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Determine ΔS° of the reaction at 570.0 K.

To determine \(\Delta S^\circ\) for the reaction, we will use the relationship between the Gibbs free energy change (\(\Delta G^\circ\)), enthalpy change (\(\Delta H^\circ\)), and entropy change (\(\Delta S^\circ\)):

\[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \]

We can find \(\Delta G^\circ\) from the equilibrium constant \(K_p\) at a given temperature \(T\), using the equation:

\[ \Delta G^\circ = -RT \ln K_p \]

From the graph provided, we have the linear relationship:

\[ \ln K_p = -21169 \left(\frac{1}{T}\right) + 34.255 \]

At \(T = 570.0 \, \text{K}\):

1. Calculate \(\frac{1}{T}\):

\[ \frac{1}{570.0} \approx 0.001754 \, \text{K}^{-1} \]

2. Substitute this value into the linear equation to find \(\ln K_p\):

\[ \ln K_p = -21169 \times 0.001754 + 34.255 \approx -37.145 + 34.255 = -2.890 \]

3. Calculate \(K_p\):

\[ K_p = e^{-2.890} \approx 0.0555 \]

4. Calculate \(\Delta G^\circ\) at 570.0 K:

\[ \Delta G^\circ = -RT \ln K_p \]
\[ R = 8.314 \, \text{J/mol} \cdot \text{K} \]
\[ \Delta G^\circ = - (8.314 \, \text{J/mol} \cdot \text{K}) (570.0 \, \text{K}) (-2.890) \]
\[ \Delta G^\circ \approx 13723.5 \, \text{J/mol} \]
\[ \Delta G^\circ \approx 13.7235 \, \text{kJ/mol} \]

5. The \(\Delta H^\circ\) was calculated previously as:

\[ \Delta H^\circ = -21169 \, \text{J/mol} \approx -21.169 \, \text{kJ/mol} \]

6. Finally, use the Gibbs free energy equation to solve for \(\Delta S^\circ\):

\[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \]
\[ \Delta S^\circ = \frac{\Delta H^\circ - \Delta G^\circ}{T} \]
\[ \Delta S^\circ = \frac{-21.169 - 13.7235}{570.0} \]
\[ \Delta S^\circ = \frac{-34.8925}{570.0} \approx -0.0612 \, \text{kJ/mol} \cdot \text{K} \]

Therefore, the entropy change \(\Delta S^\circ\) for the reaction at 570.0 K is approximately \(-0.0612 \, \text{kJ/mol} \cdot \text{K}\).

Do you have any questions or need further details? Here are some related questions you might ask:

1. How do you determine \(\Delta H^\circ\) from a plot of \(\ln K_p\) vs. \(1/T\)?
2. What is the significance of \(\Delta G^\circ\) in a chemical reaction?
3. How does temperature affect the equilibrium constant \(K_p\)?
4. Can you explain the relationship between \(\Delta G^\circ\), \(\Delta H^\circ\), and \(\Delta S^\circ\)?
5. What are the units of the gas constant \(R\) and how are they used in these calculations?

**Tip:** When solving thermodynamic problems, always ensure units are consistent to avoid calculation errors.

Learn how to calculate ΔS° for a chemical reaction at 570.0 K using the relationship between Gibbs free energy change (ΔG°), enthalpy change (ΔH°), and entropy change (ΔS°). Detailed step-by-step solution provided.

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