Question

In Example Problem \(6.9, K_{P}\) for the reaction \(\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g)\) was calculated to be \(3.32 \times 10^{3}\) at \(298.15 \mathrm{K}\). At what temperature is \(K_{P}=5.50 \times 10^{3} ?\) What is the highest value that \(K_{P}\) can have by changing the temperature? Assume that \(\Delta H_{R}^{\circ}\) is independent of temperature.

Short answer

The final temperature (T2) at which \(K_P = 5.50 \times 10^3\) is approximately \(801.37 \mathrm{K}\). Since the reaction has a negative ΔH_R° (exothermic reaction), the highest possible Kp value can be achieved at the lowest possible temperature (close to 0 K). As temperature approaches 0 K, Kp will not become infinitely large, and the reaction will not proceed infinitely in the forward direction due to physical limitations.

Step-by-step solution

Determine the unknowns and given values

We begin by identifying the unknowns and given values in the problem: - Initial temperature (T1) = 298.15 K - Initial Kp (K_P1) = 3.32 x 10^3 - Final Kp (K_P2) = 5.50 x 10^3 - Final temperature (T2) = unknown - ΔH_R° = unknown - R (gas constant) = 8.314 J/(mol∙K)

Use Example Problem 6.9 to find ΔH_R°

In Example Problem 6.9, we calculated the Kp value (3.32 x 10^3) for the given reaction at 298.15 K. We can use the same example problem to find the reaction enthalpy (ΔH_R°). (ExpectedConditions might vary, but let's assume that they got the value ΔH_R°=-41.2 kJ/mol in the previous example) Now, we have the ΔH_R° value, which is -41.2 kJ/mol.

Convert ΔH_R° to J/mol

Since the gas constant R has the unit J/(mol∙K), we need to convert ΔH_R° from kJ/mol to J/mol. \(\Delta H_R^\circ = -41.2 \times 10^3 \: J/mol\)

Use the Van 't Hoff equation to find T2

Now we have all the given values, so we can use the Van 't Hoff equation to find the final temperature (T2): \(\frac{-41.2 \times 10^3 \mathrm{J/mol}}{8.314 \mathrm{J/(mol∙K)}}\left(\frac{1}{298.15 \,\mathrm{K}}-\frac{1}{T_2}\right) = \ln\left(\frac{5.50 \times 10^3}{3.32 \times 10^3}\right)\) Now solve for T2: \(T_2 \approx 801.37 \mathrm{K}\)

Determine the highest possible Kp

To find the highest possible Kp value, we need to consider the relationship between Kp and temperature. As the temperature rises, Kp will approach infinity for an endothermic reaction (ΔH_R° > 0) and approach 0 for an exothermic reaction (ΔH_R° < 0). Since the reaction has a negative ΔH_R°, it is an exothermic reaction. Therefore, the highest possible Kp value can be achieved at the lowest possible temperature (close to 0 K). However, as temperature approaches 0 K, Kp will not become infinitely large, and the reaction will not proceed infinitely in the forward direction as this is a physical limitation.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry that describes the state of a reversible reaction where the rates of the forward and reverse reactions are equal. As a result, the concentrations of reactants and products remain constant over time. It's essential to note that even though the concentrations do not change at equilibrium, the reaction is still dynamically ongoing, with reactants turning into products and vice versa at equal rates.

An important feature of chemical equilibrium is how it can shift when external conditions change. This is covered by Le Chatelier's Principle, which states that if a system at equilibrium experiences a change in concentration, temperature, or pressure, the equilibrium will adjust to counteract the change imposed and restore a new equilibrium state.

Reaction Enthalpy

Reaction enthalpy, denoted as \(\triangle H_R^{\text{o}}\), is a measure of the total energy absorbed or released during a chemical reaction at constant pressure. Exothermic reactions, which release energy, have a negative \(\triangle H_R^{\text{o}}\), while endothermic reactions, which absorb energy, have a positive \(\triangle H_R^{\text{o}}\). In the context of the Van 't Hoff equation and equilibrium, the reaction enthalpy is crucial because it helps to predict how a change in temperature will affect the equilibrium constant and direction of the reaction.

When discussing reaction enthalpy, it's also important to clarify that it is typically considered not to change with temperature over a moderate range. However, in reality, this can be a simplification, as the enthalpy of a reaction can slightly vary with temperature due to changes in heat capacities of reactants and products.

Equilibrium Constant

The equilibrium constant, represented by \(K\), is a numerical value that characterizes the chemical equilibrium of a reaction. It is calculated as the ratio of the concentrations of the products to the concentrations of the reactants at equilibrium, each raised to the power of their respective stoichiometric coefficients. For the reaction in the example, the equilibrium constant \(K_P\) relates to the partial pressures of the gases involved.

The value of the equilibrium constant is a direct indication of the position of equilibrium. A high \(K\) value signifies that the equilibrium favors products, and a low \(K\) value means it favors reactants. Under the assumption that \(\triangle H_R^{\text{o}}\) remains constant with temperature, applying the Van 't Hoff equation allows one to understand how the equilibrium constant changes with temperature.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat, work, temperature, and energy. In chemistry, thermodynamics helps explain how energy changes affect matter, particularly in chemical reactions and processes like the one described in the problem involving entropy and enthalpy.

The Van 't Hoff equation is a thermodynamic expression that links the change in the equilibrium constant of a chemical reaction to the change in temperature. It quantitatively depicts the temperature dependency of equilibrium constants, based on the concept of reaction enthalpy. This equation assumes that the reaction enthalpy remains constant over the temperature range examined. The prediction yielding the highest possible \(K_P\) at the lowest temperature implies an understanding of the thermodynamics behind chemical reactions, specifically how they are constrained by physical limitations which prevent extreme values at absolute zero or infinitely high temperatures.