Question

A theoretical estimate of the strength of a crystalline solid is its binding energy per unit volume. Evaluate the strength of forsterite if its binding energy is \(10^{3} \mathrm{~kJ} \mathrm{~mole}^{-1}\) and its mean atomic volume is \(6.26 \times 10^{-6} \mathrm{~m}^{3} \mathrm{~mole}^{-1}\). The presence of grain boundaries and dislocations weakens a crystalline solid considerably below its theoretical strength.

Short answer

Theoretical strength of forsterite is \(1.5974 \times 10^8 \text{ J/m}^3\).

Step-by-step solution

Understand the Parameters

First, identify the given parameters for the problem. The energy per mole is given as \(10^3 \text{ kJ/mole} \) and the volume per mole is given as \(6.26 \times 10^{-6} \text{ m}^3/\text{mole} \). We need to find the strength of forsterite using these parameters.

Convert Binding Energy Units

Convert the binding energy from \( \text{kJ/mole} \) to \( \text{J/mole} \) by multiplying by \(1000 \). Therefore, \( 10^3 \text{ kJ/mole} = 10^6 \text{ J/mole} \).

Calculate Theoretical Strength

To find the theoretical strength, divide the binding energy (in J/mole) by the atomic volume (in m³/mole): \[\text{Theoretical Strength} = \frac{10^6 \text{ J/mole}}{6.26 \times 10^{-6} \text{ m}^3/\text{mole}} = \frac{10^6}{6.26} \text{ J/m}^3\] Calculate this value to find the theoretical strength.

Compute Theoretical Strength Result

Perform the division to obtain:\[\text{Theoretical Strength} = 1.5974 \times 10^8 \text{ J/m}^3\] This result represents the theoretical strength of forsterite without considering weakening factors like dislocations or grain boundaries.

Key concepts

Crystalline Strength

Crystalline strength is a measure of how much load a material can withstand before it deforms or breaks. For solid materials like minerals, the theoretical crystalline strength is an estimate of how strong the material could be if all the atoms were perfectly aligned and bonded. In nature, this is seldom the case due to imperfections or defects within the crystal lattice. Key factors influencing crystalline strength include:

• Atomic arrangement: The orderliness within the crystal structure. A more ordered array equates to higher strength.
• Faults or dislocations: Natural disruptions in the atomic pattern that significantly reduce strength.

When calculating crystalline strength, binding energy and atomic volume are crucial parameters. These variables offer insights into how atoms within the material interact and the space they occupy. With forsterite, knowing its binding energy and atomic volume helps in computing its maximum potential strength.

Binding Energy

Binding energy refers to the amount of energy required to break the bonds between atoms in a solid material. This energy is a vital indicator of the stability and resilience of the crystalline structure. A higher binding energy implies stronger atomic bonds, contributing to a material's robustness.To find binding energy per unit volume in practice:

• Convert the given energy from kilojoules per mole to joules per mole to align with standard scientific units.
• Divide this value by the atomic volume, providing a sense of the intrinsic strength of the material.

For forsterite, given its binding energy of \(10^3 \text{ kJ/mole}\), this translates to \(10^6 \text{ J/mole}\). The calculation of binding energy per volume provides an impressive theoretical strength of \(1.5974 \times 10^8 \text{ J/m}^3\). This value underscores the connectivity of the material’s atoms under ideal conditions.

Grain Boundaries

Grain boundaries are the interfaces where crystals of different orientations meet within a material. These boundaries can significantly affect the mechanical properties of the material, especially its strength and ductility. Aspects of grain boundaries affecting crystalline strength:

• Disruption in continuity: Grain boundaries break the uniformity of the crystal lattice, providing pathways for easier deformation.
• Barrier to dislocation movement: While they can weaken the overall structure, they can also hinder the movement of dislocations, sometimes enhancing strength.

In the theoretical studies of crystalline materials, ignoring grain boundaries leads to overestimations. Real-world materials invariably have these interfaces, which are considered when designing practical applications of materials such as forsterite.

Forsterite

Forsterite is a mineral from the olivine group, commonly found in the Earth's mantle. It is a magnesium silicate with the formula \(\text{Mg}_2\text{SiO}_4\). Understanding its properties, especially its crystalline structure and associated strengths, is crucial because of its prevalence and role in geodynamics.Key properties:

• High melting point: This makes it very stable under extreme conditions such as those found deeper in the Earth's layers.
• Theoretical strength: As calculated, the binding energy offers insight into its potential use in high-strength applications.

Forsterite's relevance extends to various industrial applications due to these properties. However, its real-world applications must consider factors like imperfections and grain boundaries, which affect its actual operating strength compared to the theoretical predictions.