For the following reaction:
PCl5(g)<===> PCl3(g) + Cl2(g) Kc=.058
If the initial concentration of PCl5(g) is .160M, calculate the equilibrium concentrations of all the components. (use quadratic)
You have the equation. Set up an ICE chart, substitute into Kc expression and solve.
To calculate the equilibrium concentrations of all components, we will use the quadratic equation. Here are the steps:
Step 1: Write the balanced equation:
PCl5(g) ⇌ PCl3(g) + Cl2(g)
Step 2: Define the initial concentration and the change in concentration:
Let x be the change in concentration for PCl5.
The initial concentration of PCl5 is 0.160 M.
The change in concentration for PCl3 will be x.
The change in concentration for Cl2 will also be x.
Step 3: Write the expression for the equilibrium concentrations:
PCl5: 0.160 - x
PCl3: x
Cl2: x
Step 4: Write the expression for the equilibrium constant:
Kc = [PCl3] * [Cl2] / [PCl5]
Kc = (x * x) / (0.160 - x)
Step 5: Substitute the Kc value and solve the quadratic equation:
0.058 = (x * x) / (0.160 - x)
To solve this quadratic equation, rearrange it to:
0.058 * (0.160 - x) = x * x
Expand the equation:
0.00928 - 0.058x = x²
Rearrange it to a standard quadratic form:
x² + 0.058x - 0.00928 = 0
Now, use the quadratic formula to solve for x:
x = (-0.058 ± √(0.058² - 4 * 1 * (-0.00928))) / (2 * 1)
Simplifying further:
x = (-0.058 ± √(0.003364 + 0.03712)) / 2
x = (-0.058 ± √0.040484) / 2
x = (-0.058 ± 0.2012) / 2
Next, calculate the positive root:
x = (-0.058 + 0.2012) / 2
x = 0.1432 / 2
x = 0.0716
Step 6: Calculate the equilibrium concentrations using the derived value of x:
PCl5: 0.160 - x = 0.160 - 0.0716 = 0.0884 M
PCl3: x = 0.0716 M
Cl2: x = 0.0716 M
Therefore, at equilibrium, the concentrations of PCl5, PCl3, and Cl2 are 0.0884 M, 0.0716 M, and 0.0716 M, respectively.