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 \(10^{3} \mathrm{~kJ} \mathrm{~mol}^{-1}\) and its mean atomic volume is \(6.26 \times 10^{-6} \mathrm{~m}^{3} \mathrm{~mol}^{-1}\). The presence of grain boundaries and dislocations weakens a crystalline solid considerably below its theoretical strength.

Short answer

Theoretical strength of forsterite is approximately \(1.60 \times 10^{11} \mathrm{~J} \mathrm{~m}^{-3}\).

Step-by-step solution

Understand the Problem

We need to calculate the theoretical strength of forsterite based on the given binding energy and mean atomic volume. The theoretical strength is calculated by dividing the binding energy by the mean atomic volume.

Identify Known Values

The binding energy of forsterite is given as \(10^3 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and the mean atomic volume is \(6.26 \times 10^{-6} \mathrm{~m}^3 \mathrm{~mol}^{-1}\).

Convert Binding Energy to Joules

To use consistent units, convert the binding energy from kilojoules to joules. Therefore, \(10^3 \mathrm{~kJ} \mathrm{~mol}^{-1} = 10^6 \mathrm{~J} \mathrm{~mol}^{-1}\).

Calculate Theoretical Strength

The theoretical strength is given by the formula:\[\text{Theoretical Strength} = \frac{\text{Binding Energy}}{\text{Mean Atomic Volume}}\]Substituting the values:\[\text{Theoretical Strength} = \frac{10^6 \mathrm{~J} \mathrm{~mol}^{-1}}{6.26 \times 10^{-6} \mathrm{~m}^3 \mathrm{~mol}^{-1}} = 1.60 \times 10^{11} \mathrm{~J} \mathrm{~m}^{-3}\]

Interpret the Result

The calculated theoretical strength represents the maximum strength of forsterite assuming no defects. However, the actual strength is significantly lower due to material imperfections like dislocations and grain boundaries.

Key concepts

Binding Energy

Binding energy plays a crucial role in determining the properties of crystalline solids. It is the measure of the energy required to separate a mole of a solid into individual atoms.
For forsterite, the binding energy is given as \(10^3 \text{ kJ mol}^{-1}\). This value represents the total amount of energy needed to break the bonds holding the individual atoms together in the solid lattice.

• A high binding energy indicates strong bonds within the material, contributing to its overall robustness.
• Binding energy is often measured in kilojoules per mole (kJ/mol), but converting it to joules (J) is a common practice to align with scientific calculations.
• In the problem at hand, converting \(10^3 \text{ kJ mol}^{-1}\) to \(10^6 \text{ J mol}^{-1}\) helps in using consistent units for further calculations.

Understanding binding energy is essential for estimating the theoretical limits of a material's strength.

Mean Atomic Volume

Mean atomic volume is a key factor in assessing the physical properties of crystalline substances. It refers to the average volume occupied by one mole of atoms in a crystal lattice.
In our example, the mean atomic volume for forsterite is \(6.26 \times 10^{-6} \text{ m}^3 \text{ mol}^{-1}\). This metric is significant when calculating the material's theoretical strength.

• Mean atomic volume allows us to understand the spatial arrangement and the density of atoms within the crystal.
• The smaller the atomic volume, the tighter the atoms are packed, and often, this correlates with stronger materials.
• This volume is utilized as the denominator in the theoretical strength calculation because it gives the context of energy binding per unit space in a material.

By evaluating both binding energy and mean atomic volume, one can understand the potential mechanical characteristics of a crystalline solid.

Theoretical Strength

Theoretical strength provides an estimate of the maximum stress a perfect crystalline structure can withstand without any imperfections. It is calculated by dividing binding energy by mean atomic volume.
For forsterite, the theoretical strength is calculated as \(\frac{10^6 \text{ J mol}^{-1}}{6.26 \times 10^{-6} \text{ m}^3 \text{ mol}^{-1}} = 1.60 \times 10^{11} \text{ J m}^{-3}\), representing the upper limit of its strength.

• Theoretical strength assumes no defects like dislocations or grain boundaries, portraying the ideal scenario.
• This calculated limit is often higher than the practical strength due to the presence of real-world imperfections.
• Understanding this strength helps in designing materials and predicting their behavior under stress.

Although theoretical, these computations are critical for material scientists looking to innovate and push the boundaries of material performance.

Grain Boundaries

Grain boundaries are the interfaces where different crystalline grains meet within a solid material. They play a significant role in determining the mechanical properties of materials such as strength and ductility.
In crystalline solids like metals and minerals, grain boundaries can weaken the material's strength when compared to its theoretical maximum.

• Grain boundaries often act as barriers to dislocation motion, which can advantageously strengthen or disadvantageously weaken a material depending on the context.
• They can also contribute to the scattering of free electrons in conductors, impacting electrical properties.
• The manipulation of grain boundaries is a key strategy in materials science to develop stronger and more reliable materials.

Understanding the effect of grain boundaries is crucial for assessing and enhancing the practical strength and durability of crystalline solids.

Dislocations

Dislocations are imperfections within the crystal structure that have a profound effect on the physical properties of materials. They are essentially defects where atoms are misaligned.
These imperfections allow crystals to deform more easily under stress, thus reducing their theoretical strength.

• A dislocation can move across the grains, allowing the layers of atoms to slip past each other, leading to material deformation.
• This movement is what enables metals to be shaped and bent without breaking, making dislocations useful despite their weakening effect.
• Material scientists strive to control the behavior and density of dislocations to optimize mechanical properties.

Dislocations are essential to comprehend for understanding the real-world applications and the limits of crystalline materials in various industries.