To solve this problem, we can approach it in two parts as requested:

Given Data:

• Reaction: $\frac{1}{2} \text{H}_2(g) + \frac{1}{2} \text{I}_2(s) \rightarrow \text{HI}(g)$
• Temperature, $T = 25^\circ \text{C}$ or 298 K
• Total pressure, $P = 1 \, \text{atm}$
• Mole fractions at equilibrium: $\chi_{\text{HI}} = 0.39$, $\chi_{\text{H}_2} = 0.61$

Since $\text{I}_2(s)$ is a solid, it does not appear in the equilibrium expression.

a) Calculation of the Equilibrium Constant $K_p$

1. Write the Expression for $K_p$:

   The equilibrium constant $K_p$ for this reaction is: $K_p = \frac{P_{\text{HI}}}{\left(P_{\text{H}_2}\right)^{\frac{1}{2}}}$

2. Calculate Partial Pressures Using Mole Fractions:

   Since the total pressure is 1 atm, we can directly use mole fractions as partial pressures:

   • $P_{\text{HI}} = 0.39 \times 1 = 0.39 \, \text{atm}$
   • $P_{\text{H}_2} = 0.61 \times 1 = 0.61 \, \text{atm}$

3. Plug Values into $K_p$: $K_p = \frac{0.39}{\left(0.61\right)^{\frac{1}{2}}}$

   Calculating this:

   $K_p \approx \frac{0.39}{0.781} \approx 0.499$

   So, $K_p \approx 0.499$.

b) Calculation of Standard Gibbs Energy of Formation of HI, $\Delta G^\circ_{\text{f}}(\text{HI})$

The relationship between $\Delta G^\circ_{\text{f}}$ and the equilibrium constant is: $\Delta G^\circ_{\text{f}} = -RT \ln K_p$ where:

• $R = 8.314 \, \text{J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}$
• $T = 298 \, \text{K}$

Using the $K_p$ value we found:

$\Delta G^\circ_{\text{f}} = - (8.314) \times (298) \times \ln(0.499)$

Calculating $\ln(0.499) \approx -0.695$:

$\Delta G^\circ_{\text{f}} = - (8.314) \times (298) \times (-0.695)$

$\Delta G^\circ_{\text{f}} \approx 1724 \, \text{J/mol} \text{ or } 1.724 \, \text{kJ/mol}$

Summary of Results:

• a) Equilibrium Constant $K_p \approx 0.499$
• b) Standard Gibbs Energy of Formation $\Delta G^\circ_{\text{f}}(\text{HI}) \approx 1.724 \, \text{kJ/mol}$

Would you like further details on any part of the solution, or have any questions?

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1. What are the units and significance of $K_p$ in this reaction?
2. How is Gibbs free energy related to spontaneity in reactions?
3. Why is $I_2(s)$ not included in the equilibrium expression?
4. How would changing the temperature affect $K_p$ and $\Delta G^\circ_{\text{f}}$?
5. What assumptions are made in calculating equilibrium constants for gas reactions?

Tip: When calculating equilibrium constants, always check phase states; solids and pure liquids do not appear in the expressions.