Question

Compute the binding energy of \(\mathrm{CsCl}\). Use \(\beta_{0}=\) \(5.95 \times 10^{-11} \mathrm{~Pa}^{-1}, \rho_{0}=3988 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(A=\) 1.7627. The molecular weight of \(\mathrm{CsCl}\) is 168.36 , and thermodynamic data give \(-U_{0}=660 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Short answer

Binding energy of CsCl is calculated using density, constants, and thermodynamic data.

Step-by-step solution

Understand the Formula

The binding energy is essentially the energy required to disassemble a molecule into its individual atoms. The general formula to calculate the binding energy per mole can be expressed as a factor involving the given constants and the physical properties of the compound.

Use the Densities and Constants

Binding energy per unit Volume, \( E_v \), can be estimated by:\[E_{v} = \frac{( ho_0/A)}{\beta_0}\]where \( \rho_0 \) is the density, \( A \) is the constant related to the crystal lattice, and \( \beta_0 \) is the compressibility factor. Plug in the provided values:\[E_v = \frac{(3988/1.7627)}{5.95 \times 10^{-11}}\]

Calculate Binding Energy per Volume

Calculate \( E_v \):\[E_v = \frac{3988}{1.7627} \]Calculate the above result, then divide by the compressibility factor \( 5.95 \times 10^{-11} \) Pa\(^{-1}\) to find \( E_v \).

Convert to Binding Energy per Mole

Now, convert \( E_v \) to a binding energy per mole using:\[E_{mole} = E_v \times \text{Volume per mole} = E_v \times \left(\frac{M}{\rho_0}\right)\]where \( M \) is the molar mass of \( \mathrm{CsCl} \), which is 168.36 g/mol. Convert to kg/mol for calculation.\[E_{mole} = E_v \times \left(\frac{0.16836}{3988}\right)\]

Account for Thermodynamic Contributions

Finally, incorporate the given thermodynamic data \(-U_0 = 660 \text{ kJ/mol} \):\[E_{binding} = E_{mole} + (-U_0)\]Calculate \(-U_0\) contribution in the energy calculation and add it to the previously found energy per mole.

Key concepts

Crystal Lattice Constants

Crystal lattice constants are essential in determining a material's structure and properties. In simple terms, these constants define the repeating unit in a crystal, often a cube with edges aligned with atoms or ions. This pattern is repeated throughout a solid, defining its structure. Understanding lattice constants helps us explain physical properties like density, melting points, and binding energies.

In the case of CsCl, we use a constant denoted as \(A = 1.7627\) in the binding energy calculation. This constant hints at the geometric and interaction complexities within the crystal lattice. It’s crucial to get an accurate measure of lattice constants because they will heavily influence calculated properties like binding energy.

Moreover, lattice constants can change under different conditions such as pressure or temperature. Therefore, knowing them can be important in thermodynamic applications involving phase changes or reactions.

Density of Materials

Density is a vital physical property of materials. It’s defined as mass per unit volume and is represented by \(\rho\). In this exercise, we have \(\rho_0 = 3988 \, \text{kg/m}^3\) for \(\mathrm{CsCl}\).

Density helps in understanding material stability, structural strength, and in calculations involving other properties such as binding energy. In our calculations, the density helps to convert energy per volume to energy per mole, which in turn provides us with binding energy values.

The density of a material is influenced by several things. These include the atomic or molecular weight, the arrangement of atoms (which ties back to the lattice constants), and how closely the atoms are packed together. Because these things can change with factors like temperature and pressure, so too can the density.

Compressibility Factor

Compressibility factor, often denoted as \(\beta_0\), provides information on how much a material can be compressed under pressure. For \(\mathrm{CsCl}\) in this case, it is given as \(5.95 \times 10^{-11} \, \text{Pa}^{-1}\).

If a material has a low compressibility factor, it means it's hard to compress, thereby maintaining its volume even under high pressures. This characteristic directly influences properties like binding energy because less compressible materials are often stronger and more cohesive.

To capture its essence well in such calculations, the compressibility factor is used inversely in formulas, such as in the binding energy per unit volume calculation. Knowing compressibility helps in a variety of applications, from material synthesis to hardware design, where maintaining structural integrity under stress is important.

Thermodynamics of Molecules

Thermodynamics involves studying energy changes, particularly in chemical reactions and state changes. When dealing with molecules like \(\mathrm{CsCl}\), thermodynamic data can provide valuable insights into stability and reactivity. In this problem, we take into account a thermodynamic value \(-U_0 = 660 \, \text{kJ/mol}\).

This value represents a form of stored energy within the structure, often related to bond strengths and other molecular interactions. It’s called negative because it indicates energy release when the atomic components form the compound from its elements.

This thermodynamic contribution is crucial in calculating the binding energy because it accounts for the actual energy holding the structure together, not just the energy calculated from density and compressibility factors. Including thermodynamic data ensures a more complete and realistic estimate of a molecule’s binding energy.