Conductivity (Seimen's S) is directly proportional to area of the vessel and the concentration of th

Q: Why is concentration taken in $mol/m^3$?

Because the other dimensions (area and length) are in meters, so we must use the SI unit of concentration to maintain dimensional consistency.

Q: Is 'Seimen's S' a typo for conductance?

Yes, Siemens (S) is the unit of conductance ($G$). Conductivity ($\kappa$) has units of $S\ m^{-1}$. The question loosely uses the term 'conductivity' for conductance.

Q: What is the difference between conductance and conductivity?

Conductance is an extensive property depending on the dimensions of the conductor, while conductivity is an intensive property, intrinsic to the material.

Q: How do we convert $mol/L$ to $mol/m^3$?

Since $1\ L = 10^{-3}\ m^3$, $1\ mol/L = 1\ mol / 10^{-3}\ m^3 = 1000\ mol/m^3$.

Q: Can we use $cm$ instead of $m$?

Yes, but the options are given in terms of meters ($m$), so we must stick to meters to find the correct option.

Question

Conductivity (Seimen's S) is directly proportional to area of the vessel and the concentration of the solution in it and is inversely proportional to the length of the vessel, then constant of proportionality is expressed in

Options:

(A) $S\ m\ mol^{-1}$

(B) $S^2\ m^2\ mol^{-2}$

(C) $S\ m^2\ mol^{-1}$

(D) $S^2\ m^2\ mol$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are looking at a conductivity cell. The problem states that the conductance (which it loosely calls conductivity), measured in Siemens ($S$), depends on three physical parameters of the cell and the solution inside it.

Specifically, it is directly proportional to the cross-sectional area ($A$) of the vessel and the concentration ($C$) of the solution. At the same time, it is inversely proportional to the length ($l$) of the vessel.

### The Master Equation

We can translate this physical relationship into a mathematical equation. By combining the proportionalities, we get:

$$S \propto \frac{A \cdot C}{l}$$

To remove the proportionality sign, we introduce a constant of proportionality, let's call it $\kappa$:

$$S = \kappa \cdot \frac{A \cdot C}{l}$$

Our goal is to find the dimensional unit of this constant $\kappa$.

### Dimensional Substitution

Let's plug in the standard SI units for each of the physical quantities involved in our equation.

- The unit of conductance is Siemens ($S$).
- The unit of area $A$ is square meters ($m^2$).
- The unit of concentration $C$ is moles per cubic meter ($mol\ m^{-3}$).
- The unit of length $l$ is meters ($m$).

Substituting these into our master equation, we get:

$$S = \kappa \cdot \frac{m^2 \cdot mol\ m^{-3}}{m}$$

### Final Calculation

Now, we just need to simplify the right side of the equation. In the numerator, multiplying $m^2$ by $m^{-3}$ gives us $m^{-1}$.

$$S = \kappa \cdot \frac{mol \cdot m^{-1}}{m}$$

Dividing by the meter ($m$) in the denominator further simplifies this to $m^{-2}$:

$$S = \kappa \cdot mol \cdot m^{-2}$$

Finally, to isolate the unit of our constant $\kappa$, we rearrange the equation by moving the units to the left side:

$$\kappa = \frac{S}{mol \cdot m^{-2}}$$

$$\kappa = S\ m^2\ mol^{-1}$$

This perfectly matches option (c). It's a beautiful example of how dimensional analysis can guide us straight to the answer without needing complex formulas!

Animated Solution for Chemistry - Electrochemistry: Conductivity (Seimen's S) is directly proportional to area of the vessel and the concentration of the solution in it and is inversely proportional to the length of the vessel, then constant of proportionality is expressed in

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Visualized Solution

\text{Physical Setup}

\text{Proportionality Equation}

\text{Substituting Units}

\text{Simplifying Units}

\text{Unit of Constant } \kappa

\text{The Way Forward}

The Sigma Insight: Electrolytic Conduction

Analyzing the Setup

The Master Equation

Dimensional Substitution

Final Calculation

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