Problem 50 - General Chemistry: Principles and Modern Applications

Question

Adipic acid, \(\mathrm{HOOC}\left(\mathrm{CH}_{2}\right)_{4} \mathrm{COOH},\) is among the top 50 manufactured chemicals in the United States (nearly 1 million metric tons annually). Its chief use is in the manufacture of nylon. It is a diprotic acid having \(K_{\mathrm{a}_{1}}=3.9 \times 10^{-5}\) and \(K_{\mathrm{a}_{2}}=3.9 \times 10^{-6} .\) A saturated solution of adipic acid is about \(0.10 \mathrm{M}\) \(\mathrm{HOOC}\left(\mathrm{CH}_{2}\right)_{4} \mathrm{COOH} .\) Calculate the concentration of each ionic species in this solution.

Step-by-Step Solution

Answer

The final concentrations in the solution would be the total concentration of the \(H+\) ions, total \(HOOC(CH_{2})_{4}COO^{-}\) concentration (accounting for its loss in the second ionization), and the concentration of the \((CH_{2})_{4}COO^{2-}\) ions as determined in step 4.

Step 1: Write Balanced Equations for Both Ionization Steps of Adipic Acid

Adipic acid is a diprotic acid, meaning it can lose two protons. Therefore, we need to write two separate ionization equations for the two stages. The first ionization is given by: \(HOOC(CH_{2})_{4}COOH \Rightarrow H^{+} + HOOC(CH_{2})_{4}COO^{-}\) and the second ionization by: \(HOOC(CH_{2})_{4}COO^{-} \Rightarrow H^{+} + -(CH_{2})_{4}COO^{2-}\). Each reaction shows the dissociation of adipic acid into protons and the corresponding anionic forms.

Step 2: Determine the Concentration After First Ionization

Using the formula for equilibrium constant \(K_a = [H+][A−]/[HA]\), and the approximation that \(x << 0.10 M\) where \(x\) represents the concentration change, we can find the concentrations after the first ionization. Using \(K_{a1}\) we can calculate \(x\), representing the \(H^+\) concentration, which would also be the \(HOOC(CH_{2})_{4}COO^{-}\) concentration due to the 1:1 ratio. We also find the remaining adipic acid concentration, \(HA\), as \([HA] = 0.10 M - x\). Given the small \(K_{a1}\) value, we can make the approximation that \([HA] \approx 0.10M\). By substituting \(K_{a1}\) and these concentrations into the equilibrium constant formula, we can solve for \(x\), the hydrogen ion concentration after the first ionization.

Step 3: Determine the Concentration After Second Ionization

Next, consider the ionization of the \(HOOC(CH_{2})_{4}COO^-\) ion using the second ionization equation. We use the same equilibrium constant formula but now with \(K_{a2}\). The removal of a proton from the \(HOOC(CH_{2})_{4}COO^-\) ion causes the formation of \(-(CH_{2})_{4}COO^{2-}\) ion and an additional \(H^+\). We consider the changes in concentration in the same method as in step 2 but using \(K_{a2}\). From this, we can find the new concentrations of each ion in the solution.

Step 4: Calculate Total Ion Concentrations

After determining concentrations from both ionizations, we add concentrations of like species to get total concentrations in the solution: total \(H^+\) concentration would be the sum of \(H^+\) from first and second ionization; total \((CH_{2})_{4}COO^{2-}\) concentration would equal concentration after the second ionization; total \(HOOC(CH_{2})_{4}COO^{-}\) concentration would equal its concentration after first ionization minus concentration after the second ionization as some of it further ionizes during second ionization.

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the concentrations of reactants and products remain constant over time, as the forward and reverse reactions occur at the same rate. This dynamic balance is crucial in understanding how reactions proceed and is central to the behavior of acids in solution, like adipic acid.
For a diprotic acid such as adipic acid, chemical equilibrium is reached in two stages, as it can donate two protons.
Both stages can be represented by equilibrium reactions, where the acid sequentially donates its protons to form different ions.

• In the first stage, adipic acid donates one proton, forming a monoanion and a proton ( H^+ ).
• In the second stage, the monoanion donates another proton to form a dianion and another proton.

Understanding these equilibria helps predict the concentration of ions in a saturated solution, as the system adjusts to maintain equilibrium conditions.

Ionization Constant

The ionization constant, often symbolized as K_a , is an equilibrium constant that measures the strength of an acid in solution. It indicates the degree to which an acid dissociates in water to form ions.
For diprotic acids like adipic acid, there are two ionization constants: K_{a1} and K_{a2} , each representing the ionization of one of the acid's protons.

• K_{a1} is the equilibrium constant for the first dissociation, where the acid loses its first proton. It is typically larger than K_{a2} because the first proton is often easier to remove than the second.
• K_{a2} represents the second dissociation, indicating the further ionization of the monoanion formed in the first stage.

These constants are crucial for calculating the ionic species concentrations in a solution, as they allow us to use mathematical equations to determine the extent of the reactions.

Adipic Acid

Adipic acid is a diprotic acid with the chemical formula HOOC(CH_{2})_{4}COOH . It is widely used in the manufacturing of nylon and is one of the top 50 industrial chemicals in the United States.
As a diprotic acid, it has two proton-donating groups, allowing it to undergo two separate ionization reactions in solution.
The understanding of adipic acid's behavior in water is critical in various applications, including chemical manufacturing and environmental science.

• Knowing its ionization behavior helps in predicting how it reacts in different chemical environments.
• The manufacturing processes for products like nylon can be optimized by understanding its properties in chemical reactions.

Grasping the fundamental nature of adipic acid and its applications highlights its importance in chemistry and industry.

Ionic Species Concentration

The concentration of ionic species refers to the amount of ions present in a solution. In the case of adipic acid, this involves calculating the concentrations of various anions and protons in the solution after each ionization step.

• In the first ionization, adipic acid forms H^+ ions and HOOC(CH_{2})_{4}COO^- ions.
• During the second ionization, additional H^+ ions and -(CH_{2})_{4}COO^{2-} ions are formed.

By evaluating the solution's equilibrium and using the ionization constants, we can calculate the specific ion concentrations.
These calculations help in understanding the solution's overall charge balance and its reactivity. They are significant in predicting pH, a measure of acidity, and the chemical behavior of the solution.