Conductance (Siemens, S) is directly proportional to the area of the vessel and the concentration of solution in it and is inversely proprtional to the length of the vessel, then the unit of constant of proportionlity is :

Options

A. `Smmol^(-1)`
B. `Sm^(2)mol^(-1)`
C. `S^(-2)m^(2)mol`
D. `S^(2)m^(2)mol^(-1)`.

Explanation

To solve the problem, we need to analyze the relationship given in the question regarding conductance.

### Step-by-step Solution:

1. **Understanding Conductance**:
Conductance (G) is defined as being directly proportional to the area (A) of the vessel and the concentration (C) of the solution, and inversely proportional to the length (L) of the vessel. This can be mathematically expressed as:
\[ G \propto \frac{A \cdot C}{L} \]

2. **Introducing the Proportionality Constant**:
To convert the proportionality into an equation, we introduce a constant of proportionality (k):
\[ G = k \cdot \frac{A \cdot C}{L} \]

3. **Rearranging the Equation**:
We can rearrange this equation to isolate the constant k:
\[ k = \frac{G \cdot L}{A \cdot C} \]

4. **Identifying Units**:
- The unit of conductance (G) is Siemens (S).
- The unit of area (A) is square meters (m²).
- The unit of concentration (C) is moles per cubic meter (mol/m³).
- The unit of length (L) is meters (m).

5. **Substituting Units into the Equation**:
Now we substitute the units into the equation for k:
\[ k = \frac{S \cdot m}{m^2 \cdot (mol/m^3)} \]

6. **Simplifying the Units**:
Simplifying the units:
\[ k = \frac{S \cdot m^3}{m^2 \cdot mol} = \frac{S \cdot m}{mol} \]

7. **Final Unit of k**:
Thus, the unit of the constant of proportionality (k) is:
\[ k = S \cdot m \cdot mol^{-1} \]

### Conclusion:
The unit of the constant of proportionality is \( S \cdot m \cdot mol^{-1} \).