Question

Consider two solutions of the salts \(\operatorname{NaX}(a q)\) and \(\operatorname{NaY}(a q)\) at equal concentrations. What would you need to know to determine which solution has the higher pH? Explain how you would decide (perhaps even provide a sample calculation).

Short answer

To determine which solution has a higher pH, we need to compare the basicity of the anions (X and Y) in each salt, which can be found using their Kb or pKb values. The solution with the higher Kb value (or lower pKb value) will have the higher pH. If needed, we can calculate the pH of each solution using the equation \(pOH = \frac{1}{2} (pKw - pKb)\) and then \(pH = pKw - pOH\). Comparing the calculated pH values will allow us to identify the solution with the higher pH.

Step-by-step solution

Identify the anions

First, we need to identify the anions in each salt. In the case of NaX, the anion is X, and for NaY, the anion is Y.

Determine the basicity of the anions

Next, we need to determine the basicity of the anions. We can look up the Kb values (base ionization constant) or pKb values (negative base-10 logarithm of the Kb value) of the anions in a table or other reliable source. The higher the Kb value (or lower the pKb value), the stronger the base.

Compare the basicity of the anions

Now, we compare the basicity of the anions X and Y. If one of the anions has a higher Kb value (or lower pKb value) than the other anion, the solution containing that anion will have a higher pH. Alternatively, we can also calculate the pH of each solution based on the Kb values, if needed.

Calculate the pH of each solution (if needed)

If a comparison of the Kb (or pKb) values is not sufficient to determine which solution has a higher pH, we can use the following equation to calculate the pH of each solution: \(pOH = \frac{1}{2} (pKw - pKb)\) Here, pOH is the negative base-10 logarithm of the hydroxide ion concentration, and pKw is the negative base-10 logarithm of the ion product of water (usually equal to 14 at 25°C). To find the pH of each solution, we can use the following equation: \(pH = pKw - pOH\) Thus, the pH of the NaX solution can be calculated as: \(pH_{NaX} = pKw - \frac{1}{2} (pKw - pKb_X)\) And the pH of the NaY solution can be calculated as: \(pH_{NaY} = pKw - \frac{1}{2} (pKw - pKb_Y)\)

Determine the solution with a higher pH

After calculating the pH of each solution, compare the values to determine which solution has a higher pH. The solution with the higher pH value will be the one with a higher concentration of hydroxide ions (OH-) and thus higher basicity. By following these steps, we can determine which solution, NaX or NaY, has a higher pH, considering their equal concentrations.

Key concepts

Base Ionization Constant

When dealing with basic solutions, understanding the Base Ionization Constant, often denoted as \(K_b\), is crucial. This constant measures the strength of a base in solution and reflects how well the base ionizes in water. A higher \(K_b\) value indicates a stronger base, which ionizes more completely, producing more hydroxide ions \((OH^-)\).

Unlike acids, bases accept protons, and the degree of ionization depends on the base's ability to attract them. Weak bases only partially ionize, leading to a lower \(K_b\), while strong bases ionize more extensively. To measure this, scientists often determine the concentrations of the base and ionized products at equilibrium. This gives an insightful indication of the base's potency.

Understanding these values allows chemists to predict the behavior of bases in various chemical reactions and solutions. In practice, comparing \(K_b\) values helps identify which of the bases contribute more hydroxide ions to a solution, and thus, which has a higher pH.

pKb values

The \(pK_b\) value is another handy tool for assessing base strength, offering an intuitive view via logarithmic scaling. It's calculated as the negative logarithm of the \(K_b\) value: \(pK_b = -\log(K_b)\). A lower \(pK_b\) indicates a stronger base, as it means a higher \(K_b\).

Let's say you have two bases with \(pK_b\) values of 4 and 6. The base with a \(pK_b\) of 4 is stronger, meaning it ionizes more effectively, providing more hydroxide ions in solution. This contributes to a higher solution pH.

In exercises and real-life applications, \(pK_b\) values are often more accessible and simpler to use than \(K_b\) values. Tables with \(pK_b\) values for various bases are frequently presented in chemistry resources, making it easy to compare the basicity of different compounds effectively.

Hydroxide Ion Concentration

Hydroxide ions \((OH^-)\) are key to determining a solution's basic nature and its pH. The stronger the base, the greater its ability to dissociate in water, resulting in a higher concentration of \(OH^-\) ions. As these ions increase in concentration, the pH of the solution also rises.

To measure this concentration directly, chemists use the equilibrium expression derived from the base ionization constant \(K_b\) and the initial concentration of the base. The reaction can be written generally as:

• \(B + H_2O \leftrightarrow BH^+ + OH^-\)
• \([OH^-] = \sqrt{K_b \times [B_{initial}]}\)

These equations show how hydroxide ions form in solution, which in turn allows pH to be calculated when combined with water's ion product constant \(K_w\).

The hydroxide ion concentration is essential for moving from the initial understanding of a base's strength through \(K_b\), to the practical calculation of pH in solution.

pH Calculation

Calculating pH from the hydroxide ion concentration involves a straightforward sequence of steps, valuable in predicting the behavior of basic solutions. The relationship \(pH + pOH = 14\) at 25°C allows us to target pH directly from \(OH^-\) concentration.

First, calculate the \(pOH\) of the solution by taking the negative logarithm of the \(OH^-\) concentration: \(pOH = -\log[OH^-]\). With \(pOH\) known, finding pH is as simple as rearranging the expression:

• \(pH = 14 - pOH\)

This process aligns with other equations like \(pH = pK_w - pOH\), using water's ion product \(pK_w = 14\) under typical conditions. Such calculations provide a clear measure of basicity, crucial for understanding which solutions are more alkaline in comparing bases. The pH scale runs from 0 to 14, with 7 being neutral. Values above 7 signify increasing basicity, so accurately determining pH is crucial for chemical analysis and reactions.