Question

An acid HX is \(25 \%\) dissociated in water. If the equilibrium concentration of HX is 0.30 \(M,\) calculate the \(K_{\mathrm{a}}\) value for HX.

Short answer

The short version of the answer is: The \(K_a\) value for the acid HX is 0.025.

Step-by-step solution

Calculate the concentration of the dissociated ions at equilibrium

Since the acid HX is 25% dissociated in water, we can calculate the concentration of the H+ and X- ions produced at equilibrium. Given that the equilibrium concentration of HX is 0.30 M, the concentration of the dissociated ions would be: Dissociation percentage of HX = 25 % Equilibrium concentration of HX = 0.30 M \[Percentage \: of \: dissociation = \frac{Dissociated \: ions}{Total \: ions}\] \[0.25 = \frac {[H^+]}{0.30}\] \[ [H^+] = 0.25 \times 0.30 = 0.075 M \] Since the dissociation of HX is in a 1:1 ratio (HX <--> H+ + X-), the concentration of X- is also 0.075 M.

Calculate the \(K_a\) value for HX

Now that we have the concentrations of the dissociated ions, we can use the \(K_a\) formula to find the \(K_a\) value for HX: \[K_{a} = \frac{[H^+][X^-]}{[HX]}\] Substituting the concentrations we calculated in step 1: \[K_{a} = \frac{(0.075)(0.075)}{(0.30 - 0.075)}\] Simplify the expression: \[K_{a} = \frac{0.005625}{0.225}\] Finally, calculate the \(K_a\) value: \[K_{a} = 0.025\] So, the \(K_{a}\) value for HX is 0.025.

Key concepts

Ka value calculation

When studying acids and their behavior in water, understanding the concept of the acid dissociation constant, noted as Ka, is critical. The Ka value is a measure of how well an acid can release protons (\textsc{H+}) when in solution, which greatly affects its behavior and strength. To calculate this value, we examine the chemical reaction at equilibrium where the acid (HA) dissociates into its ions (\textsc{H+} and A-).

The general expression is: \[\ K_{a} = \frac{{[H^+][A^-]}}{{[HA]}} \]
It's important to know the concentrations of the dissociated ions and the undissociated acid. In the exercise, since the acid HX is 25% dissociated at 0.30 M, the concentration of the produced ions, \textsc{H+} and X-, is 0.075 M each. Plugging these numbers into the Ka expression, we get: \[K_{a} = \frac{{(0.075)(0.075)}}{{(0.30-0.075)}}\]
After simplifying, we find the Ka value to be 0.025. This calculation is akin to solving a puzzle where the concentration of products and reactants at equilibrium give us insight into the acid's propensity to dissociate.

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction and its reverse reaction proceed at the same rate, leading to the system's concentrations of reactants and products remaining unchanged over time. It's crucial to note that reaching equilibrium does not mean the reactants and products are equal in concentration, but that their ratios remain constant.

When an acid dissociates in water, we reach a point where the rate of its dissociation to form protons and anions equals the rate at which these ions recombine to form the undissociated acid. At this point, we say the reaction has reached equilibrium. For the acid HX in our exercise, the equilibrium was established with 25% dissociation, resulting in specific concentrations of HX, H+, and X- that remain constant in the absence of external changes.

Equilibrium is vital in Ka value calculation as it provides the snapshot of concentrations needed to evaluate an acid's strength. Without understanding equilibrium, we cannot accurately determine Ka, making it a cornerstone concept in acid-base chemistry.

Acid Strength

The strength of an acid is determined by its ability to donate protons to a base. Strong acids almost completely dissociate into ions in water, implying they have higher Ka values and thus a greater tendency to lose their proton. Conversely, weak acids only partially dissociate and have lower Ka values.

In our exercise, the acid HX has a Ka value of 0.025, which indicates it is a weak acid due to its relatively low degree of dissociation at equilibrium (25%). The dissociation percentage tells us directly about the acid's strength: the higher the percentage, the stronger the acid. Knowing an acid's strength helps predict how it will react in different chemical reactions, which is essential for applications ranging from industrial processes to biological systems.

It is important to note that Ka values are affected by temperature, and this intrinsic property is regularly used by chemists to anticipate the behavior of acid reactions under various conditions. By threading the concepts of Ka, equilibrium, and acid strength together, we gain a robust understanding of acid behavior that is crucial for both theoretical studies and practical applications.