To solve this problem, we can start by understanding the concept of coupling reactions and their equilibrium constant.
Coupling reactions involve combining two or more chemical reactions to create an overall reaction. In this case, the reaction of interest can be represented as:
O2M(s) + C(s) → M(s) + CO2(g)
To determine if this reaction is in equilibrium, we need to calculate the equilibrium constant using the Gibbs free energy change. We have the value of ΔG for the spontaneous reaction involving the conversion of O2M(s) to M(s) and O2(g), which is ΔG = 290.5 kJ/mol.
The equilibrium constant (K) can be calculated using the equation:
ΔG = -RT ln(K)
Where:
ΔG is the Gibbs free energy change
R is the gas constant (8.314 J/mol·K)
T is the temperature in Kelvin
K is the equilibrium constant
Since the given value of ΔG is in kJ/mol, we need to convert it to J/mol:
ΔG = 290.5 kJ/mol x (1000 J/1 kJ) = 290,500 J/mol
Next, we need to convert the temperature from Celsius to Kelvin:
T = 25.0 °C + 273.15 = 298.15 K
Plugging these values into the equation, we have:
290,500 J/mol = -(8.314 J/mol·K)(298.15 K) ln(K)
Now we can solve for ln(K):
ln(K) = (290,500 J/mol) / (8.314 J/mol·K)(298.15 K)
ln(K) ≈ 115.074
To find K, we can take the antilog of ln(K) using the exponential function, e:
K ≈ e^115.074
K ≈ 6.059 x 10^49
Therefore, the equilibrium constant for the coupled reaction O2M(s) + C(s) → M(s) + CO2(g) is approximately 6.059 x 10^49.