Question

What are the major species present in the following mixtures of bases? a. \(0.050 \mathrm{M} \mathrm{NaOH}\) and \(0.050 \mathrm{M} \mathrm{LiOH}\) b. \(0.0010 \mathrm{M} \mathrm{Ca}(\mathrm{OH})_{2}\) and \(0.020 \mathrm{M} \mathrm{RbOH}\) What is \(\left[\mathrm{OH}^{-}\right]\) and the \(\mathrm{pH}\) of each of these solutions?

Short answer

For mixture a, the major species present are Na⁺, OH⁻, Li⁺, and OH⁻. The total concentration of OH⁻ ions is 0.100 M, and the pH of the solution is 13. For mixture b, the major species present are Ca²⁺, 2OH⁻, Rb⁺, and OH⁻. The total concentration of OH⁻ ions is 0.022 M, and the pH of the solution is approximately 12.34.

Step-by-step solution

Identify the species present

Both sodium hydroxide (NaOH) and lithium hydroxide (LiOH) are strong bases that completely dissociate in water. Thus, the major species present in this solution are Na⁺, OH⁻, Li⁺, and OH⁻ ions.

Calculate total [OH⁻]

Since both NaOH and LiOH contribute OH⁻ ions, we should add their concentrations to get the total concentration of hydroxide ions in the solution: \[ [OH^-] = [OH^-]_{NaOH} + [OH^-]_{LiOH} = 0.050\,\mathrm{M} + 0.050\,\mathrm{M} = 0.100\,\mathrm{M} \]

Calculate the pH

Now that we have the concentration of hydroxide ions, we can calculate the pOH and then the pH of the solution. Remember that pH + pOH = 14: \[ \mathrm{pOH} = -\log_{10} [OH^-] \] \[ \mathrm{pOH} = -\log_{10} (0.100\,\mathrm{M}) = 1 \] \[ \mathrm{pH} = 14 - \mathrm{pOH} = 14 - 1 = 13 \] So, the pH of this solution is 13. ##Mixture b: 0.0010 M Ca(OH)₂ and 0.020 M RbOH##

Identify the species present

Calcium hydroxide (Ca(OH)₂) is a moderately strong base and rubidium hydroxide (RbOH) is considered a strong base. They both dissociate in water. Ca(OH)₂ will release 2 moles of OH⁻ ions per mole, while RbOH will release 1 mole of OH⁻ ions per mole. So, the major species present in this solution are Ca²⁺, 2OH⁻, Rb⁺, and OH⁻ ions.

Calculate total [OH⁻]

We need to account for the amount of hydroxide ions contributed by both bases. For Ca(OH)₂, each mole contributes 2 moles of OH⁻, so: \[ [OH^-]_{Ca(OH)_2} = 2 \times 0.0010\,\mathrm{M} = 0.0020\,\mathrm{M} \] For RbOH, each mole contributes 1 mole of OH⁻: \[ [OH^-]_{RbOH} = 0.020\,\mathrm{M} \] Adding both contributions, we get the total concentration of hydroxide ions: \[ [OH^-] = [OH^-]_{Ca(OH)_2} + [OH^-]_{RbOH} = 0.0020\,\mathrm{M} + 0.020\,\mathrm{M} = 0.022\,\mathrm{M} \]

Calculate the pH

With the total concentration of hydroxide ions, we can calculate the pOH and subsequently the pH of the solution: \[ \mathrm{pOH} = -\log_{10} [OH^-] \] \[ \mathrm{pOH} = -\log_{10} (0.022\,\mathrm{M}) \approx 1.66 \] \[ \mathrm{pH} = 14 - \mathrm{pOH} = 14 - 1.66 = 12.34 \] So, the pH of this solution is approximately 12.34.

Key concepts

Strong and Weak Bases

Understanding the difference between strong and weak bases is crucial for pH calculation. A strong base is a substance that completely dissociates into its ions in an aqueous solution. Examples of strong bases include sodium hydroxide (NaOH) and potassium hydroxide (KOH). These bases break apart into their respective cations and hydroxide (OH⁻) ions readily. In contrast, a weak base does not fully dissociate in solution. Ammonia (NH₃), for instance, is a classic example of a weak base; it partially accepts protons to form ammonium ions (NH₄⁺) but does not release hydroxide ions as extensively.

For students, visualizing this process can be helped by considering strong bases as 'all in' components which break apart completely, while weak bases are like hesitant participants, only partially engaging in the ionizing process. In solutions with mixed bases, knowing which bases are strong and which are weak is essential to properly calculate pH.

Base Dissociation

Base dissociation refers to the process by which a base releases hydroxide ions (OH⁻) in solution. The degree to which a base dissociates is described by its base dissociation constant (Kb). Strong bases have a characteristic feature: they dissociate completely in water, meaning their Kb is high, usually off the scale for practical purposes. Conversely, weak bases have a smaller Kb value, reflecting their partial dissociation.

By conceptualizing the dissociation process like 'getting tickets to a show', where strong bases always 'sell out' completely while weak bases only 'sell' some of their 'tickets', students might find it easier to interpret the concept. It's important to remember that the extent of base dissociation directly affects the hydroxide ion concentration in a solution, and hence, its pH.

Hydroxide Ion Concentration

The hydroxide ion concentration (concentration of OH⁻ ions in a solution is a direct measure of its basicity. For strong bases in a mixture, such as the NaOH and LiOH combination in our exercise, the concentration is simply the sum of the concentrations from each base, as they fully dissociate. However, when dealing with weak bases, the calculation becomes more complex due to their incomplete dissociation.

To help students visualize, picture hydroxide ions like people entering a party. Strong bases send in a large, definite number, quickly filling the room, while weak bases add guests gradually. The total number of guests (OH⁻ ions) determines the overall 'vibe' (basicity) of the party (solution).

pOH and pH Relationship

The pOH of a solution is the negative logarithm of the hydroxide ion concentration, and it serves as a scale to quantify the basicity. The pH scale, conversely, measures the acidity of a solution. They are inversely related and summed up they always equal 14 at 25°C, as defined by the equation pH + pOH = 14. This is a result of the autoionization property of water, where one water molecule donates a proton to another.

As an educational image, one could liken pH and pOH to two ends of a teeter-totter. One goes up as the other goes down, but their combined 'heights' will always equal the same value. Simplifying the arithmetic, if pOH is a measure of 'party loudness' (basic strength), pH is then effectively how 'quiet' the room is in acidic terms, and the sum of these 'noise levels' is a constant.