Question

Derive Born-Lande equation and hence prove that lattice energy of an ionic crystal is inversely proportional to the interionic distances.

Short answer

The lattice energy of an ionic crystal is derived from the Born-Lande equation to be \( U = - AMz^+z^-e^2/r + BN/r^n \), which makes it inversely proportional to the interionic distance \( r \). This is because both terms with the substantial impact to the total energy in the equation are inversely proportional to \( r \).

Step-by-step solution

Components of Born-Lande Equation

The potential energy \( U \) of a pair of ions can be written as \( U = - \frac{{A}}{{r}} + \frac{{B}}{{r^n}} \), where \( A \) and \( B \) are constants, \( r \) is the interionic distance and \( n \) is the Born exponent.

Madelung Constant

To compute the total energy of ions in the lattice (lattice energy), we need to sum up the potential energy over all ion pairs in the lattice. However, we need to include the fact that negative-negative and positive-positive ion pairs will repulse each other whereas positive-negative ion pairs attraction. This can be done using a concept called the Madelung constant \( A \) that captures the fractional number of positive-negative ion pairs versus positive-positive/negative-negative ion pairs.

Deriving Born-Lande Equation

Considering the potential energy of a pair of ions and Madelung constant, we derive Madelung energy (the sum of all the potential energies) that is equal to \( U = -AMz^+z^-e^2/r + BN/r^n \), where \( z^+ \) and \( z^- \) are the charges of the positive and negative ions respectively, \( e \) is the elementary charge and other terms as defined before.

Prove the Inverse Proportionality

Now, it is clear that both terms of the equation have \( r \) in the denominator. Hence, it can be concluded that the lattice energy is inversely proportional to the interionic distance \( r \).

Key concepts

Lattice Energy

Lattice energy is a measure of the strength of the forces between the ions in an ionic solid. In essence, it's the amount of energy released when ions come together to form a lattice from a gas phase, or conversely, the energy needed to break the lattice apart into its constituent ions. The greater the lattice energy, the more stable the ionic solid and the higher the energy required to dissolve it.

Lattice energy is influenced by several factors: the charges on the ions, the size of the ions, and the structure of the lattice. For a given pair of ions, the stronger the charges and the smaller the ions, the higher the lattice energy will be. This is because smaller ions can get closer together, which enhances the electrostatic attraction between them, and higher charges also intensify this attractive force.

Understanding lattice energy not only helps in predicting the solubility and melting points of ionic compounds but also is crucial for insights into the formation and stability of solids at the microscopic level.

Interionic Distance

Interionic distance is the distance between the centers of neighboring positive and negative ions in a crystal lattice. Think of it as the 'gap' that ions must overcome for the attractive forces to hold the lattice together. This concept is pivotal when considering the interactions within a solid framework because the strength of the ionic bonds and the resulting lattice energy are inversely proportional to interionic distance.

As the distance increases, the attractive forces weaken, which reduces the lattice energy. The concept can sometimes be counterintuitive; something farther apart is weaker? Indeed, this is the case with electrostatic forces, where the energies are significantly dependent on how close the ions are to one another. Remember, just a small change in the interionic distance can have a profound impact on the strength of the ionic bond. This is a central concept when deriving relations like the Born-Lande equation, where distance plays a fundamental role in understanding the energy of a crystal lattice.

Madelung Constant

The Madelung constant is a number that reflects the strength of the electrostatic field within a crystal due to the arrangement of the ions. It is not merely a constant but is specific to the geometry of the crystal lattice, capturing the effects of both attractive and repulsive forces between ions.

To put it more plainly, since an ionic crystal is a three-dimensional structure, the ions are surrounded by other ions of opposite and like charges. The Madelung constant takes into account all these interactions to provide a single value that simplifies calculations of the lattice energy. It's like having a snapshot of the overall electric effect in the crystal, allowing us to see beyond just pair-wise interactions between ions. This mathematical neat trick is crucial in the Born-Lande equation as it acts as an aggregate measure of an ionic crystal's total electrostatic environment.

Born Exponent

The Born exponent, denoted by 'n', is a term in the Born-Lande equation that accounts for the repulsive forces that operate at very short distances between ions. These repulsive forces become significant when ions are brought extremely close together and arise due to the overlapping of electron orbitals.

Without diving deep into the complexities of quantum mechanics, you can think of the Born exponent as a variable that helps in refining our model of interionic interactions. It takes into consideration that ions are not point charges but have size and electronic structure. The value of the Born exponent generally lies between 5 and 12 and varies depending on the types of ions involved.

When considering the inverse proportionality of lattice energy to interionic distance, the Born exponent provides a necessary correction to our calculations. It's like fine-tuning a guitar string to achieve the perfect pitch — the Born exponent ensures our mathematical model sings the true note of the physical forces at play within a crystal.