terms of R:

\[ \frac{d[A]}{dt} = -2R, \quad \frac{d[B]}{dt} = -3R, \quad \frac{d[C]}{dt} = -\frac{3}{2}R \]

### Step 4: Formulate the Final Relation

To express the relationship clearly, we can set the equations equal to each other:

\[ -\frac{1}{2} \frac{d[A]}{dt} = -\frac{1}{3} \frac{d[B]}{dt} = -\frac{2}{3} \frac{d[C]}{dt} \]

Multiplying through by 6 (the least common multiple of the denominators) gives:

\[ 3 \frac{d[A]}{dt} = 2 \frac{d[B]}{dt} = 4 \frac{d[C]}{dt} \]

### Final Answer

Thus, the correct relation between the rates of reaction of species A, B, and C is:

\[ \frac{d[A]}{dt} : \frac{d[B]}{dt} : \frac{d[C]}{dt} = 2 : 3 : 4 \]