Q.Calculate the magnetic moment of a divalent ion in aqueous solution if its atomic number is 25.
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🔒 Start your 14-day free trial to unlock the full solution →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 |
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? …
The key idea is that for a divalent ion in aqueous solution, the magnetic moment depends only on the number of unpaired electrons, as orbital angular momentum is quenched.
Step 1: Atomic number 25 corresponds to manganese (Mn). The ground state electron configuration is [Ar]3d54s2.
Step 2: For a divalent ion (Mn2+), remove the two 4s electrons first. The configuration becomes [Ar]3d5.
Step 3: In the 3d5 configuration, all five electrons occupy separate orbitals with parallel spins (Hund's rule). This gives n=5 unpaired electrons. …
For a divalent ion with atomic number 25 (Mn²⁺), the magnetic moment is calculated using the spin-only formula μ=n(n+2) where n=5 unpaired electrons, giving μ=35≈5.92 BM.
The key here is to connect atomic number to electronic configuration, then to the number of unpaired electrons in the aqueous ion. Magnetic moment in transition metal ions is almost always determined by the spin-only formula because orbital angular momentum is "quenched" by the surrounding water ligands.
Let’s walk through it.
-
Identify the element and its neutral configuration.
Atomic number 25 is manganese (Mn). The ground state configuration of neutral Mn is:
1s22s22p63s23p64s23d5
Or in condensed form: [Ar]4s23d5.
-
Form the divalent ion (Mn²⁺).
When a transition metal forms a cation, electrons are removed first from the 4s orbital (higher energy than 3d in the ion, despite being filled first in the neutral atom). So Mn²⁺ loses the two 4s electrons:
[Ar]3d5.
-
Count unpaired electrons using Hund’s rule.
The 3d subshell has five orbitals. With five electrons, Hund’s rule says each orbital gets one electron with parallel spins before any pairing occurs. So all five electrons are unpaired.
n=5.
-
Apply the spin-only magnetic moment formula.
For a transition metal ion in solution, the orbital contribution is usually negligible due to interaction with water molecules (ligand field quenching). The magnetic moment is:
μ=n(n+2) BM
Substitute n=5:
μ=5(5+2)=5×7=35.
μ=n(n+2) Bohr magnetons
- Compute the numerical value. …
Method: Spin-Only Magnetic Moment Formula (for 3d transition metal ions in aqueous solution)
For first-row transition metal ions in aqueous solution, orbital angular momentum is quenched (due to ligand field effects), so the magnetic moment depends only on the number of unpaired electrons.
Steps
Step 1: Identify the ion and its electronic configuration
- Atomic number Z=25 → element is Manganese (Mn).
- Divalent ion means loss of 2 electrons: Mn2+.
- Ground state configuration of Mn: [Ar]3d54s2.
- Remove 4s electrons first (as per Aufbau for ions): Mn2+=[Ar]3d5.
Step 2: Determine number of unpaired electrons
- For 3d5 in a weak field (aqueous solution = high-spin), Hund's rule applies: All five d orbitals are singly occupied before pairing.
- Unpaired electrons n=5.
Step 3: Apply the spin-only formula
The magnetic moment μ (in Bohr magnetons, μB) is:
μ=n(n+2) μB …
Common Mistakes in Magnetic Moment Calculation (Atomic Number 25, Divalent Ion)
Mistake 1: Incorrect Electronic Configuration
The error: Students write the neutral-atom configuration (Z=25) as 1s22s22p63s23p64s23d5 correctly, but then remove the two electrons from the 3d orbitals (because 3d filled last), giving a wrong [Ar]4s23d3 for the divalent ion.
Why it's wrong: For transition metal ions, the 4s orbital empties before the 3d orbital — even though 4s fills first in the neutral atom. The correct order of removal is: 4s electrons go first, then 3d.
Correct approach:
- Neutral Mn (Z=25): [Ar]4s23d5
- Mn2+: Remove two electrons from 4s → [Ar]3d5
How to avoid: Remember the mnemonic: "Last filled, first removed" for transition metal ions. Always write the neutral configuration, then strip the outermost (highest n) s-electrons first.
Mistake 2: Wrong Number of Unpaired Electrons
The error: Students count 3 unpaired electrons (thinking 3d5 means 5 electrons paired as 2+2+1) or 7 unpaired electrons (confusing with another element).
Why it's wrong: For 3d5, Hund's rule states that electrons occupy all five d-orbitals singly before pairing. So all 5 electrons are unpaired.
Correct count: 5 unpaired electrons
How to avoid: Draw the d-orbital box diagram:
↑ ↑ ↑ ↑ ↑
dxy dyz dxz dx²-y² dz²
Each arrow is one unpaired electron. Count them — 5 unpaired.
Mistake 3: Using Wrong Formula
The error: Using μ=n(n+2) with n = total number of d-electrons instead of the number of unpaired electrons.
Why it's wrong: In the formula, n = number of unpaired electrons. For Mn2+ (3d5) the two counts happen to coincide (all 5 d-electrons are unpaired), so the error stays hidden — but for an ion like Fe2+ (3d6, only 4 unpaired) the wrong count n=6 gives 6×8=48≈6.93 BM instead of the correct 4×6=24≈4.90 BM.
Correct formula: μ=n(n+2) BM, where n=5
Calculation:
μ=5(5+2)=5×7=35≈5.92 BM
How to avoid: Always write the formula with the definition: "n = number of unpaired electrons" before plugging in.
Mistake 4: Forgetting the Unit
The error: Writing the answer as just "5.92" without units.
Why it's wrong: Magnetic moment has a specific unit — Bohr Magneton (BM). …
Showing the 12 most recent of 17 on this concept.
- GSEB Higher Secondary Certificate (HSC) Examination 2026Set ANNUAL1 markMCQQ.Which of the following ion has the maximum theoretical magnetic moment? [Fe (Z=26), Cr (Z=24), Ti (Z=22), Co (Z=27)](a) Fe3+(b) Cr3+(c) Ti3+(d) Co3+
›Reveal solutionSolution
Magnetic moment mu = sqrt(n(n+2)) BM rises with the number of unpaired electrons n; the d5 configuration allows the maximum possible unpaired electrons.
Find the d-electron configuration of each 3+ ion:
- Fe (Z=26): [Ar]3d⁶4s² → Fe3+ = [Ar]3d⁵ → 5 unpaired electrons (high spin) → μ = √(5×7) = √35 ≈ 5.92 BM
- Cr (Z=24): [Ar]3d⁵4s¹ → Cr3+ = [Ar]3d³ → 3 unpaired electrons → μ = √(3×5) = √15 ≈ 3.87 BM
- Ti (Z=22): [Ar]3d²4s² → Ti3+ = [Ar]3d¹ → 1 unpaired electron → μ = √3 ≈ 1.73 BM …
- GUJCET 2025Set 031 markMCQQ.Identify the metal whose divalent ion has 'spin only' magnetic moment 35 BM. (A) Cr (B) Mn (C) Fe (D) Co
›Reveal solutionSolution
[!TLDR]
35 BM corresponds to 5 unpaired electrons, matching Mn2+ (3d5).
Concept
The spin-only magnetic moment is μ=n(n+2) BM, where n is the number of unpaired electrons.
Solution
Solve for n:
n(n+2)=35 ⇒ n(n+2)=35 ⇒ n=5
Now find the divalent ion with 5 unpaired d electrons:
- Cr2+:3d4 → 4 unpaired …
- GSEB Higher Secondary Certificate (HSC) Examination 2025Set ANNUAL1 markMCQQ.What is the value of magnetic moment of divalent ion having atomic number 30 in aqueous solution.(a) 0 BM(b) 2.84 BM(c) 1.73 BM(d) 5.92 BM
›Reveal solutionSolution
Atomic number 30 is Zn; its divalent ion Zn2+ has a fully filled 3d10 configuration with zero unpaired electrons, so its magnetic moment is zero.
Element with Z = 30 is Zinc, electronic configuration [Ar] 3d10 4s2.
Zn2+ is formed by removing the two 4s electrons, giving [Ar] 3d10 - a completely filled d-subshell with no unpaired electrons.
…
- GUJCET 2024Set 131 markMCQQ.Which of the following ion show highest spin only magnetic moment value? (A) Co2+ (B) Mn2+ (C) Ti2+ (D) Fe2+
›Reveal solutionSolution
Spin-only moment μ=n(n+2) BM increases with the number of unpaired electrons n; Mn2+ (d5) has the most.
Concept: Count d-electrons and unpaired electrons:
- Ti2+: d2, n=2.
- Mn2+: d5, n=5 ⇒μ=5×7=5.92 BM.
- Fe2+: d6, n=4. …
- GSEB Higher Secondary Certificate (HSC) Examination 2024Set ANNUAL1 markMCQQ.What is the magnetic moment of a divalent ion in aqueous solution if its atomic number is 28?(a) 3.87 BM(b) 2.84 BM(c) 1.73 BM(d) 4.90 BM
›Reveal solutionSolution
Atomic number 28 is Nickel; its divalent ion Ni2+ has electronic configuration 3d8, giving 2 unpaired electrons, and the spin-only magnetic moment formula gives 2.84 BM.
Ni (Z=28): [Ar] 3d8 4s2
Ni2+: loses the two 4s electrons -> [Ar] 3d8
…
- GSEB Higher Secondary Certificate (HSC) Examination 2024Set ANNUAL1 markMCQQ.How many number of unpaired electrons are there in complex ion [Ni(CN)4]2-?(a) 4(b) 3(c) 2(d) 0
›Reveal solutionSolution
[Ni(CN)4]2- has Ni2+ (d8) with the strong-field ligand CN-, which forces pairing of the d-electrons into a square planar, diamagnetic (dsp2) arrangement.
Ni2+: [Ar] 3d8
CN- is a very strong field ligand (high in the spectrochemical series), causing the 3d8 electrons to pair up, freeing one 3d orbital for dsp2 hybridisation (square planar geometry). …
- GSEB Higher Secondary Certificate (HSC) Examination 2023Set ANNUAL1 markMCQQ.Which compound has magnetic moment equal to 4.90 BM?(a) Cr2(SO4)3(b) NiSO4(c) FeSO4(d) MnSO4
›Reveal solutionSolution
mu = 4.90 BM means 4 unpaired electrons; Fe2+ (d6) fits, so FeSO4.
Spin-only moment mu = sqrt(n(n+2)) BM. For mu = 4.90: n(n+2) = 24 -> n = 4 unpaired electrons.
Check the metal ions:
- Cr3+ (Cr2(SO4)3): d3 -> 3 unpaired -> 3.87 BM.
- Ni2+ (NiSO4): d8 -> 2 unpaired -> 2.83 BM. …
- GSEB Higher Secondary Certificate (HSC) Examination 2022Set ANNUAL1 markMCQQ.Among the following, which compound has the highest magnetic moment?(a) MnSO4(b) CrCl3(c) Ni(NO3)2(d) FeSO4
›Reveal solutionSolution
Magnetic moment μ = √[n(n+2)] BM, where n = number of unpaired electrons; more unpaired electrons means a higher moment.
MnSO4: Mn2+ is d5 (high spin) → 5 unpaired electrons → μ = √35 = 5.92 BM.
FeSO4: Fe2+ is d6 → 4 unpaired electrons → μ = 4.90 BM.
CrCl3: Cr3+ is d3 → 3 unpaired electrons → μ = 3.87 BM. …
- GUJCET 2021Set 151 markMCQQ.If atomic number of element is 26, then magnetic moment is ___ BM of its divalent aqueous ion? (A) 1.73 (B) 3.87 (C) 2.83 (D) 4.90
›Reveal solutionSolution
Fe²⁺ (3d6) has 4 unpaired electrons, giving spin-only μ=n(n+2)=24=4.90 BM.
Concept: Atomic number 26 = Fe. The divalent ion Fe²⁺ has configuration [Ar]3d6, with 4 unpaired electrons. …
- GUJCET 2020Set 071 markMCQQ.The divalent ion of which of the following element in aqueous solution has magnetic moment 5.92 BM? (A) Fe (B) Cr (C) Co (D) Mn
›Reveal solutionSolution
μ=n(n+2)=5.92⇒n=5 unpaired; the d5 divalent ion is Mn²⁺.
Concept — spin-only magnetic moment. μ=n(n+2) BM. 5.92=5⋅7=35, so n=5 unpaired electrons. Mn²⁺ is [Ar]3d5 (5 …
- GSEB Higher Secondary Certificate (HSC) Examination 2020Set ANNUAL1 markMCQQ.Magnetic moment of a divalent ion in aqueous solution if its atomic number is 25 ______.(a) 4.90 BM(b) 5.92 BM(c) 2.84 BM(d) 3.87 BM
›Reveal solutionSolution
Element 25 is manganese; its divalent ion Mn2+ has a 3d5 configuration with 5 unpaired electrons (a spin-only 'half-filled, maximally paramagnetic' case), giving the largest common magnetic moment among first-row transition ions.
…
- GUJCET 2019Set 131 markMCQQ.Which of the following pair has similar magnetic moment? (A) Ni2+,Co2+ (B) Fe2+,Mn2+ (C) Fe3+,Mn2+ (D) Cr3+,Mn3+
›Reveal solutionSolution
Magnetic moment depends only on the number of unpaired electrons; Fe3+ and Mn2+ are both 3d5.
Concept: Spin-only magnetic moment μ=n(n+2)BM, where n = number of unpaired electrons. Two ions have the same moment if they have the same n.
Counting unpaired d-electrons:
- Ni2+: 3d8 → 2 ; Co2+: 3d7 → 3 (not equal).
- Fe2+: 3d6 → 4 ; Mn2+: 3d5 → 5 (not equal). …
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