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The equilibrium constant, denoted as \( K_p \), provides an insightful snapshot of a chemical reaction at equilibrium. It's a way to quantify the extent to which reactants turn into products under specific conditions.
For gases, it is represented using partial pressures rather than concentrations. Calculating \( K_p \) involves using the partial pressures of products and reactants raised to the power of their coefficients in the balanced chemical equation.
In this exercise, for the equilibrium reaction: \[ \mathrm{C}_2\mathrm{H}_6(g) \rightleftharpoons \mathrm{C}_2\mathrm{H}_4(g) + \mathrm{H}_2(g) \]we calculate \( K_p \) at 1000 K using the given partial pressures.
The equation for calculating \( K_p \) is:\[ K_p = \frac{P_{\mathrm{C}_2\mathrm{H}_4} \times P_{\mathrm{H}_2}}{P_{\mathrm{C}_2\mathrm{H}_6}} \]By substituting the given pressures, we arrive at \( K_p = 0.14083 \).
This value reflects the ratio of the products' pressure to the reactant's pressure, providing insights into the position of the equilibrium.

The Van't Hoff equation sheds light on how equilibrium constants change with temperature. It's a valuable tool for predicting how a system's equilibrium position shifts as you tweak the heat settings. The equation is mathematically expressed as:
\[\ln\left(\frac{K_{P2}}{K_{P1}}\right) = \frac{-\Delta H_R^\circ}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)\]
Where:

• \( K_{P1} \) and \( K_{P2} \) are the equilibrium constants at temperatures \( T_1 \) and \( T_2 \), respectively.
• \( \Delta H_R^\circ \) is the standard enthalpy change.
• \( R \) is the gas constant.

In this problem, knowing \( K_p \) at 1000 K, \( \Delta H_R^\circ \), and the temperatures \( T_1 \) and \( T_2 \), allows us to compute \( K_p \) at 298.15 K. The formula connects temperature shifts with enthalpy change, illustrating how exothermic or endothermic reactions respond to temperature variations. By inputting these values, we calculate \( K_{P2} = 266690.5 \), indicating a significant change in equilibrium with temperature.

Gibbs free energy change, represented as \( \Delta G^\circ \), is a measure of the spontaneity of a reaction under standard conditions. A negative \( \Delta G^\circ \) suggests that a reaction is spontaneous, or favorable, while a positive value indicates non-spontaneity.
The relationship between \( \Delta G^\circ \), the equilibrium constant \( K_p \), and temperature \( T \) is given by the equation:\[\Delta G_R^\circ = -RT \cdot \ln K_p\]
Here, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( K_p \) is the equilibrium constant. Inserting these values reflects the feasible direction of a chemical process.
From the exercise, when \( K_p \) at 298.15 K is known, \( \Delta G^\circ \) is computed to be approximately -110184 J mol鈦宦�. This indicates that, at room temperature, the reaction proceeds spontaneously towards the formation of \( \mathrm{C}_2\mathrm{H}_4 \) and \( \mathrm{H}_2 \) from \( \mathrm{C}_2\mathrm{H}_6 \).