Problem 77 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

\(\mathrm{AB}, \mathrm{A}_{2}\) and \(\mathrm{B}_{2}\) are diatomic molecules. If the bond enthalpies of \(\mathrm{A}_{2}, \mathrm{AB}\) and \(\mathrm{B}_{2}\) are in the ratio \(2: 2: 1\) and enthalpy of formation \(\mathrm{AB}\) from \(\mathrm{A}_{2}\) and \(\mathrm{B}_{2}\) is \(-100 \mathrm{~kJ}\) \(\mathrm{mole}^{-1} .\) What is the bond energy of \(\mathrm{A}_{2}\) ? (a) \(200 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (b) \(100 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (c) \(300 \mathrm{~kJ} \mathrm{~mol}^{-}\) (d) \(400 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The bond energy of A2 is 200 kJ mol^{-1}.

Step 1: Understand the Relationship Between Enthalpy of Formation and Bond Enthalpies

The enthalpy of formation of AB from A2 and B2 is given by the equation: Enthalpy of formation = Bond enthalpy of A2/2 + Bond enthalpy of B2/2 - Bond enthalpy of AB. This is because A2 and B2 need to be broken into two A and B atoms, respectively, which requires half of their bond enthalpies, and then these atoms combine to form AB, releasing the bond enthalpy of AB.

Step 2: Set Up the Equation with the Given Ratio

Using the given ratio of bond enthalpies (2:2:1) for A2, AB, and B2 respectively, let the bond enthalpy of A2 be 2x, that of AB be 2x, and that of B2 be x. Replace these values in the enthalpy of formation equation from Step 1, considering the formation enthalpy given as -100 kJ mole^{-1}. This leads to -100 = 2x/2 + x/2 - 2x.

Step 3: Solve for x

Simplify and solve the equation for x to find the bond enthalpy of A2. The equation becomes -100 = x + x/2 - 2x which simplifies to -100 = x/2. Multiply both sides by 2 to solve for x.

Step 4: Determine the Bond Energy of A2

After calculating the value of x, the bond energy of A2 is 2x, so multiply the solved value of x by 2 to get the bond energy of A2.

Key Concepts

Enthalpy of Formation

Enthalpy of formation refers to the heat change that occurs when one mole of a compound is formed from its constituent elements in their standard state. It's an essential concept in thermochemistry and is typically presented in units of kJ/mol. This concept is especially pertinent when we consider chemical reactions because it helps in understanding how much energy is required or released when a new substance is created from its elemental parts. For instance, in the exercise, the enthalpy of formation of AB from diatomic molecules A₂ and B₂ gives insight into the net energy change during the formation of AB.

In solving the given problem, we consider the bond enthalpies of the diatomic molecules and apply the principle that forming a bond releases energy, whereas breaking a bond requires energy. The enthalpy of formation is calculated by considering the energy needed to break the bonds in A₂ and B₂ and the energy released when AB is formed. By manipulating the equation provided in the steps, we gain insight into the bond energy of A₂ based on the given ratio and the formation energy.

Diatomic Molecules

Diatomic molecules consist of two atoms, which may be either the same or different chemical elements, bonded together to form a stable unit. These molecules are the simplest types of molecules and serve as a fundamental component in various chemical reactions. In the case of the exercise, A₂, AB, and B₂ are all diatomic molecules, each with their unique bond enthalpy — a measure of the bond's strength.

Understanding the dynamics of diatomic molecules like A₂ and B₂ is crucial because their stability, reactivity, and the energy associated with the making or breaking of bonds dictate the course of chemical reactions they undergo. When solving homework problems, students should pay attention to the bond enthalpy values provided for these molecules, as they directly influence the overall energy change of the reaction being studied. The ratio given in the exercise (2:2:1) simplifies the calculation and directs us toward finding the bond energy of one of these diatomic molecules.

Chemical Bonding

Chemical bonding is the force that holds atoms together in molecules. There are several types of chemical bonds, including ionic, covalent, and metallic bonds. The bond energy, or bond enthalpy, is a measurement of the bond's strength and is defined as the amount of energy required to break one mole of bonds in gaseous molecules under standard conditions. In the context of the exercise, the diatomic molecules A₂, B₂, and AB are held together by chemical bonds, each with its own characteristic bond enthalpy.

When studying chemical reactions, knowing the bond enthalpies allows students to predict whether a reaction is endothermic (absorbs energy) or exothermic (releases energy). This information is pivotal in understanding the behavior of substances during a reaction and can inform practical applications, such as the release of energy in the form of heat in combustion. The exercise provided uses the concept of chemical bonding to calculate the specific bond energy of the A₂ molecule, demonstrating how the strength of chemical bonds can be quantified and compared.