Question

The molecule \(D_{2}\) (where \(D\), deuterium, is \({ }^{2} \mathrm{H}\) ) undergoes a reaction with ordinary \(\mathrm{H}_{2}\) that leads to isotopic equilibrium: $$ \mathrm{D}_{2}(g)+\mathrm{H}_{2}(g) \Longrightarrow 2 \mathrm{DH}(g) \quad K_{\mathrm{p}}=1.80 \text { at } 298 \mathrm{~K} $$ If \(\Delta H_{\mathrm{rxn}}^{\circ}\) is \(0.32 \mathrm{~kJ} / \mathrm{mol} \mathrm{DH},\) calculate \(K_{\mathrm{p}}\) at \(500 . \mathrm{K}\)

Short answer

K_p at 500 K is 1.90.

Step-by-step solution

Understand the Problem

The reaction being considered is \( \text{D}_2(g) + \text{H}_2(g) \rightarrow 2 \text{DH}(g) \). We are given the equilibrium constant \( K_p \) at 298 K (1.80) and need to find \( K_p \) at 500 K. We also have the standard enthalpy change \( \text{ΔH}_{\text{rxn}}^{\text{°}} \) for the reaction, which is \( 0.32 \text{ kJ/mol} \text{ DH}\).

Use the Van't Hoff Equation

The Van't Hoff equation relates the equilibrium constants at two different temperatures: \[ \frac{K_{p2}}{K_{p1}} = \text{exp} \biggl( \frac{ΔH_{rxn}^\text{°}}{R} \biggl( \frac{1}{T_1} - \frac{1}{T_2} \biggr) \biggr) \]. Here, \( K_{p1} \) is the equilibrium constant at the initial temperature \( T_1 \) (298 K), \( K_{p2} \) is the equilibrium constant at the final temperature \( T_2 \) (500 K), \( ΔH_{rxn}^{\text{°}} \) is the standard enthalpy change (0.32 kJ/mol), and \( R \) is the gas constant (8.314 J/mol·K).

Convert ΔH_{rxn}^{\text{°}} to Joules

Convert the enthalpy change from kJ to J: \( \text{ΔH}_{\text{rxn}}^{\text{°}} = 0.32 \text{ kJ/mol} = 320 \text{ J/mol} \).

Substitute Values

Substitute the known values into the Van't Hoff equation: \[ \frac{K_{p2}}{1.80} = \text{exp} \biggl( \frac{320 \text{ J/mol}}{8.314 \text{ J/mol·K}} \biggl( \frac{1}{298 \text{ K}} - \frac{1}{500 \text{ K}} \biggr) \biggr) \]

Perform Calculations

Calculate the exponent part first: \[ \frac{320}{8.314} \biggl( \frac{1}{298} - \frac{1}{500} \biggr) = 38.48 \biggl( 0.003356 - 0.002 \biggr) = 38.48 \times 0.001356 = 0.05215 \]. Then compute the exponential term: \( e^{0.05215} = 1.0535 \). Finally solve for \( K_{p2} \): \[ 1.80 \times 1.0535 = 1.896 \]

Key concepts

Equilibrium Constant

The equilibrium constant, often denoted as either \(K_c\) or \(K_p\), represents the ratio of the concentration of products to reactants at equilibrium, each raised to the power of their respective coefficients in the balanced chemical equation. For reactions in the gas phase, \(K_p\) is frequently used where the concentrations are represented as partial pressures.
The equilibrium constant indicates the extent of a reaction at a given temperature.

• If \(K > 1\), the reaction favors the formation of products.
• If \(K < 1\), the reaction favors the formation of reactants.

In the provided exercise, we calculated \(K_p\) at different temperatures using the Van't Hoff equation.

Enthalpy Change

Enthalpy change, symbolized as \( \text{ΔH} \), measures the heat change at constant pressure. Standard enthalpy change of reaction, denoted as \( \text{ΔH}_{\text{rxn}}^{\text{°}} \), is critical in determining how heat energy is absorbed or released.

• A negative \( \text{ΔH} \text{ (exothermic)} \) indicates heat release.
• A positive \( \text{ΔH} \text{ (endothermic)} \) indicates heat absorption.

In our problem, the enthalpy change for the reaction \( \text{D}_2(g) + \text{H}_2(g) \rightarrow 2 \text{DH}(g) \) was given as \(0.32 \text{ kJ/mol} \text{ DH} \). We converted this value to Joules (320 J/mol) to use it in the Van't Hoff equation.

Temperature Dependence of Equilibrium

The Van't Hoff equation shows how equilibrium constants change with temperature, making it a powerful tool in predicting reaction behavior. The Van't Hoff equation is:

$\backslash [\; \backslash frac\{K\_\{p2\}\}\{K\_\{p1\}\}=\backslash text\{exp\}\; \backslash biggl(\; \backslash frac\{\Delta H\_\{\backslash text\{rxn\}\}^\backslash text\{°\}\}\{R\}\; \backslash biggl(\; \backslash frac\{1\}\{T\_\{1\}\}\; -\; \backslash frac\{1\}\{T\_\{2\}\}\; \backslash biggr)\; \backslash biggr)\; \backslash biggr)\; \backslash ]$
Here, \( T_1\) and \( T_2\) are the initial and final temperatures, \( K_{p1} \) and \( K_{p2} \) are the corresponding equilibrium constants, and \( R \) is the gas constant (8.314 J/mol·K).

By understanding this equation, we determined how the equilibrium constant \( K_p \) changes from 298 K to 500 K based on the given enthalpy change. We substituted the values to find that \( K_p \) at 500 K is approximately 1.896.