A 0.411 mol sample of PCl5(g) is injected into an empty 4.65 L reaction vessel held at 250 °C. Calculate the concentrations of PCl5(g) and PCl3(g) at equilibrium.kc=1.80 at 250c

0.411mols/4.65L = 0.0884M

........PCl5 ==> PCl3 + Cl2
I......0.0884.....0......0
C.......-x........x......x
E.....0.0884-x....x......x

Substitute the E line into K expression and solve for x and 0.0884-x

To calculate the concentrations of PCl5(g) and PCl3(g) at equilibrium, we need to use the equilibrium constant expression, Kc, and the stoichiometry of the balanced chemical equation.

The balanced chemical equation for the reaction between PCl5 and PCl3 is:
PCl5(g) ⇌ PCl3(g) + Cl2(g)

Given:
- Initial moles of PCl5 = 0.411 mol
- Volume of the reaction vessel = 4.65 L
- Temperature = 250 °C
- Kc = 1.80 at 250 °C

Step 1: Convert temperature to Kelvin
To use the ideal gas law and equilibrium constant expression, we need to convert the temperature to Kelvin.

250 °C + 273.15 = 523.15 K

Step 2: Calculate the initial concentration of PCl5
The initial concentration of PCl5 can be calculated using the ideal gas law:
PV = nRT

Where:
P = Pressure (since the vessel is empty, we assume it is 1 atm)
V = Volume = 4.65 L
n = moles of PCl5 = 0.411 mol
R = Gas constant = 0.0821 L·atm/(mol·K)
T = Temperature = 523.15 K

By rearranging the ideal gas law formula, we can solve for concentration, which is moles/volume:
Concentration (mole/L) = n/V = P/(RT)

Concentration of PCl5 = (0.411 mol)/(4.65 L) = 0.0884 M

Step 3: Calculate the equilibrium concentrations of PCl5 and PCl3
Let's assume, at equilibrium, the concentration of PCl5 is (0.0884 - x) M and the concentrations of PCl3 and Cl2 are both x M.

Using the stoichiometry of the balanced chemical equation, we know that the ratio of PCl3 and Cl2 to PCl5 is 1:1.

Therefore, [PCl3(g)] = x M and [Cl2(g)] = x M.

Step 4: Calculate the equilibrium constant expression
The equilibrium constant expression (Kc) for the given reaction is:
Kc = [PCl3(g)] * [Cl2(g)] / [PCl5(g)]

Substituting the equilibrium concentrations into the equilibrium constant expression:
1.80 = (x * x) / (0.0884 - x)

Step 5: Solve for x
To solve the equilibrium constant expression, we rearrange the equation:
1.80 * (0.0884 - x) = x^2
0.15912 - 1.80x = x^2

Rearranging the equation further:
x^2 + 1.80x - 0.15912 = 0

Using the quadratic formula to solve for x, we find that the positive solution is x = 0.187 M (rounded to three decimal places).

Step 6: Calculate the equilibrium concentrations
Finally, we can calculate the equilibrium concentrations of PCl5 and PCl3.

[PCl5(g)] = 0.0884 - x = 0.0884 - 0.187 = -0.0986 M (since negative concentration is not possible, it means PCl5 is completely consumed)

[PCl3(g)] = x = 0.187 M

Therefore, at equilibrium, the concentration of PCl5 is 0 M (completely consumed) and the concentration of PCl3 is 0.187 M.