Question

Calculate \(\Delta H_{\mathrm{rxn}}\) for the reaction: $$5 \mathrm{C}(s)+6 \mathrm{H}_{2}(g) \longrightarrow \mathrm{C}_{5} \mathrm{H}_{12}(l)$$Use the following reactions and given \(\Delta H^{\prime}\) s:$$ \begin{array}{l} \mathrm{C}_{5} \mathrm{H}_{12}(l)+8 \mathrm{O}_{2}(g) \longrightarrow 5 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(g) \\\\\qquad \begin{array}{ll}\Delta H=-3244.8 \mathrm{~kJ} \\\\\mathrm{C}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g) & \Delta H=-393.5 \mathrm{~kJ} \\ 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(g) & \Delta H=-483.5\mathrm{~kJ}\end{array}\end{array}$$

Short answer

\(\Delta H_{\mathrm{rxn}} = -173.2 \mathrm{~kJ}\)

Step-by-step solution

- Write the target equation

Write down the given chemical reaction for which you need to find \(\Delta H_{\mathrm{rxn}}\): \[5 \mathrm{C}(s)+6 \mathrm{H}_{2}(g) \longrightarrow \mathrm{C}_{5}\mathrm{H}_{12}(l)\]

- Write the given equations

List the given reactions with their respective \(\Delta H'\) values: \[\begin{array}{l}\mathrm{C}_{5} \mathrm{H}_{12}(l)+8 \mathrm{O}_{2}(g) \longrightarrow 5 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(g), \ \Delta H=-3244.8 \mathrm{~kJ}\ \mathrm{C}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g), \ \Delta H=-393.5 \mathrm{~kJ}\ 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2}\mathrm{O}(g), \ \Delta H=-483.5\mathrm{~kJ}\end{array}\]

- Manipulate the given equations to match the target equation

Flip and multiply the given reactions to cancel out unwanted substances and match moles in the target equation: \[5(\mathrm{C}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)), \ \Delta H=5(-393.5 \mathrm{~kJ})\ 3(2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{H}_{2}\mathrm{O}(g)), \ \Delta H=3(-483.5\mathrm{~kJ})\ \mathrm{C}_{5} \mathrm{H}_{12}(l)+8 \mathrm{O}_{2}(g) \longrightarrow 5 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(g), \ \Delta H= (+3244.8 \mathrm{~kJ})\] Note that the last reaction is the same as the target equation but it needs to be reversed to cancel out the products and leave only the reactants.

- Calculate the enthalpy changes (\(\Delta H\)) for each manipulated equation

Calculate the \(\Delta H\) for each individual manipulated equation using the coefficients: \[\Delta H_{1}=5(-393.5 \mathrm{~kJ}) = -1967.5 \mathrm{~kJ}\ \Delta H_{2}=3(-483.5\mathrm{~kJ}) = -1450.5 \mathrm{~kJ}\ \Delta H_{3}= +3244.8 \mathrm{~kJ} (since the reaction is reversed, the sign is positive)\]

- Add up the \(\Delta H\) values

Sum the enthalpy changes for the manipulated equations to get the enthalpy change of the reaction (\(\Delta H_{\mathrm{rxn}}\)): \[\Delta H_{\mathrm{rxn}} = \Delta H_{1} + \Delta H_{2} + \Delta H_{3}= (-1967.5 \mathrm{~kJ}) + (-1450.5 \mathrm{~kJ}) + (+3244.8 \mathrm{~kJ})\]

Key concepts

Chemical Reactions

In the fascinating world of chemistry, chemical reactions are the heart of the subject, representing the processes by which substances interact to form new products. These transformations are governed by the laws of conservation of mass and energy, meaning that while the substances themselves change, the total mass and energy remain constant.

A chemical reaction involves breaking the bonds of reactants and forming new ones to create the products. This process is accompanied by energy changes, releasing or absorbing heat or other forms of energy. Every chemical reaction can be represented by a balanced chemical equation, showcasing the reactants, products, and their molar relationships.

Understanding chemical reactions not only provides insights into the composition of matter but also helps us to predict and harness energy changes associated with these reactions. For instance, the combustion of fuels, the metabolism of food in our bodies, and even the creation of materials like plastics all involve chemical reactions.

Thermochemistry

Thermochemistry is the branch of chemistry that studies the energy and heat associated with chemical reactions and phase changes. One fundamental concept in thermochemistry is the system, which can be anything we're studying, such as a beaker of chemicals. Surrounding the system is the environment, which includes everything else. In a chemical reaction, energy can be transferred between the system and its surroundings.

Enthalpy (H) is a measurement of energy in a thermodynamic system. It's indicative of the system's ability to do non-mechanical work and the energy required to create the system. Enthalpy change (H) of a reaction is the heat absorbed or released at constant pressure during a reaction.

The enthalpy change can often be visualized as the heat content difference between products and reactants. Reactions that release heat into the environment are exothermic (H is negative), while reactions that absorb heat from the environment are endothermic (H is positive). Knowing enthalpy changes in chemical processes is key for industries, for example, when maximizing energy efficiency or product yield from reactions.

Hess's Law

Hess's Law states that the total enthalpy change during the course of a chemical reaction is the same whether the reaction is made in one step or several steps. This principle is incredibly useful because it allows chemists to determine enthalpy changes (H) for reactions that are difficult to measure directly.

By using Hess's Law, we can add or subtract known reactions to create a pathway for the reaction we're interested in. This is exactly what we did in our step-by-step solution: we manipulated given reactions to find the H for an equation that's not easily measurable by itself.

In essence, Hess's Law leverages the fact that enthalpy is a state function, which means its value is determined only by the current state of the system and not the path taken to reach that state. By carefully selecting and summing the enthalpies of known reactions, we can ingeniously deduce the enthalpy change for a new reaction.