Problem 60 - Chemistry An Atoms First Approach

Question

The value of \(\Delta G^{\circ}\) for the reaction $$2 \mathrm{C}_{4} \mathrm{H}_{10}(g)+13 \mathrm{O}_{2}(g) \longrightarrow 8 \mathrm{CO}_{2}(g)+10 \mathrm{H}_{2} \mathrm{O}(l)$$ is \(-5490 .\) kJ. Use this value and data from Appendix 4 to calculate the standard free energy of formation for \(\mathrm{C}_{4} \mathrm{H}_{10}(g).\)

Step-by-Step Solution

Answer

The standard free energy of formation for gaseous butane (C\(_4\)H\(_{10}\)(g)) is 18.6 kJ/mol.

Step 1: Write down the given data and the equation

We are given: ∆G° for the reaction = -5490 kJ Values for the standard free energy of formation for other substances involved (from Appendix 4) To calculate the standard free energy of formation for C\(_4\)H\(_{10}\)(g), we will use the equation: ∆G° = ∆G°(products) - ∆G°(reactants)

Step 2: Identify the standard free energy of formation values for each substance from Appendix 4

From Appendix 4: ∆Gf° of O\(_2\)(g) = 0 kJ/mol (since it is in its standard state) ∆Gf° of CO\(_2\)(g) = -394.4 kJ/mol ∆Gf° of H\(_2\)O(l) = -237.2 kJ/mol

Step 3: Use the equation to derive the standard free energy of formation for C\(_4\)H\(_{10}\)(g)

Using the equation ∆G° = ∆G°(products) - ∆G°(reactants), substitute the given values and rearrange for the unknown value (∆Gf° of C\(_4\)H\(_{10}\)): -5490 kJ = {8 x (-394.4 kJ/mol) + 10 x (-237.2 kJ/mol)} - 2 x ∆Gf°(C\(_4\)H\(_{10}\)) - 13 x 0 Simplifying and solving for ∆Gf°(C\(_4\)H\(_{10}\)): -5490 kJ = {-3155.2 kJ/mol - 2372 kJ/mol} - 2 x ∆Gf°(C\(_4\)H\(_{10}\)) -5490 kJ = -5527.2 kJ/mol + 2 x ∆Gf°(C\(_4\)H\(_{10}\))

Step 4: Calculate the standard free energy of formation for C\(_4\)H\(_{10}\)(g)

Now, isolate ∆Gf°(C\(_4\)H\(_{10}\)) to find its value: 2 x ∆Gf°(C\(_4\)H\(_{10}\)) = 37.2 kJ/mol ∆Gf°(C\(_4\)H\(_{10}\)) = \(\frac{37.2}{2}\) kJ/mol = 18.6 kJ/mol Thus, the standard free energy of formation for gaseous butane (C\(_4\)H\(_{10}\)(g)) is 18.6 kJ/mol.

Key Concepts

Gibbs Free Energy

Gibbs Free Energy, denoted as \(\Delta G\), is a fundamental concept in thermodynamics. It helps predict the spontaneity of a process. If \(\Delta G\) is negative, the reaction proceeds spontaneously under constant temperature and pressure. If it is positive, the reaction is non-spontaneous, and external energy is required for it to proceed. This applies to chemical reactions, where knowing the Gibbs Free Energy change can predict the feasibility of a reaction occurring without outside intervention.

To understand it practically, consider a chemical reaction where the equation is given. The change in Gibbs Free Energy for this reaction is calculated using the formula:

• \(\Delta G = \Delta G\left(products \right) - \Delta G\left(reactants\right)\)

Here, the energies of products and reactants determine the overall energy balance. By calculating \(\Delta G\), chemists can ascertain whether a reaction happens spontaneously or needs energy input.

Standard Free Energy of Formation

The Standard Free Energy of Formation, symbolized as \(\Delta G_f^\circ\), is the free energy change when one mole of a compound forms from its elements in their standard states. This concept is vital to compare the energies of different substances easily. A standard state refers to a pressure of 1 bar (approximately 1 atm) and a concentration of 1 M for solutions.

This concept simplifies complex calculations by providing reference values that describe the energy needed to form compounds from elements. These values are typically tabulated in data appendices found in textbooks or chemical databases. For gaseous compounds like butane, these values help in calculating the Gibbs Free Energy change for reactions involving multiple compounds.

To calculate \(\Delta G_f^\circ\) for a compound, you'll need the standard free energies of formation for other substances in the reaction. Add these values together to find the energy change. It’s a crucial step in figuring out whether product formation is efficient and energy-favorable.

Chemical Reactions

Chemical reactions are processes involving the transformation of one or more substances into new products. These transformations are governed by the principles of thermodynamics, where energy changes play a significant role. The thermodynamic data of these reactions, such as Gibbs Free Energy, helps predict how and why reactions occur.

A balanced chemical equation is vital. It shows not only the reactants and products but also the stoichiometric proportions used in calculations. For instance, the reaction given in the exercise shows butane reacting with oxygen to produce carbon dioxide and water. The coefficients indicate mole ratios used to calculate the energy changes and other properties.

Understanding chemical reactions encompasses several concepts such as reaction equilibrium, rate of reaction, and the energy involved. For chemists, examining the change in Gibbs Free Energy across the entire reaction allows them to assess spontaneity and energy efficiency—essential for both practical applications and further theoretical research.