Materials Science And Engineering Properties
Materials Science And Engineering Properties
1st Edition
ISBN: 9781111988609
Author: Charles Gilmore
Publisher: Cengage Learning
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Chapter 3, Problem 3.9P
To determine

(a)

The comparison between the magnitude of the total surface energy to the total bond energy.

Expert Solution
Check Mark

Answer to Problem 3.9P

The magnitude of total energy of sphere is 0.16 times the total bond energy.

Explanation of Solution

Given:

The surface energy of Ni is 1.90J/m2.

The cohesive energy of Ni is 4.435eV/atom.

The lattice parameter of FCC Ni is 0.352nm.

The total radius of the sphere is 1atomicdiameter.

Formula used:

The total surface energy of the sphere is given by

E=γ(4πr2)   ....... (I)

Here, E is the total surface energy of the sphere, γ is the surface energy and r is the atomic radius.

The atomic diameter is given by

da=a2[110]   ....... (II)

Here, da is the atomic diameter of a nickel and a is the lattice parameter.

The total bond energy is given by,

Eb=NEC   ....... (III)

Here, Eb is the total bond energy, N is the number of atoms and EC is the cohesive energy of Ni atom.

The ratio of total surface energy and bond energy is given by

EEb   ....... (IV)

Calculation:

The atomic diameter of the Ni atom is calculated as

Substitute 0.352nm for a in equation (II).

da=0.352nm2[12+12+02]12=0.352nm2(2)=0.352nm2=0.249nm

The atomic diameter of Ni is equal to the radius of the sphere.

r=0.249nm

The total surface energy of the sphere is calculated as,

Substitute 0.249nm for r. 1.90J/m2 for γ in equation (I).

E=(1.90J/ m 2)[4π( 0.249nm× 10 9 m 1nm )2]=(1.90J/ m 2)(7.79× 10 19m2)=1.48×1018J

The total bond energy is calculated as

Substitute 4.435eV/atom for EC and 13 for N in equation (III).

Eb=(13atoms)(4.435eV/atom× 1.602× 10 19 J 1eV)=13(7.105× 10 19J)=9.236×1018J

Substitute 9.236×1018J for Eb and 1.48×1018J for E in equation (IV).

EEb=1.48× 10 18J9.236× 10 18JEEb=0.16E=0.16Eb

Conclusion:

Therefore, the magnitude of the total energy of the sphere is 0.16 times the total bond energy.

To determine

(b)

Whether the 13-atom cluster of atoms is stable.

Expert Solution
Check Mark

Answer to Problem 3.9P

Yes, the 13-atom cluster of atoms stable as the total energy of the sphere is negative.

Explanation of Solution

Formula used:

The total energy of the Ni atom is given by,

ET=E+Eb   ....... (V)

Calculation:

The total energy is calculated as,

Substitute 9.236×1018J for Eb and 1.48×1018J for E in equation (V).

ET=1.48×1018J+(9.236× 10 18J)=7.756×1018J

The total energy of the sphere is negative which signifies that the 13 Ni atom cluster is stable.

Conclusion:

Therefore, the 13 Ni atom cluster is stable as the total energy of the sphere is negative.

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