A 1.00 L flask containing the equilibrium mixture CO +Cl2 <-> COCl2 was found to contain 0.400 mol of COCl2, 0.100 mol of CO, and 0.500 mol of Cl2. If 0.300 mol of CO is added at constant temperature, what will be the new concentrations of each component at equilibrium?(need quadratic)
To determine the new concentrations of each component at equilibrium, we can use the quadratic equation. The initial concentrations of COCl2, CO, and Cl2 are given as 0.400 mol, 0.100 mol, and 0.500 mol, respectively.
Let's denote the change in concentration of COCl2, CO, and Cl2 as x. After the addition of 0.300 mol of CO, the new concentrations can be expressed as follows:
COCl2: 0.400 + x
CO: 0.100 + 0.300 = 0.400 + x
Cl2: 0.500 + x
The balanced equation for the reaction is CO + Cl2 <-> COCl2. According to the stoichiometry, the change in concentration of COCl2 is equal to the change in concentration of CO and Cl2.
Therefore, we can write the equation for the equilibrium expression as:
Kc = [COCl2] / ([CO] * [Cl2])
The expression for Kc can be calculated using the given equilibrium concentrations:
Kc = (0.400 + x) / ((0.400 + x) * (0.500 + x))
At equilibrium, the value of Kc remains constant. Thus, we can set up the equation:
Kc = [COCl2] / ([CO] * [Cl2]) = (0.400 + x) / ((0.400 + x) * (0.500 + x))
Now, we can substitute the given values into the equation and solve for x:
Kc = (0.400 + x) / ((0.400 + x) * (0.500 + x))
Kc * (0.400 + x) * (0.500 + x) = 0.400 + x
(0.400 + x)^2 = Kc * (0.400 + x)
Next, we can expand and rearrange the equation:
0.160 + 0.800x + x^2 = 0.400Kc + Kc * x
x^2 + (0.800 - Kc) * x + (0.160 - 0.400Kc) = 0
We now have a quadratic equation. We can substitute the value of Kc and solve for x using the quadratic formula:
x = (-B ± sqrt(B^2 - 4AC)) / 2A
Where A = 1, B = (0.800 - Kc), and C = (0.160 - 0.400Kc).
After solving for x, we can substitute the value of x into the expressions for the new concentrations:
COCl2: 0.400 + x
CO: 0.100 + 0.300 = 0.400 + x
Cl2: 0.500 + x
These will give us the new concentrations of each component at equilibrium.
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.
Kc = (COCl2)/(CO)(Cl2)
Substitute the equilibrium concns in the problem and solve for Kc. Then to part 2 of the problem.
...........CO + Cl2 ==> COCl2
initial...0.4...0.5.....0.4
change.....-x....-x......x
equil...0.4-x...0.50-x....x
Substitute from the ICE chart and solve for x, then evaluate CO, Cl2, COCl2
Post your work if you get stuck.