To find the new concentrations of each component at equilibrium, we need to apply the principles of equilibrium and use the concept of the reaction quotient (Q) to compare with the equilibrium constant (K).
Let's start by writing the balanced equation for the reaction:
CO + Cl2 ⇌ COCl2
Next, let's define the initial concentrations of each component given in the problem:
[COCl2] = 0.400 mol
[CO] = 0.100 mol
[Cl2] = 0.500 mol
Now, we need to determine the initial value of the reaction quotient (Q) using these initial concentrations. The reaction quotient is calculated by dividing the concentration of the products by the concentration of the reactants, each raised to their respective stoichiometric coefficients.
Q = ([COCl2] / [CO] * [Cl2])
Substituting the given initial concentrations, we have:
Q = (0.400 / 0.100 * 0.500)
Q = 8
Now, let's determine the change in the concentrations of CO and COCl2 when 0.300 mol of CO is added. Since we are adding more CO, its concentration will increase by 0.300 mol. However, the concentration of COCl2 will decrease by 0.300 mol as per the stoichiometry of the reaction.
[CO] (initial) = 0.100 mol
[CO] (change) = 0.300 mol
[COCl2] (change) = -0.300 mol
Now, let's calculate the new concentrations using these changes. Let's assume the change in [CO] is x, and therefore, the change in [COCl2] will be -x.
[CO] (equilibrium) = 0.100 + x
[Cl2] (equilibrium) = 0.500
[COCl2] (equilibrium) = 0.400 - x
Since we are looking for the new concentrations at equilibrium, we need to substitute these values into the reaction quotient (Q) and set it equal to the equilibrium constant (K) for the reaction.
K = ([COCl2] / [CO] * [Cl2])
Substituting the equilibrium concentrations, we have:
K = ([0.400 - x] / [0.100 + x] * [0.500])
Now, we can solve this quadratic equation for x using the given numerical values. The equation becomes:
K = (0.400 - x) / (0.100 + x) * 0.500
Simplifying the equation, we get:
2K(0.100 + x) = 0.400 - x
0.200K + 2Kx = 0.400 - x
2Kx + x = 0.400 - 0.200K
(2K + 1)x = 0.400 - 0.200K
x = (0.400 - 0.200K) / (2K + 1)
Substitute the given value of K into the equation and solve for x.
Once you find the value of x, you can substitute it back into the equations for the equilibrium concentrations to find the new values.