- Identify the thermodynamic relationship between standard Gibbs free energy and the equilibrium constant: \(\Delta G^{\circ} = -RT \ln K\).
- Convert \(\Delta G^{\circ}\) to Joules (\(-34500 \text{ J}\)) and substitute the values into the rearranged formula \(K = e^{-\frac{\Delta G^{\circ}}{RT}}\).
- Calculate the exponent: \(\frac{34500}{8.314 \times 298} \approx 13.9249\).
- Compute \(K \approx e^{13.9249} \approx 1.12 \times 10^6\), which matches the first option.
- Final Answer: \(\boxed{1.12 \times 10^6}\).
Explanation
- Understanding the Thermodynamic Relationship
To determine the equilibrium constant \(K\) for the reaction \(Cl(g) + O_3(g) \rightarrow ClO(g) + O_2(g)\) at \(298 \text{ K}\), we use the thermodynamic relationship between the standard Gibbs free energy change \(\Delta G^{\circ}\) and the equilibrium constant \(K\):
\[\Delta G^{\circ} = -RT \ln K\]
where:
- \(\Delta G^{\circ} = -34.5 \text{ kJ} = -34500 \text{ J}\)
- \(R = 8.314 \text{ J} \cdot \text{mol}^{-1} \cdot \text{K}^{-1}\) (the ideal gas constant)
- \(T = 298 \text{ K}\) (the absolute temperature)
- Calculating the Equilibrium Constant
We rearrange the equation to solve for \(K\):
\[\ln K = -\frac{\Delta G^{\circ}}{RT}\]
\[K = e^{-\frac{\Delta G^{\circ}}{RT}}\]
Substituting the given values into the expression:
\[K = e^{-\frac{-34500}{8.314 \times 298}}\]
\[K = e^{\frac{34500}{8.314 \times 298}}\]
Performing the calculation:
\[K = e^{\frac{34500}{2477.572}} \approx e^{13.9249}\]
Calculating the exponential:
\[K \approx 1,115,622.69\]
Rounding this result to three significant figures, we get \(1.12 \times 10^6\).
- Conclusion and Final Answer
Comparing our calculated value of approximately \(1.12 \times 10^6\) with the provided options:
- \(1.12 \times 10^6\)
- \(5.66 \times 10^5\)
- \(8.96 \times 10^{-7}\)
- \(4.98 \times 10^{-4}\)
- \(0.986\)
The value matches the first option. Thus, the equilibrium constant is \(\boxed{1.12 \times 10^6}\).
Examples
In atmospheric chemistry, understanding the equilibrium constant of reactions like the one between chlorine atoms and ozone is crucial for predicting the rate of ozone depletion. A high value for \(K\) indicates that the reaction strongly favors the formation of products (chlorine monoxide and molecular oxygen), which helps scientists model the impact of industrial pollutants on the ozone layer. This fundamental thermodynamic calculation allows researchers to assess the stability and persistence of chemical species in our atmosphere.