Consider the following reaction, A <===> B in a 1.00L container. At 1.77°C, the molar concentrations of A and B, respectively, 5.838M and 3.945M. Suddenly, 8.744 moles of A are added to the system. What is the change in DG of the system. Give the answer in kJ?
To find the change in Gibbs free energy (ΔG) of the system, you need to use the equation:
ΔG = ΔH - TΔS
Where:
ΔG = Change in Gibbs free energy
ΔH = Change in enthalpy
T = Temperature in Kelvin
ΔS = Change in entropy
First, we need to calculate the initial Gibbs free energy (G) of the system before the addition of A.
G = nRT ln(Q)
Where:
G = Gibbs free energy
n = Number of moles
R = Gas constant (8.314 J/(mol·K))
T = Temperature in Kelvin
Q = Reaction quotient
Since the reaction is A <===> B, the reaction quotient (Q) can be calculated using the initial molar concentrations of A and B:
Q = [B]^b /[A]^a
Where:
[A] = Molar concentration of A
[B] = Molar concentration of B
a = Stoichiometric coefficient of A in the balanced equation
b = Stoichiometric coefficient of B in the balanced equation
Given that the molar concentrations of A and B are 5.838 M and 3.945 M, respectively, and the stoichiometric coefficients of A and B are both 1 in this reaction, we have:
Q = (3.945 M)^1 / (5.838 M)^1
Q = 0.6760
Next, convert the temperature from Celsius to Kelvin:
T(K) = T(C) + 273.15
T(K) = 1.77°C + 273.15
T(K) = 274.92K
Now, calculate the initial Gibbs free energy:
G = nRT ln(Q)
G = (1.00 mol)(8.314 J/(mol·K))(274.92 K) ln(0.6760)
G = -369.8 J
Now, let's calculate the final Gibbs free energy after adding 8.744 moles of A to the system.
n = n_initial + n_added
n = 1.00 mol + 8.744 mol
n = 9.744 mol
Using the same equation, we can find the final Gibbs free energy:
G = nRT ln(Q)
G = (9.744 mol)(8.314 J/(mol·K))(274.92 K) ln(0.6760)
G = -3213.7 J
Finally, calculate the change in Gibbs free energy (ΔG) by subtracting the initial Gibbs free energy (G_initial) from the final Gibbs free energy (G_final):
ΔG = G_final - G_initial
ΔG = -3213.7 J - (-369.8 J)
ΔG = -2843.9 J
To convert ΔG from joules to kilojoules:
ΔG (kJ) = ΔG (J) / 1000
ΔG (kJ) = -2843.9 J / 1000
ΔG (kJ) = -2.844 kJ
Therefore, the change in ΔG of the system is approximately -2.844 kJ.