Q.The resistance of a conductivity cell containing $0.001\ M$ KCl solution at 298 K is $1500\ \Omega$. What is the cell constant if conductivity of $0.001\ M$ KCl solution at 298 K is $0.146 \times 10^{-3}\ S\ cm^{-1}$?

The key idea is that cell constant $G^*$ relates measured resistance $R$ to the solution's conductivity $\kappa$ by the formula:

Step 1: Write the relation between conductivity, resistance, and cell constant.

$\kappa = \frac{G^*}{R} \quad \Rightarrow \quad G^* = \kappa \times R$ …

The key idea here is simple: a conductivity cell has a fixed geometry — the distance between electrodes divided by their area — which we call the cell constant ($G^*$). You can’t measure it directly with a ruler inside the solution, but you can find it electrically.

Conductivity ($\kappa$) is an intrinsic property of the solution. Resistance ($R$) is what you measure. They are related by:

Rearranging:

That’s the whole backbone. Once you know the conductivity of a standard solution (here, $0.001\ M$ KCl at 298 K, whose conductivity is tabulated and given), and you measure its resistance in your cell, the cell constant follows immediately.

Let’s walk through it.

1. Write down what’s given.

   Resistance, $R = 1500\ \Omega$

   Conductivity, $\kappa = 0.146 \times 10^{-3}\ \text{S cm}^{-1}$

   (Notice the units: Siemens per centimetre — that’s the clue that our cell constant will come out in $\text{cm}^{-1}$.)

2. Apply the relation.

3. Do the arithmetic.

Multiply by $1500$:

4. Attach the unit. …

Method: Cell Constant from Conductivity and Resistance

This problem uses the fundamental relationship between conductivity ($\kappa$), resistance ($R$), and the cell constant ($G^*$).

Concept First

The cell constant $G^*$ (also written as $l/A$) is a fixed property of a conductivity cell — it depends only on the geometry (distance between electrodes divided by their area). It links the measured resistance $R$ to the conductivity $\kappa$ of the solution:

Rearranging:

Steps

1. Identify given data

   • Conductivity, $\kappa = 0.146 \times 10^{-3}\ \text{S cm}^{-1}$
   • Resistance, $R = 1500\ \Omega$

2. Apply the formula

3. Substitute and calculate

$G^* = 0.219\ \text{cm}^{-1}$ …

Common Mistakes & How to Avoid Them

Mistake 1: Inverting the Cell-Constant Formula

The error: Students write $G^* = \dfrac{R}{\kappa}$ instead of $G^* = \kappa \times R$, or confuse which quantity is measured and which is looked up.

Why it happens: Conductivity ($\kappa$) is related to conductance ($G = 1/R$) by $\kappa = G \times G^*$, so it's easy to rearrange incorrectly under time pressure.

How to avoid:

• Start from the definition: $\kappa = \dfrac{1}{R} \times G^*$
• Rearranging for the cell constant: $G^* = \kappa \times R$
• Sanity check: cell constant is a fixed geometric number (typically $0.1$–$10\ \text{cm}^{-1}$ for standard cells) — if your answer looks like a resistance or a conductivity, you've used the wrong formula.

Mistake 2: Mishandling the $10^{-3}$ in $\kappa = 0.146 \times 10^{-3}\ \text{S cm}^{-1}$

The error: Students drop the $\times 10^{-3}$ or misplace the decimal when multiplying by $R = 1500\ \Omega$, landing on $219\ \text{cm}^{-1}$ or $0.0219\ \text{cm}^{-1}$ instead of $0.219\ \text{cm}^{-1}$.

How to avoid:

• Convert to plain decimal or keep scientific notation consistently: $\kappa = 0.146 \times 10^{-3} = 1.46 \times 10^{-4}\ \text{S cm}^{-1}$
• Multiply: $G^* = 1.46 \times 10^{-4} \times 1500 = 1.46 \times 10^{-4} \times 1.5 \times 10^{3} = 2.19 \times 10^{-1} = 0.219\ \text{cm}^{-1}$

Mistake 3: Forgetting Cell Constant Has Units of $\text{cm}^{-1}$

The error: Students report the answer as a pure number (e.g., "0.219") with no unit, or attach the wrong unit like $\text{S cm}^{-1}$ (the unit of $\kappa$, not $G^*$).

Why it happens: $G^* = \kappa \times R$, and since $\kappa$ is $\text{S cm}^{-1}$ and $R$ is $\Omega = \text{S}^{-1}$, the Siemens cancel, leaving only $\text{cm}^{-1}$ — students often don't carry the units through the multiplication to see this.

How to avoid:

• Track units explicitly: $\text{S cm}^{-1} \times \Omega = \text{S cm}^{-1} \times \text{S}^{-1} = \text{cm}^{-1}$
• Always write the final answer as $G^* = 0.219\ \text{cm}^{-1}$, never as a bare number.

Mistake 4: Confusing the Standard KCl Solution With an Unknown

The error: Students think the $0.001\ M$ KCl and its conductivity are the unknown being solved for, rather than realising this is a standard reference solution used only to calibrate the cell.

How to avoid: …