A reaction has a standard free-energy change of –12.40 kJ mol–1 (–2.964 kcal mol–1). Calculate the equilibrium constant for the reaction at 25 °C.
dG = -RTlnK
To calculate the equilibrium constant (K) for a reaction, we can use the relationship between the standard free-energy change (ΔG°) and the equilibrium constant.
The equation is:
ΔG° = -RTln(K)
Where:
ΔG° = standard free-energy change (-12.40 kJ mol–1)
R = gas constant = 8.314 J K–1 mol–1
T = temperature in Kelvin (25°C = 298.15 K)
K = equilibrium constant (what we want to calculate)
First, we need to convert ΔG° from kJ mol–1 to J mol–1:
ΔG° = -12.40 kJ mol–1 * 1000 J kJ–1 = -12,400 J mol–1
Now, we can plug the values into the equation and solve for K:
-12,400 J mol–1 = -8.314 J K–1 mol–1 * 298.15 K * ln(K)
Divide both sides of the equation by (-8.314 J K–1 mol–1 * 298.15 K) and rearrange to solve for ln(K):
ln(K) = -12,400 J mol–1 / (-8.314 J K–1 mol–1 * 298.15 K)
ln(K) ≈ 15.24
Now, we can calculate the value of K by taking the exponential of both sides:
K = e^(ln(K))
K ≈ e^(15.24)
K ≈ 2.84 x 10^6
So, the equilibrium constant for the reaction at 25 °C is approximately 2.84 x 10^6.
To calculate the equilibrium constant (K) for a reaction using the standard free-energy change (∆Gº), you can use the following equation:
∆Gº = -RT * ln(K)
Where:
- ∆Gº is the standard free-energy change in joules/mol
- R is the gas constant (8.314 J/(mol*K))
- T is the temperature in Kelvin (K)
- ln is the natural logarithm
Given that ∆Gº is -12.40 kJ/mol, we need to convert it to J/mol by multiplying it by 1000:
∆Gº = -12.40 kJ/mol * 1000 J/kJ = -12400 J/mol
Now, we can substitute the values into the equation and solve for K:
-12400 J/mol = -8.314 J/(mol*K) * T * ln(K)
To solve for K, we need to rearrange the equation:
ln(K) = -12400 J/mol / (-8.314 J/(mol*K) * T)
Now, substitute the value of T (25 °C = 298 K) and calculate:
ln(K) = -12400 J/mol / (-8.314 J/(mol*K) * 298 K)
ln(K) = 47.54
To find K, we need to take the exponential of both sides of the equation:
K = e^(ln(K)) = e^(47.54)
Calculating this gives us the equilibrium constant (K) for the reaction at 25 °C.