Given:

The side of the copper cube is 0.10 m.

The size of grain on each side is 10×10−6 m.

The average grain boundary energy of copper is 625×10−3 J/m2.

Formula used:

The number of grain is given by,

n=VcubeVgrains   ....... (I)

Here, n is the number of grains, Vcube is the volume of cube and Vgrains is the volume of grain.

The volume of a cube is given by,

Vcube=a3   ....... (II)

Here, a is the side of the cube.

The volume of grains is given by,

Vgrain=(ag)3   ....... (III)

Here, ag is the side of each grain.

Substitute (ag)3 for Vgrain and a3 for Vcube in equation (I).

n=a3( a g )3   ....... (IV)

Calculation:

The number of grains is calculated as,

Substitute 10×10−6 m for ag and 0.10 m for a in equation (IV).

n= ( 0.10 m )3 ( 10× 10 −6  m )3=1× 10 −3 m31× 10 −15 m3=1012

Conclusion:

Therefore, the number of grains present are 1012 grains.

Given:

The side of the copper cube is 0.10 m.

The size of grain on each side is 10×10−6 m.

The average grain boundary energy of copper is 625×10−3 J/m2.

Formula used:

The total grain boundary area is given by,

Atotal=n×(N2)(ag)2   ....... (V)

Here, Atotal is the total grain boundary and N is the number of sides.

Calculation:

The total grain boundary area is calculated as

Substitute 1012 for n, 6 for N and 10×10−6 m for ag in equation (V).

Atotal=( 10 12)×(62)(10× 10 −6 m)2=( 10 12)×(3)(1× 10 −10 m2)=300 m2

Conclusion:

Therefore, the total ground boundary area is 300 m2.

Given:

The side of the copper cube is 0.10 m.

The size of grain on each side is 10×10−6 m.

The average grain boundary energy of copper is 625×10−3 J/m2.

Formula used:

The total energy due to grain boundaries is given by

EGB=γAtotal   ....... (VI)

Here, EGB is the total energy due to grain boundary and γ is the grain boundary energy.

Calculation:

The total energy due to the grain boundary is calculated as

Substitute 625×10−3 J/m2 for γ and 300 m2 for Atotal in equation (VI).

EGB=(625× 10 −3 J/ m 2)(300 m2)=188 J

Conclusion:

Therefore, the total energy present in the copper cube due to grain boundaries is 188 J.

Given:

The side of the copper cube is 0.10 m.

The size of grain on each side is 10×10−6 m.

The average grain boundary energy of copper is 625×10−3 J/m2.

Formula used:

The expression for the equivalent gravitational energy is given by

Eg=ρVcubegh   ....... (VII)

Here, Eg is the equivalent gravitational energy, ρ is the density, g is the acceleration due to gravity and h is the height.

Calculation:

The height to which the copper cube must be raised is calculated by

Substitute 188​ J for Eg, 8.96×103 kg/m3 for ρ, 10−3 m3 for Vcube and 9.81 m/s2 for g in equation (VI).

188​ J=(8.96× 103 kg/ m 3)( 10 −3 m3)(9.81 m/ s 2)hh=188​ J( 8.96× 10 3   kg/ m 3 )( 10 −3   m 3 )( 9.81 m/ s 2 )=188 J87.9 J/m=2.12 m

Conclusion:

Therefore, the height to which copper must be raised to increased energy to an equal amount of energy in the grain boundaries is 2.12 m.