Determine ΔGrxn of the reaction at 570.0 K.

To determine \(\Delta G_{\text{rxn}}\) at 570.0 K, we'll use the equilibrium constant \(K_p\) we previously calculated.

We already determined that at 570.0 K:

\[ \ln K_p = -2.890 \]

Given the relationship between the Gibbs free energy change and the equilibrium constant:

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

Where:
- \(R = 8.314 \, \text{J/mol} \cdot \text{K}\) (the universal gas constant)
- \(T = 570.0 \, \text{K}\)

Let's calculate \(\Delta G^\circ\):

\[ \Delta G^\circ = - (8.314 \, \text{J/mol} \cdot \text{K}) \times (570.0 \, \text{K}) \times (-2.890) \]

Now, performing the calculation:

\[ \Delta G^\circ = -8.314 \times 570.0 \times (-2.890) \]

First, calculate the product of the constants and the temperature:

\[ 8.314 \times 570.0 = 4740.98 \]

Then multiply by the \(\ln K_p\):

\[ 4740.98 \times (-2.890) = -13710.43 \]

Since we have two negative signs, they cancel out:

\[ \Delta G^\circ = 13710.43 \, \text{J/mol} \]

Converting this to \(\text{kJ/mol}\):

\[ \Delta G^\circ = 13.710 \, \text{kJ/mol} \]

Therefore, the Gibbs free energy change \(\Delta G_{\text{rxn}}\) of the reaction at 570.0 K is approximately \(13.710 \, \text{kJ/mol}\).

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

1. How does \(\Delta G^\circ\) relate to the spontaneity of a reaction?
2. Can you explain why the sign of \(\Delta G^\circ\) is important?
3. How do changes in temperature affect \(\Delta G^\circ\)?
4. What other thermodynamic quantities can be derived from the equilibrium constant?
5. Why is it important to convert units correctly in thermodynamic calculations?

**Tip:** Always double-check your calculations, especially when converting units, to ensure accuracy.

Learn how to calculate \( \Delta G_{\text{rxn}} \) of a reaction at 570.0 K using thermodynamic principles. This problem involves using the equilibrium constant \( K_p \) and the relationship between Gibbs free energy and temperature. Suitable for advanced undergraduate students studying thermodynamics.

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