The cell constant of a conductivity cell _______. (i) changes with change of electrolyte. (ii) changes with ch

NCERT Exemplar · Q14

Q. The cell constant of a conductivity cell _____________.

(i) changes with change of electrolyte.

(ii) changes with change of concentration of electrolyte.

(iii) changes with temperature of electrolyte.

(iv) remains constant for a cell.

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Concept understanding — Molar Conductivity

From Resistance to Conductance: Flipping the Idea

You already know resistance ( $R$ ) — it tells you how much a material opposes the flow of current. A high resistance means the wire fights the current; a low resistance means it lets current through easily.

Now flip that thought. Instead of asking "how much does it resist?", ask "how easily does it let current flow?" That's exactly what conductance measures.

Note

Conductance ( $G$ ) is the reciprocal of resistance:

$G = \frac{1}{R}$

Unit: siemens (S) — named after Werner von Siemens. 1 S = 1 A/V (ampere per volt).

If a wire has $R = 10\ \Omega$ , its conductance is $G = 0.1\ \text{S}$ . If $R = 0.5\ \Omega$ , $G = 2\ \text{S}$ — it conducts twice as well.

Ohm's Law in Conductance Form

You know $V = IR$ . Rearranging:

I = \frac{V}{R} = G V

So current = conductance × voltage. A high-conductance material draws a large current for the same voltage — it's a "good conductor."

Now, Conductivity: The Material's Intrinsic Property

Resistance depends on two things: the material itself (its "resistivity" $\rho$ ) and the geometry (length $L$ , cross-sectional area $A$ ):

R = \rho \frac{L}{A}

Conductance also depends on geometry. A thicker wire (larger $A$ ) or a shorter wire (smaller $L$ ) has higher conductance. To isolate the material's inherent ability to conduct, we define conductivity ( $\sigma$ ):

\sigma = \frac{1}{\rho}

Conductivity is the reciprocal of resistivity. It tells you how well the material itself conducts, independent of shape and size.

Unit: siemens per metre (S/m).
High $\sigma$ → good conductor (copper: $\approx 5.8 \times 10^7\ \text{S/m}$ ).
Low $\sigma$ → poor conductor / insulator (glass: $\approx 10^{-12}\ \text{S/m}$ ).

Watch out

Don't confuse conductance (property of a specific object, depends on geometry) with conductivity (property of the material, independent of geometry). A short thick copper wire has high conductance; a long thin copper wire has lower conductance — but both have the same conductivity.

The Big Picture in One Table

Quantity	Symbol	Definition	Depends on	Unit
Resistance	$R$	$V/I$	Material + geometry	$\Omega$
Resistivity	$\rho$	$R A / L$	Material only	$\Omega \cdot \text{m}$
Conductance	$G$	$1/R$	Material + geometry	S
Conductivity	$\sigma$	$1/\rho$	Material only	S/m

Intuitive Analogy

Think of a water pipe:

Resistance = how hard it is to push water through (narrow, long pipe).
Conductance = how easily water flows (wide, short pipe). …

Why this formula?

Conductance and Conductivity: Why the Formulas Hold

Let's build this from first principles — understanding the why before the what.

1. The Core Idea: How Easily Does Current Flow?

Think of a conductor (like a copper wire). When you apply a voltage across it, electrons drift through the material. Two questions arise:

How much current flows for a given voltage? → This is conductance ( $G$ ).
How well does the material itself allow current? → This is conductivity ( $\sigma$ ).

The key distinction: Conductance depends on the size and shape of the object. Conductivity is an intrinsic property of the material.

2. Ohm's Law in Terms of Conductance

You know Ohm's law:

V = IR

But we can rewrite it as:

I = \frac{V}{R}

Define conductance $G$ as the reciprocal of resistance:

G = \frac{1}{R}

So:

I = G V

Why this makes sense:

A larger $G$ means more current for the same voltage — the conductor "conducts" better.
$G$ has units of siemens (S) = $\text{A/V}$ .

3. From Resistance to Conductivity: The Geometry Factor

Resistance of a uniform conductor depends on:

Length $L$ (longer → more resistance)
Cross-sectional area $A$ (thicker → less resistance)
Material property $\rho$ (resistivity)

The formula:

R = \rho \frac{L}{A}

Now, conductivity $\sigma$ is the reciprocal of resistivity:

\sigma = \frac{1}{\rho}

So:

R = \frac{1}{\sigma} \cdot \frac{L}{A}

Why this form?

If you double the length, electrons have to travel twice as far, colliding more → resistance doubles.
If you double the area, there's twice as many "lanes" for electrons → resistance halves.

4. The Key Formula: Conductance in Terms of Conductivity

Since $G = 1/R$ , we get:

G = \sigma \frac{A}{L}

This is the central relationship. Let's see why it holds:

$\sigma$ tells you how well the material conducts (intrinsic).
$A/L$ tells you how the geometry amplifies or reduces that.

Intuition:

A fat, short wire ( $A$ large, $L$ small) has high conductance.
A thin, long wire ( $A$ small, $L$ large) has low conductance.
A material with high $\sigma$ (like copper) gives higher $G$ than one with low $\sigma$ (like iron), for the same shape.

5. Microscopic Derivation (Why $\sigma$ Exists)

At the microscopic level, conductivity arises from electron motion:

\sigma = n e \mu

Where:

$n$ = number of free electrons per unit volume
$e$ = electron charge …

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The cell constant is a fixed property of the physical geometry of a conductivity cell (electrode area and separation). It does not depend on the electrolyte, its concentration, or its temperature. The correct answer is (iv).

The core idea: What is the cell constant?

When you measure conductivity, you're not directly measuring the intrinsic conductivity ( $\kappa$ ) of the solution. Instead, you measure the resistance ( $R$ ) of the column of solution between the two electrodes. The relationship is:

R = \frac{1}{\kappa} \cdot \frac{l}{A}

Here, $l$ is the distance between the electrodes, and $A$ is the area of each electrode. The ratio $\frac{l}{A}$ is a purely geometric factor — it depends only on how the cell is built. This ratio is called the cell constant, $G^*$ :

G^* = \frac{l}{A}

So the measured resistance is:

R = \frac{G^*}{\kappa}

Once you know $G^*$ (by calibrating with a standard solution of known $\kappa$ ), you can find the conductivity of any unknown solution from its measured resistance.

Step-by-step reasoning

1. The cell constant is a physical constant of the cell.

It is determined by the distance between the two electrodes and their surface area. These are fixed once the cell is manufactured. You cannot change them by dipping the cell into a different liquid.

2. Does changing the electrolyte change $G^*$ ?

No. Swapping KCl for NaCl does not move the electrodes closer together or change their area. The cell constant remains exactly the same.

→ Option (i) is wrong.

3. Does changing the concentration change $G^*$ ?

No. Diluting the solution does not alter the geometry of the cell. The electrodes stay put.

→ Option (ii) is wrong.

4. Does temperature change $G^*$ ? …

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Here are the common mistakes students make on this question about cell constant and molar conductivity, along with how to avoid each.

1. Confusing Cell Constant with Conductivity or Molar Conductivity

The Mistake:

Students think the cell constant depends on the electrolyte, its concentration, or temperature — because they know conductivity and molar conductivity change with these factors.

Why it’s wrong:

The cell constant ( $G^*$ ) is defined as:

G^* = \frac{l}{A}

where $l$ is the distance between electrodes and $A$ is the cross-sectional area of the electrodes. It is a physical property of the cell, not of the solution.

How to avoid:

Memorise: Cell constant = geometry of the cell only.
Remember that conductivity ( $\kappa$ ) is measured using:

\kappa = G^* \times G

where $G$ is the measured conductance. The cell constant is the multiplier, not the thing being multiplied.

2. Thinking “Constant” Means Unaffected by Temperature

The Mistake:

Students see “remains constant for a cell” and think it’s wrong because the cell’s dimensions might expand with temperature.

The Reality:

Yes, $l$ and $A$ can change slightly with temperature due to thermal expansion, but in standard exam contexts (especially for this MCQ), the cell constant is treated as fixed for a given cell — independent of the electrolyte, concentration, and temperature.

How to avoid:

In exam questions, treat “remains constant for a cell” as the correct option unless the question explicitly mentions thermal expansion effects (rare in Class 12 / JEE / NEET).
Focus on the intent: the cell constant does not change when you change the solution.

3. Misreading the Options — “Changes with” vs “Remains constant”

The Mistake:

Students pick (i), (ii), or (iii) because they think “cell constant” is the same as “conductance” or “molar conductivity.”

Quick check table:

Quantity	Changes with electrolyte?	Changes with concentration?	Changes with temperature?
Cell constant ( $l/A$ )	✗ No	✗ No	✗ No (in standard context)
Conductance ( $G$ )	✓ Yes	✓ Yes	✓ Yes

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Q.The cell constant of a conductivity cell _____________.

Concept understanding — Molar Conductivity

From Resistance to Conductance: Flipping the Idea

Ohm's Law in Conductance Form

Now, Conductivity: The Material's Intrinsic Property

The Big Picture in One Table

Intuitive Analogy

Why this formula?

Conductance and Conductivity: Why the Formulas Hold

1. The Core Idea: How Easily Does Current Flow?

2. Ohm's Law in Terms of Conductance

3. From Resistance to Conductivity: The Geometry Factor

4. The Key Formula: Conductance in Terms of Conductivity

5. Microscopic Derivation (Why $\sigma$ Exists)

The core idea: What is the cell constant?

Step-by-step reasoning

Method: Definition-Based Elimination

Step 1: Recall the definition

Step 2: Identify what it depends on

Step 3: Eliminate incorrect options …

1. Confusing Cell Constant with Conductivity or Molar Conductivity

2. Thinking “Constant” Means Unaffected by Temperature

3. Misreading the Options — “Changes with” vs “Remains constant”

Q.The cell constant of a conductivity cell _____________.

The core idea: What is the cell constant?

Step-by-step reasoning

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