Problem 64 - Chemistry The Central Science

Question

From the enthalpies of reaction $$ \begin{aligned} 2 \mathrm{C}(s)+\mathrm{O}_{2}(g) & \longrightarrow 2 \mathrm{CO}(g) & \Delta H=&-221.0 \mathrm{~kJ} \\ 2 \mathrm{C}(s)+\mathrm{O}_{2}(g)+4 \mathrm{H}_{2}(g) & \longrightarrow 2 \mathrm{CH}_{3} \mathrm{OH}(g) & \Delta H=&-402.4 \mathrm{~kJ} \end{aligned} $$ calculate \(\Delta H\) for the reaction $$ \mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(g) $$

Step-by-Step Solution

Answer

The enthalpy change (ΔH) for the target reaction \(CO(g) + 2 H_2(g) \longrightarrow CH_3OH(g)\) is 90.7 kJ.

Step 1: Identify the target reaction and given reactions

The given reactions are: 1. \(2C(s) + O_2(g) \longrightarrow 2CO(g)\), ΔH = -221.0 kJ 2. \(2C(s) + O_2(g) + 4H_2(g) \longrightarrow 2CH_3OH(g)\), ΔH = -402.4 kJ We want to find the ΔH for the reaction: \(CO(g) + 2 H_2(g) \longrightarrow CH_3OH(g)\)

Step 2: Manipulate given reactions to match target reaction

We can manipulate the given reactions in order to recreate the target reaction. First, we can scale Reaction 1 by \(\frac{1}{2}\) so that it matches the target reaction for surrounding CO: \(\frac{1}{2} \cdot (2 C(s) + O_2(g) \longrightarrow 2 CO(g))\) Resulting in: \( C(s) + \frac{1}{2} O_2(g) \longrightarrow CO(g)\), ΔH = -221.0 kJ * \(\frac{1}{2}\) = -110.5 kJ Next, reverse Reaction 2 in order to have the target reaction's final product on the same side of the arrow: \(- (2C(s) + O_2(g) + 4H_2(g) \longrightarrow 2CH_3OH(g))\) Resulting in: \(2CH_3OH(g) \longrightarrow 2C(s) + O_2(g) + 4H_2(g)\), ΔH = -(-402.4 kJ) = 402.4 kJ Then, scale the resulting reaction by \(\frac{1}{2}\): \(\frac{1}{2} \cdot (2CH_3OH(g) \longrightarrow 2C(s) + O_2(g) + 4H_2(g))\) Resulting in: \(CH_3OH(g) \longrightarrow C(s) + \frac{1}{2} O_2(g) + 2H_2(g)\), ΔH = 402.4 kJ * \(\frac{1}{2}\) = 201.2 kJ

Step 3: Add manipulated reactions to find target reaction and its ΔH

Now we can add the manipulated reactions together: (C(s) + \(\frac{1}{2}\) O2(g) → CO(g), ΔH = -110.5 kJ) + (CH3OH(g) → C(s) + \(\frac{1}{2}\) O2(g) + 2H2(g), ΔH = 201.2 kJ) _______________________________________________________ Resulting in the desired reaction: \(CO(g) + 2 H_2(g) \longrightarrow CH_3OH(g)\) Now we can add the enthalpies of the manipulated reactions to find the ΔH of the target reaction: ΔH = -110.5 kJ + 201.2 kJ = 90.7 kJ The enthalpy change (ΔH) for the target reaction \(CO(g) + 2 H_2(g) \longrightarrow CH_3OH(g)\) is 90.7 kJ.

Key Concepts

Thermochemistry

Thermochemistry is a branch of chemical thermodynamics that deals with the energy and heat associated with chemical reactions and physical transformations. The energy change that occurs during a chemical reaction is measured in terms of enthalpy change (ΔH), which is the heat absorbed or released at constant pressure.

Understanding ΔH is crucial for predicting whether a reaction will be exothermic (releasing heat) or endothermic (absorbing heat). In the example given, the calculation of enthalpy changes involves understanding how energy is conserved within a chemical reaction and how it can be transferred from one form to another—principles that underscore the essence of thermochemistry. ΔH is a fundamental aspect in calculating reaction spontaneity and equilibrium, making thermochemistry a key topic in both academic studies and practical applications such as energy production and material synthesis.

To illustrate, when a substance burns in oxygen, it releases a certain amount of energy, which can be measured and is expressed as the enthalpy change for that particular reaction. This concept helps students grasp the energy flow in reactions, laying the groundwork for more complex thermodynamic concepts.

Chemical Reactions

Chemical reactions involve the transformation of one set of chemical substances to another. They are described by chemical equations that show the reactants on the left side of an arrow and the products on the right. Importantly, these reactions adhere to the principle of conservation of mass, which states that mass cannot be created or destroyed in a chemical reaction.

In order to understand a chemical reaction, it is essential to comprehend the stoichiometry—which describes the quantitative relationship between reactants and products in a balanced chemical equation—and the energy changes that accompany the reaction, encapsulated by the change in enthalpy (ΔH).

For example, the reaction to form methanol (CH_3OH) from carbon monoxide (CO) and hydrogen gas (H_2) is a synthesis reaction, where simpler molecules combine to form a more complex molecule. Explaining chemical reactions involves examining the bonds that are broken and formed during the process, which in turn relates to the energy changes observed. The step by step solution provided shows how chemical equations can be manipulated, using stoichiometry, to calculate these energy changes.

Hess's Law

Hess's Law is a statement in chemistry that the total enthalpy change for a reaction is the same, regardless of the number of steps taken to perform the reaction. This law is fundamental in thermochemistry because it allows for the calculation of ΔH for a reaction that may not be easily measurable by using known enthalpy changes of other related reactions.

Applying Hess's Law often involves breaking a reaction down into a series of steps for which ΔH values are known (or can be reliably estimated), then adding or subtracting these values as appropriate to find the overall ΔH for the reaction in question. In the exercise provided, the law is applied by manipulating given reactions to recreate the target reaction and then combining their enthalpy changes to calculate the overall ΔH.

The ability to use Hess's Law effectively requires a strong understanding of stoichiometry and the ability to manipulate chemical equations, as seen in the step by step solution. When using Hess's Law, the direction (endothermic or exothermic) and the magnitude of the enthalpy change are conserved over the different steps of the reaction, allowing for accurate calculations of energy changes that underpin much of chemical process design and analysis.