Question 17.9: Making a Simplifying Assumption to Calculate Equilibrium Con......

Plan We know from the balanced equation that when x mol of COCl_2 decomposes, x mol of CO and x mol of Cl_2 form. We use the volume (10.0 L) to convert amount (5.00 mol or 0.100 mol) to molar concentration, define x and set up the reaction table, and substitute the values into Q_c. Before using the quadratic formula, we assume that x is negligibly small. After solving for x, we check the assumption and find the equilibrium concentrations. If the assumption is not justified, we use the quadratic formula to find x.

Solution (a) For 5.00 mol of COCl_2. Writing the reaction quotient:

Calculating the initial reactant concentration, [COCl_2]_{\text{init}}:

Setting up the reaction table (Table 1), with x equal to [COCl_2]_{\text{reacting}}:

If we use the equilibrium values in Q_c with the given K_c, we obtain

              Q_c  =  \frac{[CO][Cl_2]}{[COCl_2]}  =  \frac{x^2}{0.500  −  x}  =  K_c  =  8.3×10^{−4}
Because K_c is small, the reaction does not proceed very far to the right, so let’s assume that x ([COCl_2]_{\text{reacting}}) can be neglected. In other words, we assume that the equilibrium concentration is nearly the same as the initial concentration, 0.500 M:

              [COCl_2]_{\text{init}}  −  [COCl_2]_{\text{reacting}}  =  [COCl_2]_{\text{eq}}
              0.500  M  −  x  ≈  0.500  M

Using this assumption, we substitute and solve for x:

             K_c  =  8.3×10^{−4}  ≈  \frac{x^2}{0.500}
             x^2  ≈  (8.3×10^{−4})(0.500) \text{      so      }x  ≈  2.0×10^{−2}

Checking the assumption by seeing if the error is <5%:

Solving for the equilibrium concentrations:
            [CO]  =  [Cl_2]  =  x  =  2.0×10^{−2}  M
            [COCl_2]  =  0.500  M  −  x  =  0.480  M
(b) For 0.100 mol of COCl_2. The calculation in this case is the same as the calculation in part (a), except that [COCl_2]_{\text{init}} = 0.100 mol/10.0 L = 0.0100 M. Thus, at equilibrium,

Making the assumption that 0.0100 M − x ≈ 0.0100 M and solving for x:

Checking the assumption:

Rearranging Q_c and its value at equilibrium into the form of a quadratic equation gives
           8.3×10^{−4}  =  \frac{x^2}{0.0100  −  x}\text{     or     }x^2  =  (8.3×10^{−6})  −  (8.3×10^{−4})x
and so

Solving this equation with the quadratic formula shows that
          x  =  \frac{−8.3×10^{−4}  ±  \sqrt{(8.3×10^{−4})^2  −  4(1)  (−8.3×10^{−6})}}{2(1)}
The only meaningful value of x is 2.5×10^{−3}.
Solving for the equilibrium concentrations:

          [CO]  =  [Cl_2]  =  x  =  2.5×10^{−3}  M
          [COCl_2]  =  1.00×10^{−2}  M  −  x  =  7.5×10^{−3}  M
Check Once again, use the calculated values to be sure you obtain the given K_c.

Comment The main point is that the simplifying assumption was justified at the high [COCl_2]_{\text{init}} in part (a) but not at the low [COCl_2]_{\text{init}} in part (b). The amount of [COCl_2]_{\text{reacting}} is small in (b), but since [COCl_2]_{\text{init}} is also small, [COCl_2]_{\text{reacting}} is a high percentage of [COCl_2]_{\text{init}} and cannot be neglected without introducing significant error.