Question

The \(\mathrm{p} K_{\mathrm{a}}\) of the conjugate acid of morpholine is \(8.33\). Morpholinium ion Morpholine (a) Calculate the ratio of morpholine to morpholinium ion in aqueous solution at \(\mathrm{pH} 7.0\). b) At what \(\mathrm{pH}\) are the concentrations of morpholine and morpholinium ion equal?

Short answer

Question: Calculate the ratio of morpholine to morpholinium ion in aqueous solution at pH 7.0, and determine the pH value at which the concentrations of these two species are equal. Answer: The ratio of morpholine to morpholinium ion at pH 7.0 is approximately 0.0461. The pH value at which the concentrations of morpholine and morpholinium ion are equal is 8.33.

Step-by-step solution

(Step 1: Calculate the Ka value of the conjugate acid)

Given the pKa value of the conjugate acid of morpholine, which is 8.33, we can calculate the Ka value using the following formula: Ka = 10^(-pKa) Ka = 10^(-8.33)

(Step 2: Calculate the ratio of morpholine to morpholinium ion at pH 7)

Now that we have the Ka value, we can use the buffer equation to calculate the ratio of morpholine to morpholinium ion in aqueous solution at pH 7.0. pH = pKa + log ([morpholine] / [morpholinium]) Rearrange the equation to solve for the ratio ([morpholine] / [morpholinium]): [morpholine] / [morpholinium] = 10^(pH - pKa) [morpholine] / [morpholinium] = 10^(7.0 - 8.33)

(Step 3: Calculate the final ratio at pH 7.0)

Now substitute the pH value given and compute the ratio: [morpholine] / [morpholinium] = 10^(-1.33) ≈ 0.0461 This tells us there are 0.0461 moles of morpholine per mole of morpholinium ion in aqueous solution at pH 7.

(Step 4: Determine the pH at which the concentrations of morpholine and morpholinium ion are equal)

At the point where the concentrations of morpholine and morpholinium ion are equal, their ratio will be 1. We can use the buffer equation again to find the pH value at this point. [morpholine] / [morpholinium] = 10^(pH - pKa) 1 = 10^(pH - 8.33) Now take the log and solve for pH: log(1) = pH - 8.33 0 = pH - 8.33 pH = 8.33 Thus, at pH 8.33, the concentrations of morpholine and morpholinium ion are equal.

Key concepts

pKa and Ka Relationship

Understanding the relationship between pKa and Ka is crucial for grasping the fundamentals of acid-base equilibrium. The pKa value is a logarithmic scale measure of the acidity of a substance, while Ka, the acid dissociation constant, quantifies the strength of the acid in a solution.

Here's how they are connected: the lower the pKa value, the stronger the acid, because a stronger acid more fully dissociates in solution. This is reflected in a higher Ka value. The relationship between pKa and Ka is given by the following equation:
\[ \text{pKa} = -\text{log}(\text{Ka}) \]
or conversely:
\[ \text{Ka} = 10^{-\text{pKa}} \]

So, when you're given a pKa value, such as 8.33 for the conjugate acid of morpholine, you can calculate its Ka, revealing how readily the acid gives up its proton in solution. This is the foundation for understanding acid-base behavior in aqueous solutions.

pH Calculation

Calculating pH is a key skill in acid-base chemistry, helping you to determine the acidity or basicity of a solution. The pH scale ranges from 0 to 14, with lower numbers representing acidic conditions, higher numbers corresponding to basic conditions, and 7 being neutral.The pH of a solution can be calculated if we know the concentration of the hydrogen ions (\( [H^+] \)) using the formula:\[ \text{pH} = -\text{log}([H^+]) \]In the case of a buffer solution, the pH can also be related to its components using the Henderson-Hasselbalch equation, which involves the pKa of the acid and the ratio of the concentrations of the base (\( [A^-] \)) and the acid (\( [HA] \)):\[ \text{pH} = \text{pKa} + \text{log}\bigg(\frac{[A^-]}{[HA]}\bigg) \]This equation is particularly useful in exercises like calculating the pH of a solution when given the ratio of the conjugate base to the acid, as seen in the morpholine example.

Buffer Equation

A buffer equation is astoundingly useful for predicting the pH of buffer solutions—systems that resist changes in pH when small amounts of acid or base are added. It is often portrayed by the Henderson-Hasselbalch equation:\[ \text{pH} = \text{pKa} + \text{log}\bigg(\frac{[\text{conjugate base}]}{[\text{acid}]}\bigg) \]Buffers usually contain a weak acid and its conjugate base (or a weak base and its conjugate acid). They work by either absorbing excess hydrogen ions (\( [H^+] \)) or donating them when needed.By rearranging the buffer equation, you can also calculate the ratio of the conjugate base to acid if you know the pH and the pKa. This process can elucidate the composition of a buffer at a particular pH, like we see in the example where morpholine and morpholinium ion concentrations are balanced.

Conjugate Acid-Base Pairs

Conjugate acid-base pairs are central to acid-base chemistry. Each acid (denoted as HA) has a corresponding conjugate base (\( A^- \)), which is what's left after the acid has donated a proton. Similarly, each base has a conjugate acid, formed when the base gains a proton.These pairs are vital in understanding how buffers work. When a buffer solution of a weak acid and its conjugate base is subjected to the addition of either an acid or a base, the equilibrium shifts according to the principles of Le Chatelier, allowing the buffer to maintain a relatively stable pH.In practical terms, this concept is used to calculate things like the ratio of concentrations in a buffer system. For instance, the morpholine to morpholinium ion ratio is a pair in which morpholine functions as the conjugate base to the morpholinium ion, its conjugate acid.