Q.Metallic radii of some transition elements are given below. Which of these elements will have highest density? Element: Fe, Co, Ni, Cu
Metallic radii/pm: Fe $= 126$, Co $= 125$, Ni $= 125$, Cu $= 128$

Concept: Density depends on mass per unit volume. For metals in the same period, atomic mass increases faster than atomic volume, so density generally increases across a series.

Reasoning:

1. Density $\propto \frac{\text{atomic mass}}{\text{atomic volume}}$. Atomic volume $\propto (\text{radius})^3$.
2. Atomic masses (approx.): Fe = 55.8, Co = 58.9, Ni = 58.7, Cu = 63.5 g/mol.
3. Radii: Fe = 126, Co = 125, Ni = 125, Cu = 128 pm. Volume scales as $r^3$, so Cu has the largest volume, but its mass is significantly higher.
4. Compare mass/volume ratios: Cu has the highest atomic mass with only a slightly larger radius, giving it the greatest density.

Why density depends on radius and mass

Density is mass per unit volume. For a metallic crystal, the density of the element is proportional to:

The exact formula involves packing fraction and Avogadro’s number, but for comparing elements with the same crystal structure (all four are face-centered cubic at room temperature), the packing fraction cancels out. So we only need to compare:

A higher atomic mass and a smaller radius both push density up. Let’s see which element wins.

Step-by-step comparison

1. List the given data

The radii are nearly equal — all within 3 pm of each other. So the atomic mass will be the deciding factor.

2. Compute $M / r^3$ for each

We can work with relative values since the constant factor (packing fraction, Avogadro’s number) is the same for all.

For Fe:

For Co:

For Ni:

For Cu:

3. Compare the values

• Fe: $2.79 \times 10^{-5}$ — lowest, because Fe has the smallest atomic mass and a mid-sized radius.
• Co: $3.02 \times 10^{-5}$ — higher than Fe.
• Ni: $3.00 \times 10^{-5}$ — very close to Co, slightly lower.
• Cu: $3.03 \times 10^{-5}$ — the highest value.

Copper’s atomic mass is about 7–8% higher than cobalt’s, and its radius is only 2.4% larger. The cube in the denominator means a 2.4% radius increase raises the volume by about 7.4%, but the mass increase of ~7.8% more than compensates. So Cu edges ahead.

4. Final ranking

Cu > Co > Ni > Fe in density.

Method: Density Estimation from Metallic Radius and Atomic Mass

This problem uses the relationship between density, atomic mass, and atomic radius in a periodic table trend context.

Concept Behind the Method

Density ($\rho$) is mass per unit volume. For metallic elements in the same period:

• Mass depends on atomic mass (increases across a period)
• Volume depends on atomic radius (generally decreases across a period)

Since density $\propto \frac{\text{mass}}{\text{volume}}$, and volume $\propto r^3$, we can compare density using:

Steps to Solve

Step 1: List the given data

Step 2: Calculate $r^3$ for each element

• Fe: $126^3 = 2,000,376$
• Co: $125^3 = 1,953,125$
• Ni: $125^3 = 1,953,125$
• Cu: $128^3 = 2,097,152$

Step 3: Compute $\frac{\text{Atomic mass}}{r^3}$ for comparison

• Fe: $\frac{55.85}{2,000,376} \approx 2.79 \times 10^{-5}$
• Co: $\frac{58.93}{1,953,125} \approx 3.02 \times 10^{-5}$
• Ni: $\frac{58.69}{1,953,125} \approx 3.00 \times 10^{-5}$
• Cu: $\frac{63.55}{2,097,152} \approx 3.03 \times 10^{-5}$

Step 4: Compare the values

The highest value indicates the highest density.

Order: Cu > Co > Ni > Fe

Final Answer

Cu has the highest density.

Why This Works

Across the first transition series, atomic mass increases steadily while atomic radius remains nearly constant (due to poor shielding by d-electrons). Copper has the highest atomic mass among these four, with only a slightly larger radius — giving it the highest density.

Common Mistakes & How to Avoid Them

Mistake 1: Assuming the smallest radius always gives the highest density

Why students make it:

They see Co and Ni have the smallest radii (125 pm) and assume "smaller atom = more tightly packed = denser," picking (iii) Co or (ii) Ni without checking atomic mass.

Why it's wrong:

Density depends on mass per unit volume, not radius alone. A smaller radius does increase density for a fixed mass, but here the atomic masses are also different — and that difference decides the outcome.

How to avoid:

Always write the formula:

For a spherical atom, volume $\propto r^3$. So:

Check the numbers:

Result: Cu has the highest mass per unit volume → highest density, even though its radius is the largest of the four — its atomic mass is high enough to overcome the larger volume.

Correct answer: (iv) Cu

Mistake 2: Forgetting that density depends on atomic mass, not just radius

Why students make it:

They focus only on the given radii and ignore the atomic masses (which are not directly given but must be recalled from periodic table knowledge).

How to avoid:

Always recall or note the atomic masses of the elements in the series. For 3d transition series:

• Fe ≈ 56
• Co ≈ 59
• Ni ≈ 58.7
• Cu ≈ 63.5

Cu has both the largest radius and the largest mass among these four — so the ratio, not either number alone, decides the answer.

Mistake 3: Thinking density tracks radius alone across the series

Why students make it:

They remember that atomic radii change only slightly across Fe–Co–Ni–Cu, so they assume the element with the smallest radius must be densest.

Why it's wrong:

Across the 3d series, density is set by the combination of mass and volume. Cu's radius is about 2.4% larger than Co's/Ni's, which raises its volume by about 7.4% — but Cu's atomic mass is about 7.8% higher than Co's, which more than compensates.

How to avoid:

Compare ratios directly instead of eyeballing radius or mass alone:

• Fe → Co: mass ↑ noticeably, radius ↓ slightly → density increases
• Co → Ni: mass and radius both nearly unchanged → density nearly the same
• Ni → Cu: mass ↑ more than radius ↑ → density increases again, reaching its highest value at Cu

So the peak is at Cu, not Co.

Mistake 4: Confusing metallic radius with density formula

Why students make it:

They try to use the metallic radius directly in density without converting to volume.

How to avoid:

Remember: density uses volume, which scales as $r^3$. A 2% change in radius causes ~6-7% change in volume — significant enough to matter here.

Quick Summary — Do This Instead

Final answer: (iv) Cu