Question

Propionic acid, ${\mathrm{CH}}_3{\mathrm{CH}}_2\mathrm{COOH},$ is $0.42\mathrm{\%}$ ionized in $0.80\mathrm{M}$ solution. The $K_{\mathrm{a}}$ for this acid is (a) $1.42\times {10}^{-5}$ (b) $1.42\times {10}^{-7};$ (c) $1.77\times {10}^{-5};$ (d) $6.15\times {10}^4$ (e) none of these.

Short answer

(e) None of these

Step-by-step solution

Understanding Ionization percentage and Ka

First, it is important to know that $[H^{+}]$, $C_i$ (initial concentration of the acid), and $K_a$ are all related. Since the acid is weakly ionized, the mathematical relationship is approximately $K_a=[H^{+}]/C_i$ due to the equilibrium of the ionization.

Calculating the concentration of H+

Because we know the ionization percentage (0.42%) and the initial concentration ($C_i$=0.80M), the concentration of $H^{+}$ in the solution can be found as $C_i$ multiplied by the percent ionization in decimal form. This gives: $[H+]=0.80\ast 0.0042=0.00336M$. This is the concentration of $H^{+}$ which ionized from the acid.

Calculating Ka

Now use the definition of $K_a$, $K_a=[H+]/C_i$, to find the value of Ka. This gives: $K_a=0.00336/0.80=4.20x{10}^{-}5$.

Matching the Ka value

Comparing the calculated Ka to the options given, we see that none of the provided options perfectly match the calculated result. Therefore, the right answer is 'none of these'.

Key concepts

Ionization Percentage

The concept of ionization percentage helps us understand how much of an acid dissociates in a solution. It is defined as the ratio of the concentration of ionized acid to the initial concentration of the acid, expressed as a percentage. For example, in the given problem with propionic acid, it is indicated that 0.42% of the acid is ionized in a 0.80 M solution.

To convert the ionization percentage into a usable number for calculations, you turn it into its decimal form. In this case, 0.42% equals 0.0042 in decimal. This small percentage tells us that propionic acid is a weak acid, as most of it remains undissociated in solution.

Understanding ionization percentage is crucial, especially when you're dealing with weak acids. It gives insight into the solution's acidity and the extent of dissociation, helping us predict how the acid behaves in different chemical settings.

Ka Calculation

The acid dissociation constant, denoted as $K_a$, measures the strength of an acid in solution. It provides us with a quantitative assessment of how completely an acid ionizes. In simpler terms, a higher $K_a$ value indicates a stronger acid, which means more dissociation.

For weak acids, which do not fully ionize, $K_a$ can be calculated using the equation: $$K_a=\frac{[H^{+}]}{C_i}$$ Where $[H^{+}]$ is the concentration of hydrogen ions and $C_i$ the initial concentration of the acid.

In our example, given that 0.42% ionizes, the $[H^{+}]$ ion is calculated as $[H^{+}]=0.80\times 0.0042=0.00336$ M. Now, substituting back into the formula, we find $K_a$: $$K_a=\frac{0.00336}{0.80}=4.20\times {10}^{-5}$$ This value helps chemists understand how the acid dissociates and its relative strength compared to other acids. The matched choice in the problem was not found, demonstrating how exact the calculation of $K_a$ should be for accuracy.

Weak Acid Equilibrium

In chemistry, equilibrium refers to a state where the concentrations of reactants and products remain constant over time, indicating a balance in a reversible reaction. For weak acids, like propionic acid, this concept is essential as they do not completely dissociate in water.

Weak acids establish an equilibrium between the undissociated acid and the ions it forms. The ionization of a weak acid is represented by the general equation: $$HA\leftrightharpoons H^{+}+A^{-}$$ Here, $HA$ stands for the weak acid, which dissociates into hydrogen ions $H^{+}$ and its conjugate base $A^{-}$.

When calculating $K_a$, you examine this equilibrium state to determine how much the acid ionizes. This equilibrium is dynamic, meaning the reaction continues to shift back and forth lightly instead of a static change.

Understanding this equilibrium allows chemists to predict how an acid will respond to changes in concentration, pH, or temperature, which is vital in various chemical applications ranging from biological systems to industrial processes.