Question

Using the enthalpy of reaction \(\bar{h}_{R}\) data and the \(K_{P}\) value at \(2200 \mathrm{K}\), estimate the \(K_{P}\) value of the dissociation process \(\mathrm{O}_{2} \rightleftharpoons 2 \mathrm{O}\) at \(2400 \mathrm{K}\).

Short answer

Question: Estimate the \(K_P\) value of the dissociation process of \(\mathrm{O}_2\) at \(2400 \mathrm{K}\), using the enthalpy of reaction \(\bar{h}_{R}\) data and the \(K_{P}\) value given at \(2200 \mathrm{K}\). Answer: To estimate the \(K_P\) value at \(2400 \mathrm{K}\), we can use the van't Hoff equation, as shown in the steps above. Given the values of \(K_1\), \(\bar{h}_R\), \(T_1\), and \(T_2\), we can compute the value of \(K_2\), which represents the equilibrium constant \(K_P\) at 2400 K for the dissociation process \(\mathrm{O}_2 \rightleftharpoons 2 \mathrm{O}\). Make sure to include the gas constant, \(R = 8.314 \mathrm{J/mol\,K}\), in your calculations.

Step-by-step solution

Write the van't Hoff equation

The van't Hoff equation is: \[ \frac{d\ln K_P}{dT} = \frac{\Delta H_R}{RT^2} \] where \(K_P\) is the equilibrium constant, \(\Delta H_R\) is the enthalpy of reaction, \(R\) is the gas constant (8.314 J/mol K), and \(T\) is the temperature in Kelvin.

Integrate the van't Hoff equation

We need to estimate \(K_P\) at 2400 K, given the information at 2200 K. To do this, we integrate both sides of the van't Hoff equation in the temperature interval \((T_1,T_2)\): \[ \int_{K_1}^{K_2} \frac{d\ln K_P}{dT} dT = \int_{T_1}^{T_2} \frac{\Delta H_R}{RT^2} dT \] Since \(\Delta H_R\) is constant in our problem, we can proceed with the integration.

Perform the integration

After performing the integration, we will obtain the relationship between \(K_P\) values at different temperatures: \[ \ln\frac{K_2}{K_1} = -\frac{\Delta H_R}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) \] Now, we plug in the known values: \(K_1\) (the \(K_P\) value at 2200 K), \(T_1 = 2200 \mathrm{K}\), and \(T_2 = 2400 \mathrm{K}\).

Calculate the \(K_P\) value at \(2400 \mathrm{K}\)

Rearranging the equation to solve for \(K_2\), we have: \[ K_2 = K_1 e^{(-\frac{\Delta H_R}{R})(\frac{1}{T_2} - \frac{1}{T_1})} \] Plugging in the given values of \(K_1\), \(\bar{h}_R\), \(T_1\), and \(T_2\), we can compute the value of \(K_2\), which is the equilibrium constant \(K_P\) at 2400 K for the dissociation process \(\mathrm{O}_2 \rightleftharpoons 2 \mathrm{O}\). Don't forget to provide the gas constant, \(R = 8.314 \mathrm{J/mol\,K}\).

Key concepts

Enthalpy of Reaction

Enthalpy of reaction, denoted as \(\Delta H_R\), is a critical concept in thermodynamics and chemistry. It represents the heat absorbed or released during a chemical reaction at constant pressure. This value can be either positive, for endothermic reactions where heat is absorbed, or negative for exothermic reactions where heat is released.

In the context of the van't Hoff equation, \(\Delta H_R\) shows how the equilibrium constant \(K_P\) for a gas-phase reaction changes with temperature. A higher enthalpy change typically indicates a greater temperature sensitivity of the equilibrium constant. It's important to understand that \(\Delta H_R\) is a measure of the overall energy change in the conversion of reactants to products. For the oxygen dissociation process, \(\mathrm{O}_2 \rightleftharpoons 2 \mathrm{O}\), the \(\Delta H_R\) value would indicate the energy required to break the \(O=O\) double bond and form two separate oxygen atoms.

Knowing the \(\Delta H_R\) is crucial for predicting how conditions such as temperature will affect the chemical equilibrium in a given reaction.

Equilibrium Constant

The equilibrium constant, represented by \(K_P\), is a dimensionless value that indicates the ratio of the concentrations of products to reactants at chemical equilibrium for a given reaction at a specific temperature. It reflects the extent to which reactants are converted into products. A larger \(K_P\) value implies a reaction leaning towards the products at equilibrium, while a smaller value suggests the reactants are favored.

In gas-phase reactions, as in the dissociation of oxygen, \(K_P\) refers to the partial pressure of the gases involved. Changes in temperature and pressure can influence \(K_P\), as captured by the van't Hoff equation. This equation links the change in \(K_P\) with temperature to the enthalpy change of the reaction, providing insights into the thermodynamic properties and how the reaction will respond to temperature variations.

Understanding how \(K_P\) relates to the conditions of a reaction is central to controlling and predicting the outcome of industrial chemical processes and synthesizing desired chemical products.

Dissociation Process

Dissociation processes are fundamental in chemical reactions, especially in gas-phase chemistry. They involve the separation of molecules into smaller components, such as atoms or ions. The process of dissociation can be endothermic or exothermic, and it plays a significant role in determining the physical and chemical behavior of substances at various temperatures and pressures.

An example of a dissociation process is the breaking down of diatomic oxygen \(\mathrm{O}_2\) into two oxygen atoms \(\mathrm{O}\) at high temperatures. This reaction can be represented as \(\mathrm{O}_2 \rightleftharpoons 2 \mathrm{O}\). The equilibrium constant for this dissociation reflects the extent to which \(\mathrm{O}_2\) has dissociated into oxygen atoms under given conditions.

Accurate knowledge of the dissociation process and related thermodynamic properties, such as the enthalpy change and equilibrium constant, allows scientists and engineers to predict the behavior of substances under different temperatures and pressures, which is crucial for a wide range of applications, including combustion engines and environmental modeling. The van't Hoff equation is a critical tool in quantifying the effect of these parameters on the dissociation equilibrium.