Question

An ionic bond is established between positive ion \(\mathrm{A}\) and negative ion \(B\). How would one expect the strength of the bond to be affected by each of the following changes: (a) Doubling the charge on \(\mathrm{A}\), (b) Simultaneously doubling the charge on \(\mathrm{A}\) and \(\mathrm{B}\), (c) Doubling the radius of \(\mathrm{B}\), and Simultaneously doubling the radius of \(\mathrm{A}\) and \(\mathrm{B}\) ?

Short answer

(a) Doubling the charge on ion A will result in doubling the bond strength. (b) Simultaneously doubling the charge on ion A and ion B will quadruple the bond strength. (c) Doubling the radius of ion B will result in the bond strength being quartered, i.e., it will decrease. (d) Simultaneously doubling the radius of ion A and ion B will also result in the bond strength being quartered, which means it will decrease.

Step-by-step solution

(a) Doubling the Charge on Ion A

According to Coulomb's law, force is directly proportional to the product of the charges. Thus, if we simplify Coulomb's law to \(F = k|q_A q_B|/r^2\), then we can see that if the charge on \(A\) doubles, the force or bond strength is also expected to double.

(b) Simultaneously Doubling the Charge on Ion A and Ion B

If both the charges are doubled, then, using the same simplification of Coulomb's law: \(F = k|2q_A 2q_B|/r^2 = 4k|q_A q_B|/r^2\). In this case, it's clear that the bond strength will quadruple.

(c) Doubling the Radius of Ion B

In this case, for the same simplification of Coulomb's Law, \(F = k|q_A q_B|/(2r)^2 = k|q_A q_B|/4r^2\). The force or bond strength is quartered when the distance is doubled. In physical terms, the bond strength decreases.

(d) Simultaneously Doubling the Radius of Ion A and Ion B

While the radii of the ions is not directly in Coulomb's Law, if we consider that the distance of separation between the ions' centers (which is part of the law) is the sum of their radii, then this becomes analogous to the above case. Doubling both radii is equivalent to doubling the distance between the ion centers, thus the bond strength is also expected to quarter (assuming the ions remain as individual entities, not overlapping). In other words, the bond strength decreases.

Key concepts

Coulomb's Law

Coulomb's Law is a cornerstone principle in physics that helps us understand the force between two charged particles. Named after the French physicist Charles-Augustin de Coulomb, this law quantifies the electric force between two charges. It states that the force (\( F \)) between two point charges is directly proportional to the product of the magnitudes of the charges (\( q_A \) and \( q_B \) ) and inversely proportional to the square of the distance (\( r^2 \) ) between them. The law is often expressed with the equation: \[ F = k \frac{|q_A q_B|}{r^2} \] where \( k \) is Coulomb's constant. Understanding this law is essential as it underpins our comprehension of ionic bonds and the factors influencing their strength.

When considering ionic bonds in a material, Coulomb's law enables us to predict how changes in charge or distance between ions will affect the force of attraction or repulsion between them. This understanding is crucial for fields such as chemistry and materials science where the properties of substances are determined by atomic-level interactions.

Ionic Bond

An ionic bond is a type of chemical bond formed between two ions with opposite charges. Ionic bonds are the result of the transfer of electrons from one atom to another, leading to the creation of positively charged ions (cations) and negatively charged ions (anions). This transfer results in an attractive force that holds the ions together in an ionic compound. The strength of an ionic bond is significantly influenced by the charges of the ions and the distance between them.

For instance, when the charge of one ion increases, the Coulombic attraction between the two ions increases, leading to a stronger bond. Conversely, increasing the distance between ions decreases the bond strength due to the inverse square relationship of force with distance in Coulomb's law. As ionic compounds form crystal lattices, the strength of these bonds impacts the physical properties such as melting point, hardness, and solubility.

Charge and Force Relationship

The relationship between charge and force within the context of ionic bonding is made clear by Coulomb's law. The magnitude of the force between two charged objects is directly proportional to the product of their charges. This means that, as seen in the exercise solution, if we double the charge on one ion (\( A \) ), then the force or bond strength also doubles. If the charges on both ions (\( A \) and \ (B) ) are doubled, the force increases even more substantially—by a factor of four.

This relationship indicates that the greater the charge on each ion, the stronger the attractive force that constitutes the bond. This concept is important in understanding why certain ionic compounds exhibit higher melting points or greater electrical conductivity, as these are properties that can be influenced by the strength of the force between ions.

Effect of Ion Radius on Bond Strength

The radius of an ion, or its size, also plays a critical role in determining the strength of an ionic bond as per Coulomb's law. When thinking about how bond strength is affected by ion radius, we understand that increasing the radius effectively increases the distance between the ion centers. Since the force is inversely proportional to the square of the distance between the ions, increasing the radius will decrease the bond strength.

In the context of the exercise, doubling the radius of one ion (\( B \) ) quarters the force. Doubling the radius of both ions (\( A \) and \ (B) ) has the same effect, and the reasoning is analogous to doubling the distance between the points where their charges reside. This decrease in bond strength has tangible consequences in the macroscopic properties of the material, such as lower melting points, because a lesser force is required to overcome the ionic bonds during a phase change. Understanding the interplay between ion radius and bond strength is essential for tailoring materials with specific characteristics.