Q.Which of the following complexes formed by Cu2+ ions is most stable?
Concept understanding — Magnetic Moment Calculation
From a Paperclip to a Magnet: The Intuition
You already know that a magnet can pick up iron nails. But what is actually happening inside that nail when it gets near the magnet? And why does a plastic comb, rubbed on hair, pick up tiny bits of paper — but never iron filings?
The answer lies in magnetization — the process by which a material becomes magnetic.
Think of a piece of iron as a chaotic crowd of tiny compass needles. Each needle is an atomic magnetic moment (a tiny magnet, arising from the spin of electrons). In unmagnetized iron, these needles point in random directions. Their magnetic effects cancel out, so the iron as a whole shows no net magnetism.
Now bring a strong magnet close. Its magnetic field acts like a command: "Line up!" The tiny compass needles inside the iron start rotating, aligning themselves with the external field. The more they align, the stronger the iron's own magnetic field becomes. This alignment is magnetization.
Magnetization is not the same as inducing a current. It is a purely magnetic reorientation of atomic dipoles inside a material.
The Precise Definition
Magnetization (M) is the net magnetic dipole moment per unit volume of a material. It tells you how strongly a material is magnetized — how many tiny atomic magnets are aligned, and in which direction.
If a material has N atoms per unit volume, each with an average magnetic moment μavg, then:
M=Nμavg
The SI unit of M is amperes per metre (A/m). Why? Because a magnetic dipole moment has units of A·m², and dividing by volume (m³) gives A/m.
M=volumetotal magnetic dipole moment
How Magnetization Connects to the Magnetic Field
When a material gets magnetized, it produces its own magnetic field. The total magnetic field B inside the material is the sum of:
- The external applied field H (caused by free currents, like the current in a solenoid)
- The material's response — the magnetization M
The fundamental relation is:
B=μ0(H+M)
where μ0=4π×10−7T⋅m/A is the permeability of free space.
Do not confuse H (magnetic field intensity, or "magnetizing field") with B (magnetic flux density). H is what you apply; M is what the material does; B is the total field you measure.
The Three Kinds of Magnetic Materials
Not all materials respond the same way to an external field. The magnetization M is proportional to H for most materials (at least for small fields):
M=χmH
where χm is the magnetic susceptibility — a dimensionless number that tells you how easily a material magnetizes.
| Material Type | χm | Behaviour | Example |
|---|---|---|---|
| Diamagnetic | Small and negative (≈−10−5) | Weakly repelled by a magnet; M opposes H | Water, copper, bismuth |
| Paramagnetic | Small and positive (≈10−5 to 10−3) | Weakly attracted; M aligns with H | Aluminium, oxygen gas |
| Ferromagnetic | Large and positive (≫1) | Strongly attracted; M can be huge and persists even after H is removed | Iron, nickel, cobalt |
Ferromagnetic materials have spontaneous magnetization — their atomic moments align even without an external field, forming magnetic domains. Magnetization in these materials is not linear; it saturates and shows hysteresis.
A Simple Example
Take a long iron rod placed inside a solenoid carrying current I. The solenoid produces a uniform field H inside. The iron rod becomes magnetized: its atomic moments align, producing M in the same direction as H.
If χm for iron is about 5000, then M=5000H. The total field inside the rod becomes:
B=μ0(H+5000H)=μ0(5001)H
That is why an iron core can amplify the magnetic field of a solenoid by thousands of times.
The Bottom Line
Magnetization is the measure of how much a material becomes magnetic when placed in an external field. It arises from the alignment of atomic magnetic dipoles. For linear materials, M=χmH. For ferromagnets, the response is nonlinear, strong, and can be permanent — that is how you get a bar magnet from a piece of iron.
Magnetic moment calculation using the spin-only formula is a numerically important topic in the NCERT/CBSE Class 12 Chemistry chapters on d-Block Elements and Coordination Compounds, and ‘spin only formula magnetic moment’ is a frequently searched important-question topic for board exams, JEE Main and NEET. Calculating the number of unpaired electrons correctly is a skill tested repeatedly in competitive-exam chemistry numericals.
Why this formula?
Magnetic Moment Calculation: Why the Formula Holds
Let's build this from first principles — understanding the why before the formula.
1. What is Magnetic Moment?
A magnetic moment (μ) is a measure of the strength and orientation of a magnet or current loop. It tells us how strongly an object will interact with an external magnetic field.
The core idea: any moving charge creates a magnetic field. A loop of current is like a tiny bar magnet — its magnetic moment quantifies this.
2. The Fundamental Formula: Current Loop
The Setup
Consider a planar loop of wire carrying a steady current I, enclosing an area A.
Why μ=IA?
Step 1: Force on a moving charge
A charge q moving with velocity v in a magnetic field B experiences:
F=q(v×B)
Step 2: Torque on a current loop
For a rectangular loop of sides a and b (A=ab), placed in a uniform B:
- Current I means charge flows. On side of length a, the force magnitude is F=IaB (since I=tq and v=ta).
- These forces on opposite sides form a couple (equal, opposite, not collinear).
- Torque τ=force×perpendicular distance=(IaB)×(bsinθ)
Step 3: Recognize the pattern
τ=I(ab)Bsinθ=IABsinθ
This looks exactly like:
τ=μBsinθ
Comparing, we identify:
μ=IA
Why this works: The torque on a current loop is proportional to the current and the area — this product naturally defines the magnetic moment.
3. For a Single Moving Charge (Orbital Magnetic Moment)
The Setup
An electron of charge −e moves in a circular orbit of radius r with speed v.
Why μ=2evr?
Step 1: Treat orbit as a current loop
- Time for one revolution: T=v2πr
- Current (charge per unit time): I=Te=2πrev
Step 2: Apply μ=IA
- Area of orbit: A=πr2
- So: μ=(2πrev)(πr2)=2evr
Step 3: Express in terms of angular momentum
- Orbital angular momentum: L=mvr
- Therefore: μ=2meL
Why this matters: The magnetic moment is directly proportional to angular momentum. The factor 2me is called the gyromagnetic ratio — it links mechanics to magnetism.
4. For a Solenoid (Many Turns)
The Setup
A solenoid of N turns, length l, carrying current I, cross-sectional area A.
Why μ=NIA?
Each turn contributes μturn=IA. For N identical turns in series:
μtotal=N⋅(IA)=NIA
If the solenoid has n=N/l turns per unit length:
μ=(nl)IA
Key insight: The magnetic moment adds linearly for multiple turns because each turn's torque contribution adds.
5. Summary Table of Key Results
| System | Formula | Why |
|---|---|---|
| Single current loop | μ=IA | Torque on loop ∝IA |
| Orbiting electron | μ=2meL | Current from orbital motion |
| Solenoid | μ=NIA | Sum of individual loop moments |
| General definition | μ=21∫r×JdV | For continuous current distributions |
6. Exam-Relevant Takeaway
Always remember:
- Magnetic moment always involves current × area (or equivalent)
- For particles, it's charge-to-mass ratio × angular momentum
- Direction: given by right-hand rule (curl fingers along current, thumb points along μ)
The formula isn't arbitrary — it emerges naturally from the torque a current loop experiences in a magnetic field.
The key idea is that the stability constant K (or its logarithm) directly measures how stable a complex is — a higher logK means the equilibrium lies further to the right, so the complex is more stable.
Step 1: For each complex, the given logK value is the logarithm of the formation (stability) constant.
Step 2: Compare the numerical values:
- (i) logK=11.6
- (ii) logK=27.3
- (iii) logK=15.4
- (iv) logK=8.9
Step 3: The largest logK is 27.3, corresponding to [Cu(CN)4]2−.
The most stable complex is [Cu(CN)4]2− (option ii), with logK=27.3.
The stability of a complex is directly measured by its formation constant K; the larger the logK, the more stable the complex. Here, the complex with logK=27.3 is the most stable.
The question asks which complex is most stable. In coordination chemistry, the stability of a complex is quantified by its formation constant (also called stability constant) K. The reaction given is the formation of the complex from the metal ion and ligands. A larger K means the equilibrium lies further to the right — the complex is more stable and less likely to dissociate.
The values are given as logK, so we compare these directly. No conversion is needed: the highest logK corresponds to the highest K, hence the most stable complex.
Let’s go through each option:
-
Option (i): Cu2++4NH3⇌[Cu(NH3)4]2+, logK=11.6
This is a moderately stable complex. Ammonia is a good ligand, but not exceptionally strong for copper(II).
-
Option (ii): Cu2++4CN−⇌[Cu(CN)4]2−, logK=27.3
Cyanide ion is a very strong ligand (high field strength, forms strong σ and π bonds). The logK is dramatically higher than the others — over 10 orders of magnitude larger in K than the next closest.
-
Option (iii): Cu2++2en⇌[Cu(en)2]2+, logK=15.4
Ethylenediamine (en) is a bidentate ligand, which gives a chelate effect — this usually increases stability compared to monodentate ligands like NH3. Indeed, logK=15.4 is higher than for NH3 (11.6), but still far below CN−.
-
Option (iv): Cu2++4H2O⇌[Cu(H2O)4]2+, logK=8.9
Water is a weak ligand. This is the least stable complex here.
A common mistake is to think that chelating ligands (like en) always form the most stable complexes. While the chelate effect does enhance stability, the intrinsic ligand strength matters more. Here, CN− is such a powerful ligand that it overcomes the chelate advantage.
You don’t need to calculate K from logK — just compare the logK values directly. The largest logK means the largest K, hence the most stable complex.
The most stable complex is formed with cyanide ions, option (ii).
Method: Stability Constant Comparison
The stability of a complex is directly measured by its formation constant (Kf).
A higher Kf means the complex is more stable — it forms more readily and dissociates less.
Steps
- Recall the relationship The given values are logK (base 10). The actual formation constant is:
Kf=10logK
-
Compare logK values directly
Since Kf increases with logK, the complex with the largest logK is the most stable.
-
Identify the largest logK
- (i) logK=11.6
- (ii) logK=27.3
- (iii) logK=15.4
- (iv) logK=8.9
logK=27.3 is the highest.
-
Conclude
The complex with CN− is the most stable.
Final Answer
Option (ii): [Cu(CN)4]2− is the most stable complex.
Common Mistakes & How to Avoid Them
Mistake 1: Confusing logK with K
The error: Students compare logK values directly and think the largest logK means the least stable complex.
Why it's wrong:
A higher logK means a larger equilibrium constant K, which indicates greater stability of the complex.
How to avoid:
Remember:
- logK↑⟹K↑⟹ more stable complex
- For this question: 27.3>15.4>11.6>8.9, so option (ii) is most stable.
Mistake 2: Forgetting that logK is directly proportional to stability
The error: Some students think a lower logK means the reaction "goes more to completion" — this is backwards.
Why it's wrong:
The equilibrium constant K for complex formation is:
K=[Cu2+][ligand]n[complex]
A larger K means the equilibrium lies far to the right — more complex formed, hence more stable.
How to avoid:
Write the expression for K and reason:
- Big K → products favoured → stable complex
- Small K → reactants favoured → unstable complex
Mistake 3: Ignoring the denticity of ligands
The error: Students compare logK values without considering that en (ethylenediamine) is bidentate, while NH3, CN−, and H2O are monodentate.
Why it matters:
A bidentate ligand like en forms a chelate ring, which gives extra stability (chelate effect). Even though logK for en (15.4) is less than for CN− (27.3), the chelate effect is already included in the given logK value.
How to avoid:
- The logK values already account for denticity — compare them directly.
- Do not try to "adjust" the values manually.
Mistake 4: Overthinking magnetic moment or geometry
The error: Students try to use magnetic moment or crystal field theory to decide stability.
Why it's wrong:
The question gives experimental logK values — these are the direct measure of stability. Magnetic moment tells you about unpaired electrons, not thermodynamic stability.
How to avoid:
- When logK (or K) is given, use it directly.
- Save magnetic moment reasoning for questions about geometry, spin state, or colour.
Mistake 5: Misreading the question as "least stable"
The error: Students accidentally pick the smallest logK (option iv) because they read "most stable" as "least stable".
How to avoid:
- Circle the word "most" or "least" in the question.
- Double-check: largest logK = most stable.
Final Answer
Most stable complex: Option (ii) [Cu(CN)4]2− with logK=27.3
Quick check:
- (ii) logK=27.3 → largest → most stable ✓
- (iv) logK=8.9 → smallest → least stable
- KEAM 2026Set eng-2026-04194 marksMCQQ.Some transition metal ions given below contain spin only magnetic moment (BM). Which of the following is not correctly matched? (A) Ni2+ (Z=28) 4.73 (B) Ti2+ (Z=22) 2.84 (C) Mn2+ (Z=25) 5.92 (D) Fe2+ (Z=26) 4.90 (E) Co2+ (Z=27) 3.87
›Reveal solutionSolution
[!TLDR]
Using μ=n(n+2), Ni²⁺ has 2 unpaired electrons (μ≈2.83 BM), so its listed value of 4.73 BM is the incorrect match.
Concept
The spin-only magnetic moment (from the NCERT/CBSE-aligned d-block chapter KEAM follows) is μ=n(n+2) Bohr Magnetons, where n is the count of unpaired d electrons. First find each ion's dx configuration, then n, then μ.
Solution
Evaluate each ion (remove electrons from 4s first, then 3d):
- (A) Ni²⁺ (Z=28) →3d8: 2 unpaired, μ=2⋅4=8=2.83 BM. Listed 4.73 → wrong.
- (B) Ti²⁺ (Z=22) →3d2: 2 unpaired, μ=8=2.83≈2.84 BM ✓
- (C) Mn²⁺ (Z=25) →3d5: 5 unpaired, μ=5⋅7=35=5.92 BM ✓
- (D) Fe²⁺ (Z=26) →3d6: 4 unpaired, μ=4⋅6=24=4.90 BM ✓
- (E) Co²⁺ (Z=27) →3d7: 3 unpaired, μ=3⋅5=15=3.87 BM ✓
Only Ni²⁺ fails: its correct value is 2.83 BM, not 4.73 BM.
[!ANSWER]
(A) Ni²⁺ (Z=28) — the mismatch; its spin-only moment is ≈ 2.83 BM.
- KEAM 2026Set pha-2026-0418F4 marksMCQQ.The calculated magnetic moment of two dipositive ions of 3d series element is 4.9 BM. The ions are (A) Ti2+ and Sc2+ (B) Mn2+ and Cr2+ (C) V2+ and Ti2+ (D) Cr2+ and Fe2+ (E) Fe2+ and Ni2+
›Reveal solutionSolution
CH3CH2CH(C2H5)CH2CH(CH3)CH2CH3 is 3-ethyl-5-methylheptane.
Reasoning
Select the longest continuous chain: it contains 7 carbons (heptane). Numbering to give the lowest locants:
- an ethyl group at C-3,
- a methyl group at C-5.
Either direction gives locants {3,5}; alphabetically 'ethyl' is cited first and takes the lower number (3). Name: 3-ethyl-5-methylheptane.
(Options naming a 'hexane' parent are wrong because the longest chain has 7 carbons.)
✓Final answerThe correct option is (D).
- KEAM 2026Set pha-2026-0419F4 marksMCQQ.Which of the following pair of transition metal ions are diamagnetic? (A) Ti2+ and Mn2+ (B) Mn2+ and Ni2+ (C) V2+ and Cr2+ (D) Co2+ and Ni2+ (E) Sc3+ and Zn2+
›Reveal solutionSolution
Diamagnetic means no unpaired electrons. Sc3+ is 3d0 and Zn2+ is 3d10 — both have all electrons paired.
Sc3+: [Ar]3d0 (0 unpaired) and Zn2+: [Ar]3d10 (0 unpaired) are both diamagnetic. The other options (Ti2+, Mn2+, Ni2+, V2+, Cr2+, Co2+) all have unpaired d-electrons and are paramagnetic.
✓Final answerThe correct option is (E).
- KEAM 2025Set eng-2025-04254 marksMCQQ.The magnetic moment of a divalent ion in aqueous solution is 3.87 BM. The number of unpaired electrons present in it is (A) 4 (B) 5 (C) 3 (D) 2 (E) 1
›Reveal solutionSolution
The spin-only magnetic moment is μ=n(n+2) BM. Setting μ=3.87 gives n(n+2)=15, so n=3 unpaired electrons.
The spin-only formula relates magnetic moment to the number of unpaired electrons:
μ=n(n+2) BM
Squaring the given moment:
n(n+2)=(3.87)2≈15
Solving, n=3 (since 3×5=15), giving μ=15=3.87 BM. Hence the divalent ion has 3 unpaired electrons.
✓Final answerThe correct option is (C).
- KEAM 2025Set eng-2025-04264 marksMCQQ.Which of the following transition metal has the highest magnetic moment? (A) Sc3+ (B) Ti3+ (C) Cr2+ (D) Fe2+ (E) Mn2+
›Reveal solutionSolution
Magnetic moment rises with unpaired electrons; Mn2+ (d5) has 5 unpaired e⁻, the maximum here.
Spin-only magnetic moment μ=n(n+2) BM, so more unpaired electrons (n) means a larger moment.
- Sc3+: d0, n=0.
- Ti3+: d1, n=1.
- Cr2+: d4, n=4.
- Fe2+: d6, n=4.
- Mn2+: d5, n=5, giving μ=5×7=35=5.92 BM.
Mn2+ has the most unpaired electrons and hence the highest magnetic moment.
✓Final answerThe correct option is (E).
- KEAM 2025Set eng-2025-04274 marksMCQQ.The transistion metal ion with the highest magnetic moment is (A) Fe2+ (B) Mn2+ (C) Ni2+ (D) Co2+ (E) Cr2+
›Reveal solutionSolution
Magnetic moment rises with the number of unpaired electrons. Mn2+ (d5) has 5 unpaired electrons, more than any other ion listed, so it has the highest moment.
Reasoning
Spin-only magnetic moment: μ=n(n+2)BM, where n = number of unpaired electrons.
Electronic configurations (high spin):
- (A) Fe2+: d6 → 4 unpaired
- (B) Mn2+: d5 → 5 unpaired
- (C) Ni2+: d8 → 2 unpaired
- (D) Co2+: d7 → 3 unpaired
- (E) Cr2+: d4 → 4 unpaired
Mn2+ has the most unpaired electrons (5):
μ=5(5+2)=35=5.92BM
which is the highest of all the ions.
✓Final answerThe correct option is (B).
- KEAM 2025Set eng-2025-04294 marksMCQQ.Which of the following metal ion is diamagnetic? (A) Zn2+ (B) Ni2+ (C) Co2+ (D) Cu2+ (E) Mn2+
›Reveal solutionSolution
Zn2+=3d10 (fully filled, no unpaired electrons) is diamagnetic; the others have unpaired d-electrons and are paramagnetic.
Magnetic behaviour depends on unpaired electrons. The d-electron counts of the M2+ ions are:
- Zn2+: [Ar]3d10 — 0 unpaired electrons → diamagnetic.
- Ni2+: 3d8 — 2 unpaired.
- Co2+: 3d7 — 3 unpaired.
- Cu2+: 3d9 — 1 unpaired.
- Mn2+: 3d5 — 5 unpaired.
Only Zn2+, with a completely filled 3d10 subshell, has all electrons paired and is diamagnetic.
✓Final answerThe correct option is (A).
- KEAM 2025Set pha-2025-0429F4 marksMCQQ.What is the magnetic moment of divalent ion with three unpaired electrons? (A) 2.84 BM (B) 5.92 BM (C) 3.87 BM (D) 4.90 BM (E) 1.73 BM
›Reveal solutionSolution
The spin-only magnetic moment is μ=n(n+2) BM. For 3 unpaired electrons, μ=15≈3.87 BM.
Reasoning
The spin-only formula gives the magnetic moment from the number of unpaired electrons n:
μ=n(n+2) BM
With n=3:
μ=3(3+2)=15=3.87 BM
(The '+2' charge is not needed; only the unpaired-electron count matters.)
✓Final answerThe correct option is (C).
- KEAM 2023Set eng-2023-P1-A14 marksMCQQ.Geometry, hybridisation and magnetic moment of [MnBr4]2−,[FeF6]4−, and [Ni(CN)4]2− ions, respectively, are: (A) Tetrahedral, square planar, octahedral; sp3,dsp3,sp3d2; 5.9, 0, 4.9 (B) Tetrahedral, octahedral, square planar; sp3,sp3d2,dsp2; 5.9, 4.9, 0 (C) Octahedral, square planar, tetrahedral; sp3d2,dsp2,sp3; 4.9, 0, 5.9 (D) Square planar, tetrahedral, octahedral; sp3d2,sp3,dsp2; 0, 4.9, 5.9 (E) Tetrahedral, octahedral, square planar; sp3,sp3d2,dsp2; 0, 5.9, 4.9.
›Reveal solutionSolution
[MnBr4]2- is tetrahedral (sp3, μ=5.9), [FeF6]4- octahedral (sp3d2, μ=4.9), and [Ni(CN)4]2- square planar (dsp2, μ=0).
Concept and Intuition
Geometry and magnetic moment depend on the metal d-count, ligand field strength, and coordination number. Weak-field ligands give high-spin outer-orbital complexes; strong-field CN- pairs electrons giving diamagnetic dsp2 square-planar Ni2+.
Step-by-Step Solution
- [MnBr4]2-: Mn2+ is d5; weak-field Br-, CN 4 → tetrahedral, sp3, 5 unpaired → μ = √(5·7) ≈ 5.9 BM.
- [FeF6]4-: Fe2+ is d6; weak-field F-, CN 6 → octahedral high-spin, sp3d2, 4 unpaired → μ ≈ 4.9 BM.
- [Ni(CN)4]2-: Ni2+ is d8; strong-field CN-, CN 4 → square planar, dsp2, 0 unpaired → μ = 0.
- Matches option B: tetrahedral, octahedral, square planar; sp3, sp3d2, dsp2; 5.9, 4.9, 0.
Common Mistakes
- Assigning dsp2 to weak-field tetrahedral/octahedral complexes; only strong-field CN- forces the low-spin square-planar Ni case.
✓Final answerThe correct option is (B) — Tetrahedral, octahedral, square planar; sp3, sp3d2, dsp2; 5.9, 4.9, 0.
ANSWER: B
- KEAM 2021Set eng-2021-P1-A14 marksMCQQ.The magnetic moment of a trivalent ion of a metal with Z=24 in aqueous solution is (A) 3.87 BM (B) 2.84 BM (C) 1.73 BM (D) 4.90 BM (E) 5.92 BM
›Reveal solutionSolution
The magnetic moment of Cr3+ is 3.87 BM.
Concept and Intuition
The spin-only magnetic moment depends on the number of unpaired electrons: μ=n(n+2) BM.
Step-by-Step Solution
- Z=24 is Cr, configuration [Ar]3d54s1.
- Cr3+: remove three electrons → [Ar]3d3.
- 3d3 has n=3 unpaired electrons.
- μ=3(3+2)=15=3.87 BM.
Common Mistakes
- Counting electrons from 4s first but miscounting unpaired d-electrons for Cr3+.
✓Final answerThe correct option is (A) — 3.87 BM.
ANSWER: A
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