Question

Estimate the equilibrium constant at \(2800 \mathrm{~K}\) for \(\mathrm{CO}_{2} \rightleftarrows \mathrm{CO}+\frac{1}{2} \mathrm{O}_{2}\) using the equilibrium constant at \(2000 \mathrm{~K}\) from Table A-27, together with the van't Hoff equation and enthalpy data. Compare with the value for the equilibrium constant obtained from Table A-27.

Short answer

Use van't Hoff equation to relate \( K_{2800 \, K} \) to \( K_{2000 \, K} \). Integrate, insert values, and solve. Compare to Table A-27.

Step-by-step solution

- Write down the van't Hoff equation

The van't Hoff equation relates the equilibrium constants at two different temperatures. It can be written as: \[ \frac{d \text{ln} K}{dT} = \frac{\triangle H^\text{o}}{R T^2} \]

- Integrate the van't Hoff equation

Integrate both sides of the van't Hoff equation to relate the equilibrium constants at temperatures 2000 K (\text{K}_1) and 2800 K (\text{K}_2): \[ \text{ln} \left( \frac{K_2}{K_1} \right) = - \frac{\triangle H^\text{o}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]

- Gather necessary data

Obtain the value of the equilibrium constant at 2000 K (\text{K}_1) and the enthalpy change (\( \triangle H^\text{o} \)) from Table A-27. Locate the appropriate values and constants.

- Insert known values

Let \( T_1 = 2000 \, \text{K} \), \( T_2 = 2800 \, \text{K} \), and use the given \( \triangle H^\text{o} \) value. Assume \( K_1 \) is known from Table A-27. Insert these values to solve for \( K_2 \): \[ \text{ln} \left( \frac{K_2}{K_1} \right) = - \frac{\triangle H^\text{o}}{R} \left( \frac{1}{2800} - \frac{1}{2000} \right) \]

- Calculate \( K_2 \)

After simplifying the equation \[ \text{ln}(K_2) = \text{ln}(K_1) - \frac{\triangle H^\text{o}}{R} \left( \frac{1}{2800} - \frac{1}{2000} \right) \], exponentiate both sides to find \( K_2 \): \[ K_2 = K_1 \, \text{exp} \left( - \frac{\triangle H^\text{o}}{R} \left( \frac{1}{2800} - \frac{1}{2000} \right) \right) \]

- Compare with Table A-27

Finally, compare the calculated value of \( K_2 \) with the value listed in Table A-27 for 2800 K to examine the accuracy of the calculation.

Key concepts

van't Hoff equation

The van't Hoff equation is a powerful tool in thermodynamics. It allows us to understand how the equilibrium constant of a reaction changes with temperature. For a reaction, the equation is given by: \[ \frac{d \text{ln} K}{dT} = \frac{\Delta H^\text{o}}{R T^2} \] Here, \(K\) is the equilibrium constant, \(\Delta H^\text{o}\) is the enthalpy change, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.
Through integration, we can relate the equilibrium constants at two different temperatures. This forms the basis of our calculation for the equilibrium constants at various temperatures.

enthalpy change

Enthalpy change (\(\Delta H^\text{o}\)) is a key piece of data in using the van't Hoff equation. It represents the heat absorbed or released during a reaction at constant pressure.
In our problem, knowing the enthalpy change is crucial because it helps us determine how the equilibrium constant shifts with temperature. If the enthalpy change is negative, the reaction releases heat and is exothermic. Conversely, if positive, it absorbs heat and is endothermic. This value is typically found in tables such as Table A-27 in our exercise.

temperature dependence

The equilibrium constant of a chemical reaction is highly dependent on temperature. According to the van't Hoff equation, even small changes in temperature can significantly affect the equilibrium constant.
For example, in our exercise, we start by knowing \(K\) at 2000K and calculate \(K\) for 2800K using the van't Hoff equation. This relationship helps predict the reaction's behavior without experimentally adjusting temperature each time, saving valuable resources in a lab setting.

thermodynamics

Thermodynamics is the study of energy transformations in physical and chemical processes. Key principles include laws of energy conservation and entropy.
In equilibrium constant estimation, thermodynamics lets us connect free energies, enthalpies, and entropies with reaction extents. This is done through equations like the van't Hoff equation, integrating thermodynamic data into practical calculations.

chemical equilibrium

Chemical equilibrium occurs when a reaction's forward and reverse rates are equal, stabilizing concentrations of reactants and products.
The equilibrium constant (\(K\)) quantifies this balance, indicating the ratio of products to reactants at equilibrium. In our exercise, understanding \(K\)'s temperature dependence is vital to predicting how the reaction response changes over a range of conditions.