Lambda⁰m(NH_4OH) is equal to _______. (i) Lambda⁰m – Class 12 Chemistry

NCERT Exemplar · Q16

\Lambda^0_{m(NH_4OH)}

(i)

\Lambda^0_{m(NH_4OH)} + \Lambda^0_{m(NH_4Cl)} - \Lambda^0_{m(HCl)}

(ii)

\Lambda^0_{m(NH_4Cl)} + \Lambda^0_{m(NaOH)} - \Lambda^0_{m(NaCl)}

(iii)

\Lambda^0_{m(NH_4Cl)} + \Lambda^0_{m(NaCl)} - \Lambda^0_{m(NaOH)}

(iv)

\Lambda^0_{m(NaOH)} + \Lambda^0_{m(NaCl)} - \Lambda^0_{m(NH_4Cl)}

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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 limiting molar conductivity of a weak electrolyte like $NH_4OH$ can be found by combining the $\Lambda_m^0$ values of strong electrolytes that share its ions. Using Kohlrausch’s law of independent ion migration, the correct expression is $\Lambda_{m(NH_4OH)}^0 = \Lambda_{m(NH_4Cl)}^0 + \Lambda_{m(NaOH)}^0 - \Lambda_{m(NaCl)}^0$ , which corresponds to option (ii).

The key idea here is Kohlrausch’s law: at infinite dilution, each ion contributes a fixed amount to the molar conductivity, independent of the other ion it travels with. So the limiting molar conductivity of any electrolyte is simply the sum of the limiting conductivities of its constituent ions.

For a weak base like $NH_4OH$ , we cannot measure $\Lambda_m^0$ directly by extrapolation (because it doesn’t fully dissociate even at low concentrations). But we can build it from the $\Lambda_m^0$ values of strong electrolytes that contain the same ions — $NH_4^+$ and $OH^-$ — by adding and subtracting known values to cancel out the unwanted ions.

Let’s see how.

Write what we want in terms of ions.

$\Lambda_{m(NH_4OH)}^0 = \lambda_{NH_4^+}^0 + \lambda_{OH^-}^0$

That’s our target.
Find strong electrolytes that give us these ions.
- $NH_4Cl$ gives $\lambda_{NH_4^+}^0 + \lambda_{Cl^-}^0$
- $NaOH$ gives $\lambda_{Na^+}^0 + \lambda_{OH^-}^0$
- $NaCl$ gives $\lambda_{Na^+}^0 + \lambda_{Cl^-}^0$
Combine them to isolate the target sum.

If we add the first two and subtract the third:

(\lambda_{NH_4^+}^0 + \lambda_{Cl^-}^0) + (\lambda_{Na^+}^0 + \lambda_{OH^-}^0) - (\lambda_{Na^+}^0 + \lambda_{Cl^-}^0)

The $\lambda_{Na^+}^0$ and $\lambda_{Cl^-}^0$ cancel perfectly, leaving:

\lambda_{NH_4^+}^0 + \lambda_{OH^-}^0 = \Lambda_{m(NH_4OH)}^0

Translate back to electrolyte notation. So: …

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Common Mistakes & How to Avoid Them

Mistake 1: Not Understanding Kohlrausch’s Law

Students often try to memorise the answer without knowing why the formula works.

How to avoid:

Kohlrausch’s Law states that at infinite dilution, molar conductivity is the sum of independent ionic contributions:

\Lambda_m^0 = \lambda_+^0 + \lambda_-^0

For a weak base like $NH_4OH$ , you cannot measure $\Lambda_m^0$ directly (it doesn’t fully dissociate). So you build it from strong electrolytes whose $\Lambda_m^0$ values are known.

Mistake 2: Forgetting to Cancel Ions Correctly

Students pick options without checking if the unwanted ions cancel out.

How to avoid:

Write each electrolyte as its ions, then cancel common ions.

For correct option (ii):

\Lambda_{m(NH_4Cl)}^0 = \lambda_{NH_4^+}^0 + \lambda_{Cl^-}^0

\Lambda_{m(NaOH)}^0 = \lambda_{Na^+}^0 + \lambda_{OH^-}^0

\Lambda_{m(NaCl)}^0 = \lambda_{Na^+}^0 + \lambda_{Cl^-}^0

Now do:

(\lambda_{NH_4^+} + \lambda_{Cl^-}) + (\lambda_{Na^+} + \lambda_{OH^-}) - (\lambda_{Na^+} + \lambda_{Cl^-})

Cancel $\lambda_{Na^+}$ and $\lambda_{Cl^-}$ → you get:

\boxed{\lambda_{NH_4^+}^0 + \lambda_{OH^-}^0 = \Lambda_{m(NH_4OH)}^0}

Mistake 3: Confusing Addition/Subtraction Signs

Students misplace the minus sign, especially in options like (iii) or (iv).

How to avoid:

Always write the ionic breakdown before deciding the sign. The target is $NH_4^+ + OH^-$ .

You need $NH_4^+$ from $NH_4Cl$
You need $OH^-$ from $NaOH$
You must subtract the common ion pair ( $Na^+ + Cl^-$ ) that appears twice — that’s $NaCl$ .

So the correct structure is:

\Lambda_{m(NH_4Cl)}^0 + \Lambda_{m(NaOH)}^0 - \Lambda_{m(NaCl)}^0

Mistake 4: Picking Option (i) Without Checking

Option (i) looks similar but uses $HCl$ instead of $NaOH$ and $NaCl$ . …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

AHSEC Higher Secondary (HS) Final Examination 2026Set ANNUAL2 marks
$Q. State the variation of conductivity and molar conductivity of acetic acid with dilution.$
›Reveal solutionSolution

Diluting acetic acid lowers its conductivity but raises its molar conductivity (increasing dissociation).

Conductivity (κ) measures the conductance of ions present in unit volume of solution. On dilution the number of ions per unit volume decreases, so conductivity always decreases with dilution.

Molar conductivity (Λ_m = κ × 1000 / c) measures the conducting power of all the ions produced by one mole of electrolyte. On dilution the volume containing one mole increases, and for a weak electrolyte like acetic acid the degree of dissociation (α) rises steeply, releasing many more ions. Therefore molar conductivity increases with dilution, and the increase is very sharp near infini …

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AHSEC Higher Secondary (HS) Final Examination 2024Set ANNUAL2 marks
$Q. Define conductivity and molar conductivity for the solution of an electrolyte. Discuss their variation with concentration.$
›Reveal solutionSolution
Conductivity falls as a solution is diluted (fewer ions per unit volume), while molar conductivity rises on dilution (each mole's ions become more independent / more of a weak electrolyte dissociates).

Conductivity (κ, specific conductance):

Conductivity is defined as the conductance of a solution of unit length (1 cm) and unit cross-sectional area (1 cm²) — i.e. it is the reciprocal of resistivity (ρ):

κ = 1/ρ

Its SI unit is S m⁻¹ (commonly expressed as S cm⁻¹). It measures how well the solution as a whole (per unit volume) conducts electricity.

Molar conductivity (Λm):

Molar conductivity is the conducting power of all the ions produced by dissolving 1 mole of an electrolyte in solution, and is related to conductivity by:

Λm = κ × 1000 / C (C = molar concentration in mol L⁻¹, κ in S cm⁻¹)

Its unit is S cm² mol⁻¹.

Variation with concentration:

Conductivity (κ) always decreases as the solution is diluted, because the number of ions per unit volume of solution decreases with dilution, even though the degree of dissociation (for weak electrolytes) increases.

Molar conductivity (Λm) always increases as concentration decreases (i.e. on dilution), because it is normalised per mole of electrolyte: …
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AHSEC Higher Secondary (HS) Final Examination 2023Set ANNUAL2 marks
$Q. The following limiting molar conductivities are given as: lambda-m-zero(H2SO4) = x S cm^2 mol^-1, lambda-m-zero(K2SO4) = y S cm^2 mol^-1, lambda-m-zero(CH3COOK) = z S cm^2 mol^-1. Calculate limiting molar conductivity of acetic acid. OR The cell potential for the following cell is 0.576V at 298K. Calculate the pH of the solution: Pt | H2(g) | H+(aq) || Cu2+(0.01M) | Cu(s). Given, E-zero(Cu2+/Cu) = 0.34V.$
›Reveal solutionSolution

Option 1 combines the three given conductivities via Kohlrausch's law to cancel out K⁺ and SO₄²⁻; Option 2 uses the Nernst equation for the H₂|H⁺ vs Cu²⁺|Cu cell to back-calculate [H⁺].

Option 1 — Limiting molar conductivity of acetic acid via Kohlrausch's law:

By Kohlrausch's law of independent migration of ions, each limiting molar conductivity splits into ionic contributions:

$2\lambda^0(H^+) + \lambda^0(SO_4^{2-}) = x \quad \text{...(i), from } H_2SO_4$

$2\lambda^0(K^+) + \lambda^0(SO_4^{2-}) = y \quad \text{...(ii), from } K_2SO_4$

$\lambda^0(CH_3COO^-) + \lambda^0(K^+) = z \quad \text{...(iii), from } CH_3COOK$

Subtracting (ii) from (i): $2\lambda^0(H^+) - 2\lambda^0(K^+) = x - y \Rightarrow \lambda^0(H^+) = \lambda^0(K^+) + \dfrac{x-y}{2}$

We want $\lambda_m^0(CH_3COOH) = \lambda^0(CH_3COO^-) + \lambda^0(H^+)$ . From (iii), $\lambda^0(CH_3COO^-) = z - \lambda^0(K^+)$ , so:

$\lambda_m^0(CH_3COOH) = \left[z - \lambda^0(K^+)\right] + \left[\lambda^0(K^+) + \frac{x-y}{2}\right] = z + \frac{x-y}{2}$

(Equivalently, $\lambda_m^0(CH_3COOH) = \lambda_m^0(CH_3COOK) + \lambda_m^0(\tfrac12H_2SO_4) - \lambda_m^0(\tfrac12K_2SO_4)$ .)

Option 2 — pH from cell potential:

Cell: $Pt|H_2(g)|H^+(aq)\ ||\ Cu^{2+}(0.01M)|Cu(s)$ , with anode = H₂/H⁺ (oxidation, $E^0=0V$ ) and cathode = Cu²⁺/Cu (reduction, $E^0=0.34V$ ).

…

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AHSEC Higher Secondary (HS) Final Examination 2020Set ANNUAL2 marks
$Q. Define molar conductivity of a solution. Explain how molar conductivity changes with change in concentration of solution for a weak and a strong electrolyte. (1+1=2)$
›Reveal solutionSolution

Molar conductivity is conductivity per mole of dissolved electrolyte; it rises on dilution for both strong and weak electrolytes, but far more steeply for weak electrolytes because dilution increases their degree of dissociation.

Definition of molar conductivity

Molar conductivity (Λm) is the conducting power of all the ions produced by dissolving one mole of an electrolyte in solution. It is related to the specific conductivity (κ) and molar concentration (C) by:

Λm = κ × 1000 / C (with κ in S cm⁻¹ and C in mol L⁻¹, giving Λm in S cm² mol⁻¹)

Variation with concentration — strong electrolytes

For strong electrolytes (fully ionised at all concentrations, e.g. NaCl, KCl), Λm increases only slowly as concentration decreases (i.e. on dilution). This is because, even though the number of ions per unit volume stays proportional to concentration (the electrolyte is always ~100% ionised), interionic attractive forces between the oppositely charged ions reduce their mobility at higher concentrations; diluting the solution weakens these interionic forces, allowing ions to move a little more freely, so Λm rises gradually. Λm for a strong electrolyte varies linearly with √C (Debye-Hückel-Onsager equation), so Λm° (the limiting molar conductivity at infinite dilution) can be found by extrapolating the Λm vs √C plot to C = 0.

Variation with concentration — weak electrolytes …

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AHSEC Higher Secondary (HS) Final Examination 2020Set ANNUAL2 marks
$Q. (a) Following reactions occur at cathode during the electrolysis of aqueous silver chloride: Ag+(aq) + e- -> Ag(s), E° = +0.80V; H+(aq) + e- -> 1/2 H2(g), E° = 0.00V. On the basis of their standard reduction electrode potential (E°) values, which reaction is feasible at the cathode and why?$
(1)
(b) State Kohlrausch law of independent migration of ions. Write an expression for the molar conductivity of acetic acid at infinite dilution according to Kohlrausch law. (1/2+1/2=1)
›Reveal solutionSolution

(a) The half-reaction with the more positive standard reduction potential occurs preferentially at the cathode. (b) Kohlrausch's law lets us calculate the limiting molar conductivity of any electrolyte, including weak ones like acetic acid, by summing the independent ionic contributions.

(a) Which reaction occurs at the cathode?

Given: Ag+(aq) + e- → Ag(s), E° = +0.80 V and H+(aq) + e- → ½H2(g), E° = 0.00 V

At the cathode, reduction occurs, and between two competing reduction half-reactions, the one with the higher (more positive) standard reduction potential is thermodynamically more favourable and occurs preferentially, since it has the greater tendency to be reduced (to gain electrons). Since E°(Ag+/Ag) = +0.80 V is more positive than E°(H+/H2) = 0.00 V, silver ions are reduced in preference to hydrogen ions at the cathode:

Ag+(aq) + e- → Ag(s)

(b) Kohlrausch's law of independent migration of ions

At infinite dilution, when dissociation of an electrolyte is complete and interionic interactions vanish, each ion migrates independently of the other ion with which it is associated, and each ion makes its own definite contribution to the total molar conductivity of the electrolyte, regardless of the nature of the other ion present. Mathematically, for an electrolyte that dissociates into ν+ cations and ν- anions:

Λm° = ν+ λ°+ + ν- λ°-

where λ°+ and λ°- are the limiting (infinite-dilution) molar conductivities of the cation and anion respectively.

Molar conductivity of acetic acid at infinite dilution …

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Q. $\Lambda^0_{m(NH_4OH)}$ is equal to ______________.

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)

Method: Kohlrausch’s Law of Independent Migration of Ions

Steps

Common Mistakes & How to Avoid Them

Mistake 1: Not Understanding Kohlrausch’s Law

Mistake 2: Forgetting to Cancel Ions Correctly

Mistake 3: Confusing Addition/Subtraction Signs

Mistake 4: Picking Option (i) Without Checking

Q.Λm(NH4OH)0\Lambda^0_{m(NH_4OH)}Λm(NH4​OH)0​ is equal to ______________.

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Q. $\Lambda^0_{m(NH_4OH)}$ is equal to ______________.

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