Question

If pH of \(0.001 \mathrm{M}\) potassium propionate solution be \(8.0\), then the dissociation constant of propionic acid will be (a) \(10^{-3}\) (b) \(10^{-2}\) (c) \(10^{-2.5}\) (d) \(10^{-5}\)

Short answer

The dissociation constant of propionic acid (Ka) will be \(10^{-11}\).

Step-by-step solution

Understanding the Relationship Between pH and pKa

The pH of a solution can be related to the pKa (acid dissociation constant) and the concentrations of the acid and its conjugate base via the Henderson-Hasselbalch equation: pH = pKa + log([A-]/[HA]), where [A-] is the concentration of the conjugate base, [HA] is the concentration of the acid.

Using the Given pH to Find pKa

Since the solution is of potassium propionate, the conjugate base of propionic acid, we have a basic solution. With a pH of 8.0, we can directly infer that pKa = pH - log([A-]). Given [A-] as the concentration of potassium propionate (0.001 M), we can calculate pKa.

Calculating the pKa

Substitute the known values into the Henderson-Hasselbalch equation: pKa = 8.0 - log(0.001).

Logarithm Calculation and Finding pKa

log(0.001) is equivalent to -3. Thus pKa = 8.0 - (-3) = 8.0 + 3 = 11.0.

Determine Ka from pKa

Ka is found by taking the antilog (inverse of log) of the negative pKa: Ka = 10^{-pKa}. Thus, Ka = 10^{-11}.

Key concepts

Henderson-Hasselbalch Equation

Understanding the relationship between pH and pKa is essential in the study of acid-base chemistry. The Henderson-Hasselbalch equation is a critical tool for this purpose. It is given by:
\[ \text{pH} = \text{pKa} + \log\left(\frac{[A^-]}{[HA]}\right) \]
The equation describes how the pH of a solution is affected by the acid dissociation constant (pKa) and the ratio of the concentrations of its anion ([A^-]) to the undissociated acid ([HA]). In simpler terms, it shows how the concentrations of an acid and its conjugate base determine the pH of the solution. This equation is particularly useful when dealing with buffer solutions, which resist changes in pH when small amounts of acid or base are added.
To improve your understanding, remember that the Henderson-Hasselbalch equation assumes that the concentrations of the acid and its conjugate base are known or can be determined and that the solution follows the behavior of an ideal, dilute solution. Keeping these assumptions in mind will help you apply the equation correctly in various scenarios.

pH Calculation

Calculating the pH of a solution is a fundamental aspect of chemistry that can tell us whether the solution is acidic, basic, or neutral. The pH scale ranges from 0 to 14, where a pH less than 7 indicates an acidic solution, a pH of 7 is neutral, and a pH greater than 7 signifies a basic solution. The pH is defined as the negative logarithm of the hydrogen ion concentration:
\[ \text{pH} = -\log([H^+]) \]
In practice, figuring out the pH of a basic solution involves knowing the concentration of the hydroxide ion ([OH^-]) and then finding the corresponding [H^+] concentration through the water autoionization constant (Kw), which allows one to use the above equation. When dealing with buffered solutions, like the potassium propionate example, we utilize the Henderson-Hasselbalch equation to relate pH with pKa and the concentration ratio of the conjugate base and acid.

Chemical Equilibrium

The concept of chemical equilibrium is central to understanding reactions that can occur in both the forward and reverse directions. At equilibrium, the rates of the forward and reverse reactions are equal, resulting in constant concentrations of reactants and products over time. This state can be represented with an equilibrium constant (Keq), which is the ratio of the concentration of products to reactants, each raised to the power of their stoichiometric coefficients.
For acid-base reactions, the acid dissociation constant (Ka) is a special case of the equilibrium constant, describing the extent of an acid's dissociation in water. The higher the Ka, the stronger the acid, as it implies a greater concentration of hydrogen ions. Conversely, the pKa is the negative logarithm of the Ka and helps in comparing the strengths of acids easily. A smaller pKa value indicates a stronger acid. Remember that equilibrium calculations assume a closed system where temperature remains constant, as Keq values are dependent on temperature.

Logarithm in Chemistry

The logarithm is a mathematical concept that's widely used in chemistry, particularly in the context of pH calculations and the Henderson-Hasselbalch equation. A logarithm, in simple terms, is the inverse of an exponentiation: it tells us the power to which a number, called the base, must be raised to produce another number.
In chemistry, the most common logarithm is the base 10 logarithm, denoted as 'log.' When we say pH is the negative log of the hydrogen ion concentration, we mean it tells us the power of 10 we need to get that concentration. For example, a [H^+] concentration of \( 10^{-3} \) M would have a pH of 3 because \( \log(10^{-3}) = -3 \) and \( -(-3) = 3 \). Logarithms help in transforming multiplicative relationships into additive ones, making it easier to handle numbers that span several orders of magnitude, such as hydrogen ion concentrations in a pH calculation.