Question

An ionic bond is formed between a cation \(\mathrm{A}^{+}\) and an anion \(\mathrm{B}^{-}\). Based on Coulomb's law $$ E \propto \frac{Q_{1} \times Q_{2}}{d} $$ how would the energy of the ionic bond be affected by the following changes: (a) doubling the radius of \(\mathrm{A}^{+}\) (b) tripling the charge on \(\mathrm{A}^{+},(\mathrm{c})\) doubling the charges on \(\mathrm{A}^{+}\) and \(\mathrm{B}^{-},(\mathrm{d})\) decreasing the radii of \(\mathrm{A}^{+}\) and \(\mathrm{B}^{-}\) to half their original values?

Short answer

(a) Energy is halved, (b) energy is tripled, (c) energy is four times larger, (d) energy is doubled.

Step-by-step solution

Understanding Coulomb's Law

Coulomb's Law describes the electric force between two point charges, where \( E \) represents the energy of interaction, \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, and \( d \) is the distance between them. The energy is directly proportional to the product of the charges and inversely proportional to the distance.

Effect of Doubling the Radius of A+

Doubling the radius of \( \mathrm{A}^{+} \) increases the distance \( d \), thereby reducing the magnitude of \( E \). Since \( d \) doubles, the energy due to the attraction force becomes half.

Effect of Tripling the Charge on A+

Tripling the charge on \( \mathrm{A}^{+} \) implies that \( Q_1 \) becomes three times larger. Since \( E \) is directly proportional to the product of the charges, \( E \) will become three times larger.

Effect of Doubling the Charges on A+ and B-

Doubling both \( Q_1 \) and \( Q_2 \) implies that the product \( Q_1 \times Q_2 \) becomes four times larger \((2 \times 2 = 4)\). Therefore, \( E \) is four times larger.

Effect of Halving the Radii of A+ and B-

Halving both the radii means that the distance \( d \) is also halved. Since \( E \) is inversely proportional to \( d \), halving \( d \) makes \( E \) two times larger.

Key concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in understanding the energy associated with ionic bonds. It explains the interaction between electric charges. According to the law, the force between two charges is:

• Directly proportional to the product of the magnitudes of the charges (i.e., the larger the charges, the greater the force).
• Inversely proportional to the square of the distance between them—in the case of energy in an ionic bond, it simplifies to being inversely proportional to the distance when considering energy directly.

The formula used in the context of ionic bonds is expressed as:\[ E \propto \frac{Q_1 \times Q_2}{d} \]where:

Through Coulomb's Law, we understand how alterations in charge and distance affect the energy within an ionic bond.

Cation and Anion Interaction

Ionic bonds involve the attraction between cations (positively charged ions) and anions (negatively charged ions). This attraction is rooted in the opposite charges, which naturally draw the particles together, leading to high lattice energies in ionic crystals. The strength of this interaction is influenced by several factors:

• Magnitude of Charges: Greater charges typically result in stronger attractions and thus more energy within the bond. For example, if the charge on a cation \( \mathrm{A}^{+} \) is tripled, the attraction increases, as explained through Coulomb's Law.
• Distance between Ions: The attraction force diminishes with greater distances. Thus, if the radius of \( \mathrm{A}^{+} \) or \( \mathrm{B}^{-} \) increases, the interaction weakens.

Understanding these interactions helps in predicting ionic bond strength and stability.

Effect of Charge and Radius on Ionic Bonds

Both charge and radius significantly influence the dynamics of ionic bonds. Adjustments in these parameters can dramatically change the energy of the bond:

Charge Effects

• Increasing Charge: When the charge on ions is increased, like tripling the charge on \( \mathrm{A}^{+} \), it enhances the attraction due to a higher \(Q_1 \times Q_2\) product, thereby increasing bond energy.
• Doubling Charges on Both Ions: The effect multiplies, as seen when both \( \mathrm{A}^{+} \) and \( \mathrm{B}^{-} \) have their charges doubled, resulting in energy being quadrupled.

Radius Effects

• Increasing Radius: Increases in radius (or distance) lower the attraction force, reducing bond energy, as illustrated when the radius of \( \mathrm{A}^{+} \) is doubled, halving the energy.
• Decreasing Radius: Conversely, reducing the radius increases the attraction and energy, evident when both radii are halved, doubling the bond energy.

By understanding how charge and radius work in concert or opposition, one can predict how certain changes will affect the ionic interaction's energy.

Energy of Ionic Bond

The energy of an ionic bond is a reflection of how tightly bonded the ions are within a crystal lattice. It is directly influenced by changes in the charges and distances between the ions. Factors affecting this energy are:

Increases in Energy

• Higher ionic charges usually lead to increases in energy, as more significant charge interactions lead to stronger forces keeping the ions together.
• Shorter distances between ions (i.e., smaller radii) enhance the interaction, resulting in more binding energy.

Decreases in Energy

• Lower ionic charges or increased distances will typically result in a reduction of energy as the interaction weakens.
• Doubling the distance (such as doubling the ionic radii) tends to halve the energy because the ions do not attract each other as strongly.

The energy in an ionic bond is crucial as it helps in determining the compounds' physical properties, such as melting points and solubility.