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 use the concept of stoichiometry in chemical reactions. The rate of a reaction can be expressed in terms of the change in concentration of the reactants and products over time.

### Step-by-Step Solution:

1. **Identify the Stoichiometric Coefficients:**
- From the balanced equation, we can see the stoichiometric coefficients:
- For A: 2
- For B: 3
- For C: \( \frac{3}{2} \)

2. **Write the Rate Expressions:**
- The rate of the reaction can be expressed in terms of the change in concentration of each reactant:
\[ \text{Rate} = -\frac{1}{2} \frac{d[A]}{dt} = -\frac{1}{3} \frac{d[B]}{dt} = -\frac{2}{3} \frac{d[C]}{dt} \]

3. **Relate the Rates:**
- From the rate expressions, we can derive the relationships:
- For A:
\[ \frac{d[A]}{dt} = -2 \times \text{Rate} \]
- For B:
\[ \frac{d[B]}{dt} = -3 \times \text{Rate} \]
- For C:
\[ \frac{d[C]}{dt} = -\frac{3}{2} \times \text{Rate} \]

4. **Express the Rates in Terms of a Common Rate:**
- Let’s denote the rate of the reaction as \( r \):
- For A:
\[ r = -\frac{1}{2} \frac{d[A]}{dt} \implies \frac{d[A]}{dt} = -2r \]
- For B:
\[ r = -\frac{1}{3} \frac{d[B]}{dt} \implies \frac{d[B]}{dt} = -3r \]
- For C:
\[ r = -\frac{2}{3} \frac{d[C]}{dt} \implies \frac{d[C]}{dt} = -\frac{3}{2}r \]

5. **Summarize the Relationships:**
- Therefore, we can summarize the relationships between the rates of change of concentrations:
\[ \frac{d[A]}{dt} : \frac{d[B]}{dt} : \frac{d[C]}{dt} = -2 : -3 : -\frac{3}{2} \]

6. **Final Relation:**
- To express this in a more standard form, we can multiply through by a common factor (e.g., 6) to eliminate fractions:
\[ 6 \cdot \left(-2 : -3 : -\frac{3}{2}\right) = -12 : -18 : -9 \]
- Thus, the final relation can be simplified to:
\[ 12 : 18 : 9 \]

### Conclusion:
The correct relation between the rates of reaction of species A, B, and C is:
\[ \frac{d[A]}{dt} : \frac{d[B]}{dt} : \frac{d[C]}{dt} = 2 : 3 : \frac{3}{2} \]