Respuesta :
Answer:
(a) [A] = 0.13 M, [B]= 0.23 M and [C] = 0.17 M.
(b) Option B.
Explanation:
The reaction given:
A(aq) + B(aq) ⇄ C(aq) (1)
Initial: 0.30M 0.40M 0M (2)
Equilibrium: 0.3 - x 0.4 - x x (3)
The equation of Gibbs free energy of the reaction (1) is the following:
[tex] \Delta G^{\circ} = - RTLn(K_{eq}) [/tex] (4)
where ΔG°: is the Gibbs free energy change at standard conditions, R: is the gas constant, T: is the temperature and [tex]K_{eq}[/tex]: is the equilibrium constant
(a) To calculate the concentrations of A, B, and C at equilibrium, we need first determinate the equilibrium constant using equation (4), with ΔG°=-4.51x10³J/mol, T=25 + 273 = 298 K, R=8.314 J/K.mol:
[tex] K_{eq} = e^{-\frac{\Delta G^{\circ}}{RT}} = e^{-\frac{-4.51\cdot 10^{3} J/mol}{8.314 J/K.mol \cdot 298 K}} = 6.17 [/tex] (5)
Now, we can calculate the concentrations of A, B, and C at equilibrium using the equilibrium constant calculated (5):
[tex] K_{eq} = \frac{[C]}{[A][B]} = \frac{x}{(0.3 - x)(0.4 - x)} [/tex] (6)
Solving equation (6) for x, we have two solutions x₁=0.69 and x₂=0.17, and by introducing the solution x₂ into equation (3) we can get the concentrations of A, B, and C at equilibrium:
[tex] [A] = 0.3 - x_{2} = 0.3 - 0.17 = 0.13 M [/tex]
[tex] [B] = 0.4 - x_{2} = 0.4 - 0.17 = 0.23 M [/tex]
[tex] [C] = x = 0.17 M [/tex]
Notice that the solution x₁=0.69 would have given negative values of the A and B concentrations.
(b) If the reaction had a standard free-energy change of +4.51x10³J/mol, the equilibrium constant would be:
[tex] K_{eq} = e^{-\frac{\Delta G^{\circ}}{RT}} = e^{-\frac{4.51\cdot 10^{3} J/mol}{8.314 J/K.mol \cdot 298 K}} = 0.16 [/tex]
By solving the equation (6) for x, with the equilibrium constant calculated, we can get again two solutions x₁ = 6.9 and x₂= 0.017, and by introducing the solution x₂ into equation (3) we can get the concentrations of A, B, and C at equilibrium:
[tex] [A] = 0.3 - x_{2} = 0.3 - 0.017 = 0.28 M [/tex]
[tex] [B] = 0.4 - x_{2} = 0.4 - 0.017 = 0.38 M [/tex]
[tex] [C] = x = 0.017 M [/tex]
Again, the solution x₁=6.9 would have given negative values of the A and B concentrations.
Hence, the correct answer is option B: there would be more A and B but less C.
I hope it helps you!