Tough questions series – 2011 CHEM 13 NEWS Exam

(This is a reprint from the December 2011 and January 2012 issue of Chem 13 News, page 11.)

The subject of this article is question #8 from the 2011 CHEM 13 NEWS Exam. Only 16% of students answered the question correctly and, more surprisingly, 62% of students did not answer it at all. 

In a particular solution, [Br⁻] = 0.020 mol L⁻¹ and [CrO₄^2-] = 0.0030 mol L⁻¹. Finely-divided solid silver nitrate, AgNO₃, is slowly added to the solution. What is [Br⁻] when Ag₂CrO₄(s) just begins to precipitate? K_sp  for Ag₂CrO₄ is 1.9×10⁻¹² and AgBr  is 5.2×10⁻¹³.                                                                                          
A    2.1×10⁻⁸ mol L⁻¹                

B    6.0×10⁻⁸ mol L⁻¹        

C    2.7×10⁻⁷ mol L⁻¹

D   5.2×10⁻¹³ mol L⁻¹

E   6.4×10⁻⁴ mol L⁻¹

An important step in solving this problem is to decide whether AgBr or Ag₂CrO₄ precipitates first. If Ag₂CrO₄ precipitates first, then [Br⁻] is 0.020 mol L⁻¹ when Ag₂CrO₄ just begins to precipitate. This value is not among the choices, and so it is reasonable to assume that AgBr, not Ag₂CrO₄, precipitates first. When solid Ag₂CrO₄ just begins to form, we must have K_sp(Ag₂CrO₄) = [Ag⁺]²[CrO₄^2-], with [CrO₄^2-] = 0.0030 mol L⁻¹. Thus:   

What is [Br⁻] at this point? Because K_sp(AgBr) = [Ag⁺] [Br⁻], we find that [Br⁻] = K_sp(AgBr) / 2.5×10⁻⁵ = 2.1×10⁻⁸ mol L⁻¹. Thus, the answer is “A”. Notice that [Br⁻] is reduced from 0.020 mol L⁻¹ to about 2×10⁻⁸ mol L⁻¹ by the time Ag₂CrO₄ precipitates. In other words, 99.9999% of the Br⁻ will have precipitated from solution at the instant Ag₂CrO₄ begins to form.

Although we did not do it above, it is a simple matter to demonstrate that AgBr precipitates first. AgBr begins to precipitate from the solution when [Ag⁺] = K_sp(AgBr) / 0.020 = 2.6×10⁻¹¹ mol L⁻¹ and, as shown above, Ag₂CrO₄ begins to precipitate when [Ag⁺] is 2.5×10⁻⁵ mol L⁻¹. The precipitation of AgBr requires a lower concentration of Ag⁺ and thus, AgBr precipitates first. However, it may not be obvious that the controlled addition of AgNO₃ results in the precipitation of AgBr exclusively, or that the precipitation of AgBr keeps [Ag⁺] quite low, at least until [Br⁻] is also quite low. Figure 1 shows that [Ag⁺] is rather low, less than 1×10⁻⁸ mol L⁻¹, until [NO₃⁻] is almost 0.020 mol L⁻¹, the initial concentration of Br⁻. (Keep in mind that [NO₃⁻] provides a measure of how much AgNO₃ has been added to the solution, because NO₃⁻ is not involved in any reaction.) Figure 2 shows that [CrO₄²⁻] does not decrease (i.e. Ag₂CrO₄ does not precipitate) until [NO₃⁻] exceeds 0.020 mol L⁻¹. 

Figure 1: The solid curve shows how [Ag⁺] changes as AgNO₃ is added to the solution. The dashed vertical lines at 0.020 and 0.026 mol L⁻¹ are the values of [NO₃⁻] required by stoichiometry to precipitate first all of the Br⁻ and then all of the CrO₄²⁻. The rapid increase in [Ag⁺] terminates abruptly at [NO₃⁻] = 0.020 mol L⁻¹ because, at that point, Ag₂CrO₄ begins to precipitate. The dotted curve shows how [Ag⁺] would change if CrO₄²⁻ was not present.

     
Figure 2: This figure shows how [Br ⁻] and [CrO₄²⁻ ] change as AgNO₃ is added to the solution. The vertical scale is the same as in figure 1. The significance of the dashed vertical lines at 0.020 and 0.026 mol L⁻¹ is explained in the caption of Figure 1. The rapid decrease in [Br ⁻] terminates abruptly at [NO₃⁻] = 0.020 mol L⁻¹ because, at that point, Ag₂CrO₄ begins to precipitate.