- Weak acids and weak bases react only a small amount with water.
- ICE tables are a useful tool when determining the pH of a weak acid or weak base solution.
- In an ICE table, use x to represent the small unknown amount of weak acid or weak base that reacts with water.
- Solve for the value of x using an equilibrium equation.
- If x is less than 5% of the starting concentration of weak acid (or weak base), then it is valid to make the assumption that the initial and equilibrium concentrations of the weak acid (or weak base) are approximately equal to each other.
Weak acids and weak bases only react a small amount with water before reaching equilibrium. Thus, unlike with strong acids/bases in the previous section, we cannot simply assume that all of the acid (or base) added to water reacts to form H3O+ (or HO−). The same ICE table approach introduced in the previous section, however, will continue to allow us to determine the pH of solutions containing weak acids or weak bases. We know that some (but not all) of a weak acid (or weak base) will react in water, so to solve for [H3O+] we need to know how much of the acid (or base) reacts with water so that we can put this value in the change row of the ICE table. Since the exact amount that reacts depends on the particular weak acid or base, we can use the variable “x” to represent the amount that reacts in our ICE table, then solve for x using the Ka (or Kb) of the acid (or base). The Ka (or Kb) is the equilibrium constant and relates to how far the reaction between a weak acid (or base) and water proceeds.
Let’s try an example: What is the pH if 0.79 moles of CH3COOH () are added to 1.3 L of water?
First, it is helpful to calculate the concentration of CH3COOH:
Notice that this is the exact same concentration that we used for our HCl (strong acid) example in the previous section. The key difference in our approach here is that in the change row of the ICE table we will use the variable “x” to indicate that a small unknown amount of the weak acid reacts with water, since it is a bad assumption to treat all of the weak acid as reacting. We can use the Ka given in the question to solve for the value of x. We'll use the same steps when constructing the ICE table as we did for the strong acid example on the previous page:
- Write out the reaction between CH3COOH and water:
- Draw an ICE table and add initial amounts:
- This is a weak acid, so a small unknown amount, x, reacts with water:
- Add the initial and change amounts in each column to get the equilibrium concentrations:
- From the ICE table, the equilibrium concentration of H3O+ is x. Use the Ka equation to solve for x, then use x to solve for the pH.
The Ka equation for the reaction between CH3COOH and water isRelating this equation to the equilibrium concentration of each species in the ICE table gives
Note that we can use values from our ICE table directly in the Ka equation because we completed our ICE table using concentration values. If you complete your ICE table using moles instead of concentration, always convert to concentration before plugging any values into an equilibrium equation. The next step is to solve the equation
for x, since this will give us the equilibrium concentration of H3O+ that we can use to calculate the pH. It is possible to solve this equation precisely using the quadratic formula. However, since x will be small compared to the starting concentration for most weak acids (and bases), it’s often valid to simplify the solution by making the assumption that the starting and equilibrium concentrations of the weak acid (or base) are essentially the same (e.g. in this case
). So long as less than 5% of the original amount of weak acid (or base) reacts, this assumption is valid (e.g. the assumption is valid if
in this case). For this example, we’ll make the simplifying approximation that
and thus
At this point we can check whether the assumption that x is small was good: if less than 5% of the original amount of weak acid reacted, then the assumption that x is very small was good. Here,
That is, x is only 0.88% of the starting amount of acid, and thus the assumption that x is small was good. Had this test failed (i.e. if x was >5% of the original value), then we would need to go back and solve the question using the quadratic formula instead of the simplified version. Now that we know x, which is the equilibrium value of [H3O+], we can calculate pH by
Notice that adding the same amount of strong acid to water (the example in the previous section) gave a much more acidic solution with pH 0.16.
Several examples are given below which involve some of the variations on this question type that you may encounter. Try applying the approach above to each of the examples below yourself before expanding the see the detailed solutions.Example 1: What is the pH of a 1.29 M solution of weak acid NH4+ at 298 K? The Kb for NH3 at this temperature is 1.79 x 10−5.View solution:
NH4+ is a weak acid, so when constructing the ICE table a small unknown amount, x, will react in the change row:
From the above ICE table, the equilibrium concentration of H3O+ is x, which gives the pH via pH = -log(x). We can use the equilibrium equation for this reaction to solve for x:
Notice that the equilibrium constant for this reaction, Ka, was not given in the question. However, the value of Ka for NH4+ can easily be calculated from the Kb of its conjugate base, NH3, which was given in the question:
The equation that we need to solve to find x, then, is
Let's assume that x is sufficiently small such that it is valid to assume that
and thus this equation simplifies to
and thus
This value of x is much less than 5% of the starting amount of acid, 1.29 M, and thus the assumption that x is small was valid. Finally, this concentration of H3O+ gives the pH via
Example 2: What is the pH of a 1.29 M solution of NH3 (Kb=1.79 x 10−5) at 298 K?
View solution:
Based on the small Kb value (<1), NH3 is a weak base. Thus, when constructing the ICE table a small unknown amount, x, will react in the change row:
To calculate the pH, we could determine pOH using
then convert to pH using
. Based on the ICE table above, the equilibrium concentration of HO− is x. To solve for x, we can use the equilibrium equation for the above reaction:
Let's assume that x is sufficiently small such that it is valid to assume that
and thus
This value of x is much less than 5% of the starting amount of weak base, 1.29 M, and thus the assumption that x is small was valid. This concentration of HO− gives the pOH via
The pH of this solution is thus
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