Stearic acid, nature's most common fatty acid, dimerizes when dissolved in hexane:
2C17H35COOH Picture (C17H35COOH)2; ΔH°rxn = -172 kJ
The equilibrium constant for this reaction at 28°C is 2900. Estimate the equilibrium constant at 38°C.
To estimate the equilibrium constant at 38°C, we can use the Van 't Hoff equation. The Van 't Hoff equation relates the change in equilibrium constant (lnK) with the change in temperature (ΔT) using the equation:
ln(K2/K1) = (ΔH°/R) * (1/T1 - 1/T2)
Where:
K1 = equilibrium constant at initial temperature (28°C)
K2 = equilibrium constant at final temperature (38°C)
ΔH° = standard enthalpy change of the reaction (-172 kJ)
R = gas constant (8.314 J/mol·K)
T1 = initial temperature in Kelvin (28°C -> 301 K)
T2 = final temperature in Kelvin (38°C -> 311 K)
Let's substitute the given values into the equation and solve for ln(K2/K1):
ln(K2/2900) = (-172000 J/mol / 8.314 J/mol·K) * (1/301 K - 1/311 K)
Simplifying the expression inside the parentheses:
ln(K2/2900) = (-172000 J/mol / 8.314 J/mol·K) * (0.0033 K^-1)
ln(K2/2900) = -20.902
Now, we can solve for K2 by taking the exponential of both sides:
K2/2900 = e^(-20.902)
K2 = 2900 * e^(-20.902)
Using a calculator, we find:
K2 ≈ 7.75 x 10^(-10)
Therefore, the estimated equilibrium constant at 38°C is approximately 7.75 x 10^(-10).
To estimate the equilibrium constant at a different temperature, we can use the Van 't Hoff equation, which relates the equilibrium constant (K) at different temperatures with the enthalpy change (ΔH°rxn) for the reaction:
ln(K2/K1) = (-ΔH°rxn/R) * (1/T2 - 1/T1)
Where:
K1 and K2 are the equilibrium constants at temperatures T1 and T2 respectively,
ΔH°rxn is the enthalpy change for the reaction,
R is the gas constant (8.314 J/(mol·K)),
T1 and T2 are the initial and final temperatures respectively.
In this case, we know that the equilibrium constant (K1) at 28°C is 2900. We need to find the equilibrium constant (K2) at 38°C.
Step 1: Convert temperatures to Kelvin
To use the equation, we need to convert the temperatures to Kelvin.
T1 (28°C) = 28 + 273.15 = 301.15 K
T2 (38°C) = 38 + 273.15 = 311.15 K
Step 2: Calculate the equilibrium constant at 38°C
Using the Van 't Hoff equation, we have:
ln(K2/2900) = (-(-172000 J)/(8.314 J/(mol·K))) * (1/311.15 K - 1/301.15 K)
Simplifying the equation:
ln(K2/2900) = (20736.74732) * (0.003332067153 )
Calculating the natural logarithm (ln) of both sides:
ln(K2/2900) = 0.069089794866
Now, we can solve for K2 by taking the exponential of both sides:
K2/2900 = e^(0.069089794866)
K2 = 2900 * e^(0.069089794866)
Using this equation, we can estimate the equilibrium constant (K2) at 38°C by substituting the values into a calculator:
K2 ≈ 2900 * e^(0.069089794866)
Calculating this expression gives us the estimated value of the equilibrium constant at 38°C.