© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Write the Balanced Chemical Equation
The reaction is:
\text{PCl}_5(g) \rightleftharpoons \text{PCl}_3(g) + \text{Cl}_2(g)
We are given:
K_c = 1.844
Initial moles of PCl5 = 3.0 moles (in a 1 L container).
Step 2: Define the Change in Moles
Let the amount of PCl5 that dissociates at equilibrium be x . Then:
• Moles of PCl5 remaining at equilibrium = 3 - x .
• Moles of PCl3 formed at equilibrium = x .
• Moles of Cl2 formed at equilibrium = x .
Step 3: Express the Equilibrium Constant
Since K_c for the reaction is given by:
K_c = \frac{[\text{PCl}_3][\text{Cl}_2]}{[\text{PCl}_5]}
and in terms of moles (because volume is 1 L, concentration = moles / 1 L), this becomes:
K_c = \frac{x \cdot x}{3 - x} = \frac{x^2}{3 - x} = 1.844.
Step 4: Form the Quadratic Equation
Rearrange the expression:
x^2 = 1.844 \times (3 - x).
So,
x^2 + 1.844x - 5.532 = 0.
Step 5: Solve the Quadratic Equation
The quadratic equation is:
x^2 + 1.844x - 5.532 = 0.
Use the quadratic formula:
x = \frac{-b \pm \sqrt{b^2 + 4ac}}{2a},
where a = 1 , b = 1.844 , and c = -5.532.
Substitute the values:
x = \frac{-1.844 \pm \sqrt{(1.844)^2 + 4 \times 1 \times 5.532}}{2}.
Simplify under the square root:
(1.844)^2 = 3.399,\quad 4 \times 5.532 = 22.128,\quad
\sqrt{3.399 + 22.128} = \sqrt{25.527} \approx 5.052.
Hence,
x = \frac{-1.844 + 5.052}{2} \approx 1.604.
(We take the positive solution to represent the physically meaningful concentration of products forming.)
Step 6: Compute the Equilibrium Moles of PCl5
Moles of PCl5 at equilibrium = 3 - x = 3 - 1.604 = 1.396.
Step 7: Express the Final Answer
The question requests the moles of PCl5 in the form of " \_\_\_\_ \times 10^{-3} " and then to round off to the nearest integer.
1.396 moles can be written as 1396 × 10−3 moles.
When rounding 1396 to the nearest integer, we get 1396.
Thus, the final number of moles of PCl5 is:
\boxed{1396 \times 10^{-3}.}
