Question

Using the data provided, calculate the multiple bond energy \(\left(\mathrm{kJ} \mathrm{mol}^{-1}\right)\) of a \(\mathrm{C} \equiv \mathrm{C}\) bond in \(\mathrm{C}_{2} \mathrm{H}_{2}\). That energy is (take the bond energy of a C \(\mathrm{H}\) bond as \(350 \mathrm{~kJ} \mathrm{~mol}^{-1}\).) \(2 \mathrm{C}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{2}(\mathrm{~g}) \quad \Delta H=225 \mathrm{~kJ} \mathrm{~mol}^{-1}\) $$ 2 \mathrm{C}(\mathrm{s}) \longrightarrow 2 \mathrm{C}(\mathrm{g}) \quad \Delta H=1410 \mathrm{~kJ} \mathrm{~mol}^{-1} $$ \(\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{H}(\mathrm{g}) \quad \Delta H=330 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (A) 1165 (B) 837 (C) 865 (D) 815

Short answer

The multiple bond energy of a \(\mathrm{C} \equiv \mathrm{C}\) bond in \(\mathrm{C}_2\mathrm{H}_2\) is \(2675\ \mathrm{kJ/mol}\).

Step-by-step solution

Understand the given information

The exercise provides us with three pieces of information:1. The enthalpy change \(\Delta H\) for the formation of acetylene (\(\mathrm{C}_2\mathrm{H}_2\)) from solid carbon (\(\mathrm{C}(\mathrm{s})\)) and hydrogen gas (\(\mathrm{H}_2(\mathrm{g})\)) is 225 kJ/mol.2. The enthalpy change for converting solid carbon to gaseous carbon is 1410 kJ/mol for two moles of carbon.3. The enthalpy change to dissociate one mole of hydrogen gas into atomic hydrogen is 330 kJ/mol.The bond energy for a C−H bond is given as 350 kJ/mol.

Write out the balanced chemical equation for acetylene formation

The balanced reaction for the formation of acetylene is:\[2 \mathrm{C}(\mathrm{s}) + \mathrm{H}_2(\mathrm{g}) \rightarrow \mathrm{C}_2\mathrm{H}_2(\mathrm{g})\]We now use Hess's Law to find out the enthalpy change involved in the formation of C≡C bond.

Calculate the total energy input

The total energy put into the system includes the energy to convert carbon from solid to gas and to dissociate H2 gas into atomic hydrogen:\[\text{Energy input} = 2 \times \text{Carbon sublimation energy} + \text{Hydrogen bond dissociation energy}\]\[\text{Energy input} = 2 \times 1410\ \mathrm{kJ/mol} + 330\ \mathrm{kJ/mol}\]\[\text{Energy input} = 2820 + 330\]\[\text{Energy input} = 3150\ \mathrm{kJ/mol}\]

Calculate energy released in forming C-H bonds in acetylene

There are two C−H bonds in acetylene, each releasing 350 kJ/mol during formation:\[\text{Energy released from 2 C-H bonds} = 2 \times 350\ \mathrm{kJ/mol}\]\[\text{Energy released} = 700\ \mathrm{kJ/mol}\]

Calculate the net energy change for acetylene formation

The net energy change for the formation reaction is the total energy input minus the energy released in forming C-H bonds, added to the reaction enthalpy (\(\Delta H\)) provided.\[\text{Net energy change} = \text{Energy input} - \text{Energy released in forming C-H bonds} + \text{Reaction enthalpy}\]\[\text{Net energy change} = 3150\ \mathrm{kJ/mol} - 700\ \mathrm{kJ/mol} + 225\ \mathrm{kJ/mol}\]\[\text{Net energy change} = 2675\ \mathrm{kJ/mol}\]

Determine the multiple bond energy of the C≡C bond

The net energy change calculated in step 5 represents the energy for forming one mole of C≡C bonds (since this is the reaction for forming one mole of acetylene which contains one mole of C≡C bonds):\[\text{Bond energy of C≡C bond} = \text{Net energy change}\]\[\text{Bond energy of C≡C bond} = 2675\ \mathrm{kJ/mol}\]

Key concepts

Hess's Law

Hess's Law, a fundamental principle in chemical thermodynamics, provides a path to capture the total enthalpy change in a chemical reaction, irrespective of the steps that take place. According to this law, the total enthalpy change in a chemical process is the same no matter how the process is carried out, as long as the initial and final states are the same. This means that if a reaction can be expressed as the sum of a series of steps, the enthalpy change of the entire reaction is the sum of the enthalpy changes of the individual steps.

This principle simplifies enthalpy change calculations significantly. It allows chemists to use tabulated thermochemical data for bonds and reactions to compute enthalpy changes for reactions for which direct measurement isn't easy. This law is especially important when direct routes of a reaction are unknown or difficult to experimentally determine. In our exercise, we employed Hess's Law to break down the formation of acetylene into steps involving the sublimation of carbon, dissociation of hydrogen, and formation of carbon-hydrogen bonds.

Bond Energy

Understanding Bond Energy

Bond energy is a measure of the strength of a chemical bond. It's defined as the amount of energy required to break one mole of molecules into its individual atoms in the gas phase. Conversely, it is also the energy released when one mole of bonds are formed from the corresponding atoms. It's an average value derived from many different compounds, making it a useful, though approximate, tool for understanding reaction energetics.

In our exercise, we specifically looked at the bond energies of C−H and C≡C bonds. These values are critical for calculating the enthalpy change during the formation of acetylene, and they help us understand how much energy is stored within the bonds of molecules. It's important to remember that bond energies for different types of bonds, even between the same elements, can vary significantly, reflecting the diversity of chemical interactions.

Chemical Thermodynamics

Role in Reactivity and Stability

Chemical thermodynamics is the branch of physical chemistry that deals with energy changes, particularly heat and work, associated with chemical reactions and the properties of matter. The first law of thermodynamics ensures conservation of energy in chemical processes, while the second law explains the spontaneous direction of reactions – towards increasing disorder, or entropy. Enthalpy (\( \text{H} \)) is a central concept in chemical thermodynamics, providing insights into the heat content of a system at constant pressure.

In practical applications, chemical thermodynamics helps us predict whether a reaction will occur spontaneously and it assesses how external conditions like temperature and pressure can influence reactivity and stability. The exercise showcases the application of these principles to calculate the bond energy associated with a specific chemical bond within a reactive molecule.

Enthalpy of Formation

Calculating Reaction Enthalpy

The enthalpy of formation is defined as the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states. It serves as a reference point for the thermochemistry of reactions where it is used to calculate the heat absorbed or released during a reaction. If we know the enthalpy of formation for the reactants and products of a reaction, we can calculate the reaction enthalpy by subtracting the sum of the enthalpies of the reactants from the sum of the enthalpies of the products.

In our case, we used given enthalpy changes for sublimation, bond dissociation, and bond formation to determine the enthalpy change of the overall reaction, which is effectively the formation enthalpy of the C≡C bond in acetylene. Enthalpy of formation values are often found in tables in chemistry references and are crucial for solving a variety of thermodynamic problems.