Question

Estimate the enthalpy of reaction for the equilibrium reaction \(\mathrm{CH}_{4}+2 \mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O}\) at \(2500 \mathrm{K}\), using \((a)\) enthalpy data and \((b) K_{P}\) data. Obtain enthalpy and entropy properties from EES.

Short answer

Question: Estimate the enthalpy of reaction for the given equilibrium reaction at 2500 K using enthalpy data and Kp data from EES. Answer: To estimate the enthalpy of reaction at 2500 K, first calculate the enthalpy change of the reaction using enthalpy data obtained from EES for each substance. Next, use Kp data from EES and the van't Hoff equation to find the relationship between Kp and enthalpy change. Finally, compare the enthalpy change of the reaction calculated from enthalpy data with the enthalpy of reaction estimated using Kp data. The values should be close to each other, confirming the validity of the estimation.

Step-by-step solution

Obtain enthalpy data from EES for each substance at the given temperature

To calculate the enthalpy change of the given reaction, we need to first find the enthalpy, \(h\), of each substance (CH4, O2, CO2, and H2O) at the given temperature T = 2500 K from EES. Note down the values for future use in the calculations.

Calculate the enthalpy change of the reaction using enthalpy data

Using the enthalpy data obtained in step 1, we can calculate the enthalpy change of the reaction, \(\Delta H_{rxn}\) formula: \(\Delta H_{rxn} = \sum h_{products} - \sum h_{reactants}\) \(\Delta H_{rxn} = (1 mol \cdot h_{CO_{2}} + 2 mol \cdot h_{H_{2}O}) - (1 mol \cdot h_{CH_{4}} + 2 mol \cdot h_{O_{2}})\) Using the enthalpy values obtained in step 1, calculate the value of \(\Delta H_{rxn}\).

Obtain Kp data from EES

Using the EES, obtain the Kp value for the given reaction at the given temperature, T = 2500 K. Note down the Kp value for future use in calculations.

Use the van't Hoff equation to find the relationship between Kp and enthalpy change

The van't Hoff equation relates the equilibrium constant (\(K_p\)) to the enthalpy change (\(\Delta H_{rxn}\)) and temperature (T): \(\log K_p = -\frac{\Delta H_{rxn}}{2.303 \times R \times T} + C\) Here, R = 8.314 J/mol.K (gas constant), T = 2500 K, and C is a constant. We will use this equation to estimate the enthalpy of reaction using the Kp value obtained in step 3.

Estimate the enthalpy of reaction using Kp data

Substitute the Kp value obtained in step 3, temperature T = 2500 K, and the gas constant R into the van't Hoff equation: \(\log K_p = -\frac{\Delta H_{rxn}}{2.303 \times 8.314 \times 2500} + C\) Now, we can solve for \(\Delta H_{rxn}\): \(\Delta H_{rxn} = -2.303 \times 8.314 \times 2500 \times \log K_p + C'\) Here, C' is a constant derived from C. Finally, compare the enthalpy change of the reaction (\(\Delta H_{rxn}\)) calculated in step 2 (from enthalpy data) with the enthalpy of reaction estimated using Kp data in step 5. The values should be close to each other, which will confirm the validity of the estimation.

Key concepts

Chemical Thermodynamics

Chemical thermodynamics is a subsection of thermodynamics that deals with the energy changes associated with chemical reactions and the composition of substances. It is a fundamental concept in chemistry that helps scientists and engineers understand how energy is transferred within systems and how it influences the direction and extent of chemical reactions.

At the heart of chemical thermodynamics is the first law of thermodynamics, which states that energy can neither be created nor destroyed, only transformed from one form to another. In the context of chemical reactions, this transforming energy is often referred to as the enthalpy change (\text{\text{\text{\text{\text{\text{\text{\text{enthalpy change (\(\Delta H\)))}, and it signifies the heat absorbed or released during a reaction at constant pressure. Understanding the concept of enthalpy is crucial for determining whether a reaction is exothermic or endothermic.

Importance of Enthalpy in Chemical Reactions

• Exothermic Reactions: These are reactions that release energy into the surroundings, often in the form of heat, resulting in a negative \(\Delta H\). A common example is the combustion of methane.
• Endothermic Reactions: Reactions that absorb energy from the surroundings, which often results in a temperature decrease in the immediate environment and a positive \(\Delta H\).

The determination of enthalpy changes is essential not only for predicting reaction outcomes but also for the industrial design of chemical processes, where energy efficiency is a critical consideration.

Van't Hoff Equation

The van't Hoff equation is a powerful tool in chemical thermodynamics, serving as a bridge between thermodynamics and chemical equilibrium. It describes how the equilibrium constant (\text{\text{\text{\text{\text{\text{\text{equilibrium constant (\(K\))})) for a reaction varies with temperature. This relation is particularly useful for understanding how temperature impacts the position of equilibrium and hence the yield of reaction products.The van't Hoff equation can be represented in its integrated form as:\[log K = -\frac{\Delta H}{2.303 \times R \times T} + C\]where:

• \(K\) is the equilibrium constant,
• \(\Delta H\) is the enthalpy change of the reaction,
• \(R\) is the universal gas constant,
• \(T\) is the absolute temperature, and
• \(C\) is the integration constant, related to the entropy change of the reaction.

Applying the Van’t Hoff Equation

The equation allows us to calculate one of the terms if the others are known. In the context of the exercise, knowing \(K\) and \(T\), one can deduce \(\Delta H\), the enthalpy of reaction, which is required to understand how energy changes during the reaction. Additionally, it provides insight into the nature of the reaction, whether it is more product-favored or reactant-favored at a given temperature.

Equilibrium Constant

The equilibrium constant, often denoted as \text{\text{\text{\text{\text{\text{\text{equilibrium constant (\(K\))))}, is a numerical value that expresses the ratio of concentrations of products to reactants for a reversible reaction at equilibrium. It is a pivotal concept in chemistry because it quantifies the extent of a reaction, or in other words, how far the reaction will proceed before reaching a state of balance.Depending on the circumstances of the reaction (e.g., whether it involves gases or solutions), different forms of the equilibrium constant are used, such as \(K_p\) for partial pressures or \(K_c\) for concentrations.

Significance of \text{\text{\text{\text{\text{Significance of \(K\) in Reactions

• If \(K\) is much greater than 1, the reaction favors the formation of products, suggesting that at equilibrium, there are more products than reactants.
• If \(K\) is less than 1, the equilibrium favors the reactants, indicating that there are more reactants than products at equilibrium.

The equilibrium constant is not static; it can change with temperature, as per the van't Hoff equation, which provides a quantitative means of correlating thermodynamics with reaction equilibria. Being able to calculate the equilibrium constant from experimental data or predict how it will change with temperature is an indispensable skill for chemists and chemical engineers.

Engineering Equation Solver (EES)

The Engineering Equation Solver (EES) is a software tool widely used by engineers and scientists for solving complex mathematical problems that arise in thermodynamics, heat transfer, fluid mechanics, and many other engineering disciplines. EES is particularly valued for its ability to perform iterative calculations with high precision and deal with systems of equations involving both numerical and symbolic computations.

Advantages of EES in Calculating Thermodynamic Properties

• It contains a vast database of thermodynamic properties for a wide range of substances, saving considerable time that would be otherwise spent on manual look-ups in data tables.
• The software supports automatic unit conversion, which minimizes human error related to unit inconsistencies.
• EES can efficiently deal with complex scenarios, such as those involving multiple variables and intricate dependencies between them.

In the context of the given exercise, EES provides a quick and reliable way to obtain the enthalpies of various substances at a specific temperature, which is crucial for calculating the enthalpy of the reaction. Moreover, it enables the computation of the equilibrium constant \(K_p\), which further allows for the application of the van't Hoff equation to estimate changes in the enthalpy of reaction due to alterations in temperature. EES thus contributes significantly to streamlining the process of analyzing and solving problems in chemical thermodynamics.