Problem 110 - Chemistry The Central Science In Si Units

Question

We can use Hess's law to calculate enthalpy changes that cannot be measured. One such reaction is the conversion of methane to ethane: $$ 2 \mathrm{CH}_{4}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{6}(g)+\mathrm{H}_{2 (g) $$ Calculate the \(\Delta H^{\circ}\) for this reaction using the following thermochemical data: $$ \begin{aligned} \mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) & \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(I) & & \Delta H^{0}=-890.3 \mathrm{~kJ} \\ 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) & \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) & \Delta H^{0} &=-571.6 \mathrm{~kJ} \\ 2 \mathrm{C}_{2} \mathrm{H}_{6}(g)+7 \mathrm{O}_{2}(g) & \longrightarrow 4 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(I) & \Delta H^{0}=&-3120.8 \mathrm{~kJ} \end{aligned} $$

Step-by-Step Solution

Answer

\(-84.8\, kJ\) is the enthalpy change for the conversion of methane to ethane.

Step 1: Balance the Target Reaction

The given target reaction is \( 2 \mathrm{CH}_{4}(g) \rightarrow \mathrm{C}_{2} \mathrm{H}_{6}(g) + \mathrm{H}_{2}(g) \). This reaction is already balanced with respect to all elements.

Step 2: Understand the Given Reactions

We have been provided three reactions with their enthalpy changes. These reactions are: (1) \( \mathrm{CH}_{4}(g) + 2 \mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) + 2 \mathrm{H}_{2} \mathrm{O}(l) \), (2) \( 2 \mathrm{H}_{2}(g) + \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(l) \), and (3) \( 2 \mathrm{C}_{2} \mathrm{H}_{6}(g) + 7 \mathrm{O}_{2}(g) \rightarrow 4 \mathrm{CO}_{2}(g) + 6 \mathrm{H}_{2} \mathrm{O}(l) \) with corresponding \( \Delta H^{0} \) values.

Step 3: Rearrange Reactions for Desired Product

To obtain the desired reaction, modify provided reactions such that \( 2\mathrm{CH}_{4} \) is consumed and \( \mathrm{C}_{2}\mathrm{H}_{6} \) and \( \mathrm{H}_{2} \) are produced. Reverse reaction (1) to decompose methane and multiply reaction (3) by \( \frac{1}{2} \) to produce ethane. This gives: \( 2 \mathrm{CO}_{2}(g) + 4 \mathrm{H}_{2} \mathrm{O}(l) \rightarrow 2 \mathrm{CH}_{4}(g) + 4 \mathrm{O}_{2}(g) \), and \( \mathrm{C}_{2}\mathrm{H}_{6}(g) + \frac{7}{2} \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{CO}_{2}(g) + 3 \mathrm{H}_{2} \mathrm{O}(l) \).

Step 4: Apply Hess's Law for Calculation

Using Hess's Law, express the \( \Delta H^0 \) of the target reaction as a sum of the \( \Delta H^0 \) changes of the individual reactions. Reverse reaction (1) yields \( +2 \times 890.3 \) kJ. Multiply reaction (3) by \( \frac{1}{2} \) to yield \( -1560.4 \) kJ. Sum these with \( (2 \times 571.6) \) to account for the \( \mathrm{H}_{2} \rightarrow \mathrm{H}_{2} \mathrm{O} \) changes in reaction (2), which gives \( -1143.2 \) kJ.

Step 5: Calculate Final ΔH° Value

Combine all values from Step 4. \( (2 \times 890.3) + ( -1560.4 ) + (-1143.2) = -84.8 \) kJ. Therefore, \( \Delta H^{0} \) for the conversion of methane to ethane is \(-84.8\) kJ.

Key Concepts

Enthalpy Change

Enthalpy change, represented by \( \Delta H \), is the measure of heat energy absorbed or released during a chemical reaction at constant pressure. It's an essential concept in thermochemistry that helps us understand how energy is transferred within chemical systems. When a reaction releases heat, indicated by a negative \( \Delta H \), it is exothermic. Conversely, when heat is absorbed, \( \Delta H \) is positive, denoting an endothermic process. Understanding enthalpy is vital for predicting reaction behavior and energy requirements. For instance, industries often need to calculate energy changes to optimize processes. In the context of Hess's Law, the concept of enthalpy change allows us to manipulate and combine thermochemical equations to find unknown \( \Delta H \) values of reactions. This is extremely useful when dealing with reactions that are challenging to measure directly. Through Hess's Law, we can use known enthalpy changes of simpler reactions to derive the \( \Delta H \) of complex reactions, as we do in the conversion of methane to ethane.

Thermochemical Equations

Thermochemical equations are chemical equations that include enthalpy change, providing a complete picture of a reaction's energetics. When writing these equations, it's important to depict both the reactants and the products along with their physical states because these affect the energy change. For example, the equation \( \mathrm{CH}_{4}(g) + 2 \mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) + 2 \mathrm{H}_{2} \mathrm{O}(l) \) with \( \Delta H = -890.3 \) kJ informs us that methane combusting with oxygen releases a significant amount of energy in the form of heat.In calculating an overall \( \Delta H \) for a complex reaction, we rearrange and combine thermochemical equations according to Hess's Law. We may reverse equations and multiply them by necessary factors. For instance, reversing an equation like the one above changes the sign of \( \Delta H \), which is crucial for accurate energy calculations.

Chemical Reactions

Chemical reactions involve the transformation of substances through the breaking and forming of chemical bonds, resulting in one or more new products. The driving force of these reactions is often energy change, which could lead to energy release or absorption. In the case of methane's conversion to ethane, balancing the chemical equation ensures that the number of atoms for each element is equal on both sides of the equation. This step helps in the isolation of each substance's role in the reaction. Understanding chemical reactions includes recognizing the types—such as synthesis, decomposition, combustion, and more—and how they influence the system's energy dynamics. Recognizing how various reactions interrelate can highlight pathways where Hess’s Law applies, allowing us to compute otherwise challenging enthalpy changes by leveraging simpler known reactions. Through these insights, we see that reactions are not isolated events but part of a larger energetic framework governed by the laws of thermodynamics.