In this example, we calculate the equilibrium concentrations from the equilibrium constant and initial concentrations,
solving for x using the quadratic formula.
Our example reads, "A reaction vessel initially contains 0.30 molar bromine gas, 0.40 molar chlorine gas, and 0.40 molar BrCl gas.
Calculate the concentration of BrCl at equilibrium."
We can begin by creating a RICE table with the reaction at the top and the initial concentrations in the second row.
To determine the change in concentration, we need to calculate the reaction quotient
to determine which direction the reaction will shift to reach equilibrium.
When calculating the reaction quotient, Q, we use the equilibrium constant expression but we substitute in the current concentrations.
Completing the calculation, our reaction quotient is 1.3.
Since the reaction quotient is smaller than our equilibrium constant value, we need more product: we need to shift right to reach equilibrium.
This means we will have -x, -x, and +2x for our change row.
The coefficients 1, 1, and 2 come from the coefficients in the balanced chemical equation.
The equilibrium row is the initial row plus the change row.
Now we need to solve for x using that equilibrium constant expression.
The value of KC was given in the problem, and we can plug in the equilibrium row of our RICE table for the concentrations.
We can not use the perfect squares method on this problem, so we'll have to use the quadratic formula to solve for x.
To use the quadratic formula, we need to get our equation in the form 0 = AX squared + BX + C.
We can first multiply both sides of the equation by the denominator, 0.30-x times 0.40-x.
We will need to use the FOIL method, which stands for First, Outer, Inner, Last, on each side.
On the left, 6.0 times 0.30 times 0.40 equals 0.72.
Then 6.0 times 0.30 times -x equals -1.8x.
Then 6.0 times 0.40 times -x equals -2.4x.
And finally on the left, 6.0 times -x times -x equals 6x squared.
We can follow the same process for the right side and we'll get 0.16 + 0.80x + 0.80x + 4x squared.
We can move all numbers over to the left side and combine like terms.
We should get 2x squared -5.8x + 0.56 = 0.
The quadratic formula is -B plus or minus the square root of B squared minus 4AC all divided by 2A.
A is 2, B is -5.8, and C is 0.56.
Plugging these into the quadratic formula, we'll get x equals 0.10 or 2.8.
One of these two numbers will produce at least one negative concentration when plugged back into our equilibrium row of the RICE table, so it is not valid.
The valid number for x should produce all positive concentrations when substituted back into the equilibrium row.
If we plug in 0.10 for x in the equilibrium row, we'll get concentrations of 0.20, 0.30, and 0.60 molar, so this is our correct value of x.
If we plug in 2.8 for x, we'll give a negative concentration for both bromine and chlorine gas, so this is not a valid solution.
For our problem, we need the concentration of BrCl at equilibrium.
Plugging in 0.10 for x, we get 0.60 molar; this is the solution to our problem.
We can double check to make sure we solved for x correctly by substituting our value of x back into the equilibrium constant expression.
When we substitute 0.10 in for x, we should get 6.0, which is our value of KC given in the problem.
This means we have solved for x using the quadratic equation correctly.