Chapter 12: Problem 15
Compute the theoretical density of diamond given that the \(\mathrm{C}-\mathrm{C}\) distance and bond angle are \(0.154 \mathrm{~nm}\) and \(109.5^{\circ}\), respectively. How does this value compare with the measured density?
Short Answer
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Question: Calculate the theoretical density of a diamond lattice structure based on the given carbon-carbon bond distance of 0.154 nm and bond angle of 109.5°, and compare it to the experimentally measured density of 3.51 g/cm³.
Step by step solution
01
Calculate the length of the unit cell
In a diamond structure, each carbon atom is bonded to four other carbon atoms, forming a tetrahedron. The bond length between carbon atoms is given as \(0.154 \mathrm{nm}\). Since the bond angle in a tetrahedron is \(109.5^{\circ}\), we can use the cosine rule to find the length of the unit cell.
\(a^2 = b^2 + c^2 - 2bc\cos(\alpha)\)
In the case of a tetrahedron, \(b = c\). Therefore, the equation becomes:
\(a^2 = 2b^2 - 2b^2\cos(\alpha)\)
We are given \(b = 0.154 \mathrm{nm}\) and \(\alpha = 109.5^{\circ}\). Therefore, we can calculate the length of the unit cell.
02
Calculate the volume of the unit cell
Since the unit cell of a diamond is a cube, we can calculate the volume using the formula:
\(V_{unit \ cell} = a^3\)
Here, \(a\) is the length of the unit cell found in Step 1.
03
Determine the number of carbon atoms in the unit cell
In the diamond structure, there are eight carbon atoms at the corners of the unit cell, and six carbon atoms at the face centers. Since each corner atom is shared by eight unit cells and each face atom is shared by two unit cells, the formula for the number of carbon atoms in one unit cell (n) is:
\(n = \frac{8}{8} + \frac{6}{2} = 1 + 3 = 4\)
04
Calculate the theoretical density of diamond
To calculate the theoretical density of diamond, we need to know the mass of a carbon atom. We can find this using the atomic weight of carbon (\(12.01\, \mathrm{g/mol}\)) and Avogadro's number \((6.022 \times 10^{23}\,\mathrm{atoms/mol})\).
Mass of one carbon atom = \(\frac{12.01 \mathrm{g/mol}}{6.022 \times 10^{23}\,\mathrm{atoms/mol}}\)
The theoretical density of diamond can be calculated using the formula:
density = \(\frac{n \cdot \text{mass of one carbon atom}}{V_{unit \ cell}}\)
05
Compare with the measured density
After calculating the theoretical density of diamond, we can compare it to the measured value. The measured density of diamond is around \(3.51\, \mathrm{g/cm^3}\). Calculate the percent difference between the theoretical and measured densities to evaluate how close they are.
Percent difference = \(\frac{|\text{theoretical density} - \text{measured density}|}{(\text{theoretical density} + \text{measured density}) / 2} \times 100\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Diamond Cubic Structure
The beauty and strength of a diamond lie in its atomic blueprint, known as the diamond cubic structure. This unique arrangement forms a repeating lattice that's both elegant and rigid. Picture each carbon atom as a central point in a four-sided pyramid or tetrahedron. These carbon atoms are bonded strongly to four others, leading to a three-dimensional network of tightly packed atoms.
This structure is not just visually stunning under a microscope; it also provides diamonds with their renowned hardness. The tetrahedrons link together to form a larger cubic unit cell, which repeats itself throughout the entire structure of the gem. Understanding this architecture at the molecular level is essential for anyone delving into the properties of diamond and calculating its theoretical density.
This structure is not just visually stunning under a microscope; it also provides diamonds with their renowned hardness. The tetrahedrons link together to form a larger cubic unit cell, which repeats itself throughout the entire structure of the gem. Understanding this architecture at the molecular level is essential for anyone delving into the properties of diamond and calculating its theoretical density.
Cosine Rule in Crystallography
Crystallography can sometimes feel like navigating a maze of angles and lengths, but the cosine rule brings order to the chaos, serving as a guiding star. When dealing with the intricate connections within crystal structures, such as the bonds in a diamond, we rely on the cosine rule to calculate unknown distances. This rule is a cornerstone in the triangle of crystallography equations.
In simple terms, the cosine rule allows us to find the length of one side of a triangle if we know the lengths of the other two sides and the angle enclosed between them. For the diamond cubic structure, it's the secret weapon to find the length of the unit cell edge, which is the first step in unlocking the door to establishing the structure's theoretical density.
In simple terms, the cosine rule allows us to find the length of one side of a triangle if we know the lengths of the other two sides and the angle enclosed between them. For the diamond cubic structure, it's the secret weapon to find the length of the unit cell edge, which is the first step in unlocking the door to establishing the structure's theoretical density.
Carbon Atoms in Diamond
Of Tetrahedrons and Carbon Allotropes
Carbon is a cosmic chameleon, able to take on many forms, or allotropes – diamond being one of the most celebrated. In a diamond, each carbon atom forms a tetrahedral bond with four neighbors. Wonderfully simple, yet complex, the diamond cubic structure contains exactly four carbon atoms per unit cell. This precise arrangement is crucial for two things: the diamond’s mesmerizing sparkle and the theoretical density calculations.Counting Carbon’s Contributions
Understanding the count is critical—it allows us to factor in only the discrete number of carbon atoms contributing to one unit cell. Remember, atoms at the corners of the cell are shared among eight others, while those on the faces share the limelight with just one other cell. This careful accounting ensures that the theoretical density reflects the true amount of carbon packed into a single unit cell of diamond.Comparison of Theoretical and Measured Densities
Embarking on the journey of calculating theoretical density is like assembling a puzzle where precision is key. The measured density of a diamond is known from empirical data, but the thrill lies in computing the theoretical density using crystallographic insights and atomic weights.
Once the theoretical density is in hand, we can't help but compare it to what has been actually measured in the lab. Discrepancies between these two numbers are not merely numbers; they narrate stories of inherent defects, impurities, or possible errors in measurement or calculation. A percentage difference gives us a quantified expression of this variance and helps assess the accuracy and reliability of our theoretical model when held up against the litmus test of reality.
Once the theoretical density is in hand, we can't help but compare it to what has been actually measured in the lab. Discrepancies between these two numbers are not merely numbers; they narrate stories of inherent defects, impurities, or possible errors in measurement or calculation. A percentage difference gives us a quantified expression of this variance and helps assess the accuracy and reliability of our theoretical model when held up against the litmus test of reality.