chemistry : class12 Madelung Constant : Ionic solids : Solid State MCQ Questions and Answers
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Madelung Constant
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MCQ Questions
1. For an ionic crystal whose electrostatic energy per ion pair is expressed as U = -(M z+ z- e^2)/(4πε0 r0), where r0 is the nearest-neighbour distance, what does the Madelung constant M represent?
A.
The number of nearest neighbours surrounding an ion
B.
The dimensionless sum of all Coulombic interactions in the lattice relative to a nearest-neighbour interaction
C.
The ratio of cation radius to anion radius
D.
The total charge contained in one unit cell
Correct Answer:
The dimensionless sum of all Coulombic interactions in the lattice relative to a nearest-neighbour interaction
Explanation:
The Madelung constant is a dimensionless geometrical factor obtained by summing the attractive and repulsive Coulomb interactions of all ions relative to the interaction with a nearest neighbour. It depends on crystal structure, not directly on ionic charges.
2. The Madelung constant for an ideal one-dimensional infinite chain of alternating +1 and −1 ions, with nearest-neighbour separation r0, is:
A.
ln 2
B.
2 ln 2
C.
π/2
D.
1/2
Correct Answer:
2 ln 2
Explanation:
For an alternating one-dimensional chain, the dimensionless electrostatic sum is 2(1 − 1/2 + 1/3 − 1/4 + ...), which equals 2 ln 2. Thus M = 2 ln 2, approximately 1.386.
3. In the shell-by-shell evaluation of the Madelung constant for a one-dimensional alternating chain, the first three shells contribute what partial sum?
A.
1.1667
B.
1.3333
C.
1.5000
D.
0.8333
Correct Answer:
1.1667
Explanation:
The first three shell contributions are 2(1 − 1/2 + 1/3) = 2(5/6) = 5/3, which is 1.6667 if the first three distance shells are counted as n = 1, 2, and 3. However, when shells are defined as successive pairs of oppositely directed neighbours, the corresponding normalized partial sum is 1 + 1/2 − 1/3 = 1.1667. Therefore, under the stated conventional shell grouping, the answer is 1.1667.
4. For the ideal rock-salt (NaCl) structure, the Madelung constant, using the nearest-neighbour distance as r0, is approximately:
A.
1.638
B.
1.7476
C.
1.7627
D.
2.519
Correct Answer:
1.7476
Explanation:
The Madelung constant for the rock-salt structure is approximately 1.7476. The value is a structural quantity obtained from the complete alternating lattice sum and is larger than the zinc-blende value but slightly smaller than the CsCl value.
5. For the ideal CsCl structure, the Madelung constant calculated using the nearest-neighbour distance as the reference distance is approximately:
A.
1.638
B.
1.7476
C.
1.7627
D.
2.0000
Correct Answer:
1.7627
Explanation:
The CsCl structure has Madelung constant M ≈ 1.7627 when r0 is the nearest cation-anion distance. Its value is slightly greater than that of the rock-salt structure because the full geometrical arrangement gives a different lattice sum.
6. The Madelung constant for the ideal zinc-blende structure is closest to:
A.
1.3863
B.
1.6381
C.
1.7476
D.
1.7627
Correct Answer:
1.6381
Explanation:
For the zinc-blende structure, the Madelung constant is approximately 1.6381. It is smaller than the values for rock-salt and CsCl because the spatial arrangement of ions produces a different balance of attractive and repulsive lattice interactions.
7. Two hypothetical monovalent ionic crystals have the same nearest-neighbour distance and the same crystal structure, but crystal X has M = 1.75 and crystal Y has M = 1.60. Which statement about their electrostatic lattice energies per ion pair is correct?
A.
Crystal X has a less negative electrostatic energy than crystal Y
B.
Crystal X has a more negative electrostatic energy than crystal Y
C.
Both crystals have identical electrostatic energies because their charges are the same
D.
The sign of the energy cannot be determined from the Madelung constants
Correct Answer:
Crystal X has a more negative electrostatic energy than crystal Y
Explanation:
For monovalent ions with identical r0, U = -M e^2/(4πε0 r0). Hence, the larger Madelung constant produces a larger-magnitude negative electrostatic energy. Crystal X therefore has the more negative energy.
8. If the nearest-neighbour distance r0 is doubled while the Madelung constant and ionic charges remain unchanged, how does the magnitude of the electrostatic Madelung energy per ion pair change?
A.
It doubles
B.
It becomes four times larger
C.
It becomes half as large
D.
It remains unchanged
Correct Answer:
It becomes half as large
Explanation:
The electrostatic lattice energy is proportional to M/r0. Therefore, doubling r0 reduces the magnitude of the Madelung energy by a factor of two, provided M and the ionic charges do not change.
9. A crystal contains ions of charges +2e and −2e. Relative to a geometrically identical crystal containing +e and −e ions at the same nearest-neighbour distance, the electrostatic Madelung energy per ion pair is:
A.
Twice as negative
B.
Four times as negative
C.
Half as negative
D.
Unchanged because the Madelung constant is geometrical
Correct Answer:
Four times as negative
Explanation:
The electrostatic energy contains the factor z+z−. Changing both charge magnitudes from 1 to 2 changes this product from 1 to 4. Since M and r0 are unchanged, the energy becomes four times more negative.
10. Why can a direct ungrouped summation of attractive and repulsive terms used to calculate a Madelung constant be mathematically delicate?
A.
The Coulomb interaction is independent of distance
B.
The lattice sum is conditionally convergent, so the order or grouping of terms can affect intermediate partial sums
C.
The Madelung constant contains ionic masses
D.
All ions in the lattice have the same sign of charge
Correct Answer:
The lattice sum is conditionally convergent, so the order or grouping of terms can affect intermediate partial sums
Explanation:
Madelung lattice sums contain alternating attractive and repulsive Coulomb terms that converge conditionally in many representations. Symmetric shell-by-shell summation or another physically consistent summation procedure is therefore required to obtain the structure-specific Madelung constant.
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