Question

The \(\mathrm{p} K_{\mathrm{a}}\) of a weak acid is 5 . What is the \(\mathrm{pH}\) when the concentration of the acid form is \(0.5 \mathrm{M}\) and the concentration of the conjugate base form is \(0.05 \mathrm{M}\) ?

Short answer

The pH of the solution is 4.

Step-by-step solution

Write the Henderson-Hasselbalch Equation

To find the pH of a buffer solution, we can use the Henderson-Hasselbalch equation, which states \( \text{pH} = \mathrm{p}K_{\mathrm{a}} + \log \left( \frac{[\text{Base}]}{[\text{Acid}]} \right) \).

Substitute Known Values into the Equation

We have \( \mathrm{p}K_{\mathrm{a}} = 5 \), the concentration of base \( [\text{Base}] = 0.05 \mathrm{M} \), and the concentration of acid \( [\text{Acid}] = 0.5 \mathrm{M} \). Substitute these values into the Henderson-Hasselbalch equation:\[ \text{pH} = 5 + \log \left( \frac{0.05}{0.5} \right) \].

Calculate the Logarithmic Quotient

Calculate the logarithm of the ratio by first simplifying the fraction: \( \frac{0.05}{0.5} = 0.1 \). Now, find \( \log(0.1) \), which is \( -1 \).

Solve for the pH

Substitute the result from Step 3 back into the Henderson-Hasselbalch equation:\[ \text{pH} = 5 + (-1) = 5 - 1 = 4 \].

Confirm the Calculation

Verify each step to ensure accuracy: the initial values were correctly substituted, the fraction was simplified, and the logarithm was calculated correctly. The calculation \( 5 - 1 = 4 \) confirms the pH of the solution.

Key concepts

Buffer Solutions

Buffer solutions are fascinating chemical systems that maintain a stable pH level, even after the addition of small amounts of acid or base. They are mixtures, typically made of a weak acid and its conjugate base, or a weak base and its conjugate acid. These systems are crucial in many natural and laboratory settings where controlling the acidity or alkalinity is important.

Buffers work by neutralizing added acids or bases. Here's how they perform this balancing act: the weak acid in the buffer system can donate hydrogen ions (H⁺) to counteract added bases, while the conjugate base can absorb hydrogen ions to neutralize added acids. This dual ability makes buffer solutions incredibly effective in maintaining a consistent pH.

In practice:

• When adding an acid to a buffer, the base component of the buffer neutralizes the H⁺ ions.
• When adding a base, the acidic component donates H⁺ to neutralize added OH⁻ ions.

This adaptability is why buffers are essential for many biological processes, like maintaining blood pH within the narrow range necessary for proper physiological functions.

pH Calculation

Calculating the pH of a buffer solution is made simpler with the Henderson-Hasselbalch equation, a fundamental formula in chemistry. This equation links the pH of a buffer solution with the pKa (the acid dissociation constant) and the concentrations of the acid and its conjugate base.

The equation is expressed as:
\[\text{pH} = \mathrm{p}K_{\mathrm{a}} + \log \left( \frac{[\text{Base}]}{[\text{Acid}]} \right)\]
This equation is powerful because it allows us to quickly compute the pH without the need for complicated equilibrium calculations, assuming that the concentrations of the acid and base are much greater than the ions generated by reactions.

In our specific example:

• \( \mathrm{p}K_{\mathrm{a}} = 5 \)
• Concentration of acid, \([\text{Acid}] = 0.5 \text{ M}\)
• Concentration of base, \([\text{Base}] = 0.05 \text{ M}\)

Using these values, the equation becomes \[\text{pH} = 5 + \log \left( \frac{0.05}{0.5} \right)\]

Breaking this down further, you simplify \( \frac{0.05}{0.5} \) to 0.1, and \( \log(0.1) \) equals \(-1\). Thus, the pH calculation is straightforward: \( 5 + (-1) = 4 \). With the pH calculated, we better understand the acidity of the solution.

Weak Acids

Weak acids play a pivotal role in buffer solutions. Unlike strong acids, which dissociate completely in water, weak acids only partially dissociate. This partial dissociation is crucial because it allows weak acids to establish an equilibrium between their undissociated form and their ions.

For an acid labeled as HA, it exists in an equilibrium where:

• HA ↔ H⁺ + A⁻

This equilibrium is influenced by the acid's dissociation constant (Ka), which indicates the strength of the acid. However, when we discuss pH calculations, we often use pKa, which is the negative logarithm of Ka (\( \mathrm{p}K_{\mathrm{a}} = -\log(\mathrm{K}_{\mathrm{a}}) \)). The lower the pKa, the stronger the acid.

Weak acids are perfect for buffer systems because their partial dissociation allows for a reversible reaction, which is key to maintaining pH stability. In the exercise, our weak acid has a pKa of 5, indicative of moderate acidity, making it ideal for forming a buffer solution with its conjugate base. This balance between acid and base is essential for the buffer's effectiveness in pH regulation.