# Solution of a Weak Base

### In-class Problem Set #1

**1. Calculate the pH of a solution that is 0.155 M in ammonia.**

As students begin to ponder this question, and as the instructor begins to circulate among the groups, some things to ask are:

*What is ammonia? Is it an acid or a base? Is it strong or weak?*

After about five minutes, everyone should have identified ammonia as a weak base and have the correct chemical formula. I write the correct chemical formula on the board and that it is a weak base. With this information, they can next be asked:

*What does ammonia react with? Can you write the correct chemical equation representing this reaction?*

Students may not recall that the solution contains water and that the water is necessary in the proper reaction. Once groups have written the correct reaction, I call timeout, write it on the board, and indicate how we can use this to describe the general reaction of a base – a base reacts with water to produce its protonated form and hydroxide ion. I then ask the following question:

*What is the K expression for this equation?*

Which invariably leads to the question:

*Should [H _{2}O] appear in this expression?*

Using a timeout, we spend about five minutes as a class discussing what [H_{2}O] is and why it shouldn’t appear in the equation. It may be worth mentioning other species that do not appear in the *K* expression such as other solvents and solids.

*What subscript do we attach to this K?*

Students usually realize that it is K_{b}. The equilibrium constant tables that we use in the course only have pK_{a} values so there is only a value for the ammonium ion. This leads to the following questions:

*What is the relationship between K _{a}, K_{b}, pK_{a}, and pK_{b}? What reaction is the K_{a} expression describing? What is the relationship between Kb, K_{w} and this reaction?*

Allow the students several minutes to discuss this and to try to figure out the relationships. At some point in the discussion the concept of a conjugate pair will come up and this should be summarized by the instructor at the board. Also, the relationship between pK and K will come up in the discussion. Some students will remember that Ka times Kb for a conjugate pair equals Kw (or at least remember that there is some connection between these three equilibrium constants even if they do not remember the exact relationship). When groups get to the correct expression, I ask them to:

*Prove that Ka times Kb for a conjugate pair equals Kw.*

Some groups immediately write the two K expressions and multiply them together. Others need prompting to do this. It is then worth summarizing at the board the proof that Ka times Kb for a conjugate pair equals Kw. Students can then be asked:

*What is the value of K _{b} for ammonia? What does the magnitude of K_{b} tell you about the strength of ammonia as a base?*

It may take the groups a few minutes arrive at a *K _{b}* value that they all agree on. They should be able to recognize that a small

*K*value implies that the base is weak and therefore not very reactive. This is a good time to talk about what the magnitude of any

_{b}*K*value tells you about the reaction. At this point, students can now perform the actual calculation.

*Instruct students to create a table of concentrations below each species in the reaction where the first row is the starting concentration and the second row is the concentration at equilibrium. Ask them just to fill in the first row.*

Groups usually correctly write the concentrations of ammonia and ammonium, but will probably have a “0” as the initial concentration of OH^{-}.

*Is the initial concentration of OH ^{-} really 0? *

Students will remember that water dissociates and it is worth spending a brief amount of time discussing the auto-protolysis of water and how the initial concentrations of both OH^{-} and H_{3}O^{+} must be 10^{-7} in order for *K _{w}* to have a value of 10

^{-14}.

*Instruct them to fill in the next row on the table. *

Students should very quickly identify that the concentration of NH_{3} will decrease by *x* while the concentration of NH_{4}^{+} and OH^{-} will increase by *x*.

*What happens if these values were substituted directly into the **K _{b} expression*?

They will probably recognize that this would result in having to solve a quadratic.

*Ask the students to think about the magnitude of x as it might compare to the initial concentration of ammonia and the initial concentration of hydroxide.? *

Most realize it will be small compared to the initial amount of ammonia and large compared to the initial amount of hydroxide. It is helpful to then briefly explain why these two conclusions are valid.

*If x is small relative to the initial concentration of ammonia and large relative to the initial concentration of hydroxide, are there any approximations that can be made?*

Groups will usually realize right away that the value of x can be ignored relative to the amount of initial ammonia and that the initial amount of hydroxide can be ignored relative to x. It is then worth summarizing these conclusions and explaining that we will use a 5% or less criteria throughout the semester to determining whether simplifying approximations are valid. I usually remind them of our overall approximation in which we are using concentration instead of activity and ask them to consider whether the use of concentration is truly warranted for an initial ammonia concentration of 0.155 M.

*What is the final concentration of OH ^{-}?*

*Are the two approximations valid?*

Some groups may need to be reminded how to calculate a percent.

*What is the relationship between OH ^{-} and pH?*

*What is the pH of the solution at equilibrium? Is it basic and does the value seem about right given the magnitude of the Kb value? What would we expect for a pH if the Kb was smaller and what would we expect if the Kb was larger?*