Question

The heat of combustion of cyclopropane, \(\left(\mathrm{CH}_{2}\right)_{3}\), to give carbon dioxide and liquid water is \(499.8 \mathrm{kcal} \mathrm{mol}^{-1}\). Show how this value, assuming normal \(\mathrm{C}-\mathrm{H}\) bond strengths, can be used to calculate the average \(\mathrm{C}-\mathrm{C}\) bond energy of cyclopropane.

Short answer

The C-C bond energy in cyclopropane is approximately 272.9 kcal/mol.

Step-by-step solution

Write the Combustion Equation

The combustion of cyclopropane can be represented by the following balanced chemical equation: \[ \text{C}_3\text{H}_6 + \frac{9}{2} \text{O}_2 \rightarrow 3 \text{CO}_2 + 3 \text{H}_2\text{O}. \] This equation shows that each molecule of cyclopropane combusts with 4.5 molecules of oxygen to form 3 molecules of carbon dioxide and 3 molecules of water.

Determine Known Bond Energies

List the bond energies involved in the reaction. Assume average bond energies: \( \text{C-H} = 99 \ \text{kcal/mol} \), \( \text{C=O} (in \ \text{CO}_2) = 187 \ \text{kcal/mol} \), \( \text{O=O} = 119 \ \text{kcal/mol} \), \( \text{O-H} = 111 \ \text{kcal/mol}. \)

Calculate Total Energy of Bonds Broken

To find the energy required to break the bonds in cyclopropane, calculate: 3 C-C bonds, 6 C-H bonds, and 4.5 O=O bonds are broken. The total bond energy of the reactants is: \[ E_{\text{broken}} = 6(99) + 3E_{\text{C-C}} + 4.5(119) \] where \( E_{\text{C-C}} \) is the unknown bond energy of the C-C bond in cyclopropane.

Calculate Total Energy of Bonds Formed

Calculate the energy released when new bonds are formed in the products. The formation involves 6 C=O bonds (3 \( \text{CO}_2 \)) and 6 O-H bonds (3 \( \text{H}_2\text{O} \)), totaling: \[ E_{\text{formed}} = 6(187) + 6(111). \]

Apply the Heat of Combustion to the Energy Equation

The heat of combustion is defined as the difference between the energy of bonds formed and the energy of bonds broken, equal to 499.8 kcal/mol. Set up the equation: \[ 6(187) + 6(111) - (6(99) + 3E_{\text{C-C}} + 4.5(119)) = 499.8. \] Simplify to find \( E_{\text{C-C}} \): \[ 1782 + 666 - (594 + 3E_{\text{C-C}} + 535.5) = 499.8. \]

Solve for the C-C Bond Energy in Cyclopropane

Solve the equation for \( E_{\text{C-C}} \): \[ 2448 - 1129.5 - 3E_{\text{C-C}} = 499.8. \] Further simplify: \[\begin{align*} 1318.5 - 3E_{\text{C-C}} &= 499.8, \ 3E_{\text{C-C}} &= 818.7, \ E_{\text{C-C}} &= 272.9 \text{kcal/mol}. \end{align*}\] Thus, the C-C bond energy in cyclopropane is approximately 272.9 kcal/mol.

Key concepts

Cyclopropane

Cyclopropane is a type of cycloalkane that is composed of three carbon atoms arranged in a triangle, each bonded to two hydrogen atoms. This small, three-membered ring structure results in significant angle strain due to bond angles of 60 degrees, which is much less than the tetrahedral angle of approximately 109.5 degrees in regular alkanes.
Such strain makes cyclopropane quite reactive compared to other larger cyclic and linear alkanes. Despite this strain, cyclopropane is stable under normal conditions but tends to be more strained and thus more interesting to chemists.
It serves as an important compound in studies of bond angles and angle strain in organic chemistry. The reactivity of cyclopropane can also be harnessed in synthetic organic chemistry, particularly in reactions where the ring-opening leads to more stable products.

Heat of Combustion

The heat of combustion of a compound like cyclopropane is the amount of heat energy released when a specified amount is burned completely in oxygen. This is a crucial concept in thermochemistry, reflecting the stability of a compound.
In the case of cyclopropane, its combustion equation shows the complete reaction with oxygen to produce carbon dioxide and water. The considerable angle strain in cyclopropane's structure contributes to its relatively high heat of combustion. This value is an indicator of how much energy can be released in converting cyclopropane to more thermally stable products, like CO2 and H2O.
Understanding the heat of combustion is valuable for calculating bond energies and understanding reaction energetics in organic molecules.

Chemical Bonds

Chemical bonds are the connections between atoms in a molecule. They dictate the molecule's structure and reactivity. In cyclopropane, each carbon atom forms bonds with two other carbon atoms (C-C) and two hydrogen atoms (C-H).
C-H bonds are relatively strong and uniform in energy, whereas C-C bonds in cyclopropane are notably weaker than in other alkanes due to the angle strain of the cyclopropane ring structure. This is important when considering bond energies during chemical reactions.
Bond strength, often quantified by bond energy, is crucial in chemical reactions because breaking and forming bonds involves changes in energy. Understanding bond energies helps chemists predict the stability and reactivity of molecules like cyclopropane.

Organic Chemistry

Organic chemistry is the branch of chemistry that studies carbon compounds, particularly hydrocarbons and their derivatives. Cyclopropane is one such compound, serving as a fundamental example of a strained ring in organic chemistry.
The study of organic chemistry encompasses various properties of compounds, including their structure, reactivity, and interactions. Cyclopropane, with its unique three-membered ring, poses interesting challenges and opportunities for study, particularly in understanding the effects of strain on reactivity and stability.
Organic chemistry is vital in synthesizing new compounds and materials, ranging from pharmaceuticals to plastics. Cyclopropane's properties are harnessed in many applications, and its study assists in understanding broader concepts such as bond energies and reaction mechanisms within the field.