Question

Calculate the \(K_{\mathrm{a}}\) of the following acids using the given information. a. 0.00330 \(\mathrm{M}\) solution of benzoic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right), \mathrm{pOH}=10.70\) b. 0.100 \(\mathrm{M}\) solution of cyanic acid \((\mathrm{HCNO}), \mathrm{pOH}=11.00\) c. 0.150 \(\mathrm{M}\) solution of butanoic acid \(\left(\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{COOH}\right), \mathrm{pOH}=11.18\)

Short answer

The Ka values for the given acids are as follows: a. For benzoic acid: \(Ka = 4.21 \times 10^{-5}\) b. For cyanic acid: \(Ka = 1.01 \times 10^{-4}\) c. For butanoic acid: \(Ka = 1.56 \times 10^{-5}\)

Step-by-step solution

Write the dissociation reaction

Write the balanced chemical equation for the dissociation of the given acid in water. a. For benzoic acid (C6H5COOH): \[ C_{6}H_{5}COOH \rightleftharpoons C_{6}H_{5}COO^{-} + H^{+} \] b. For cyanic acid (HCNO): \[ HCNO \rightleftharpoons H^{+} + CNO^{-} \] c. For butanoic acid (C3H7COOH): \[ C_{3}H_{7}COOH \rightleftharpoons C_{3}H_{7}COO^{-} + H^{+} \]

Convert pOH to pH

Use the given pOH values to calculate the pH for each solution by using the relationship \(pH + pOH = 14\). a. For \(\text{pOH} = 10.70\): \[ \text{pH} = 14 - pOH = 14 - 10.70 = 3.30 \] b. For \(\text{pOH} = 11.00\): \[ \text{pH} = 14 - pOH = 14 - 11.00 = 3.00 \] c. For \(\text{pOH} = 11.18\): \[ \text{pH} = 14 - pOH = 14 - 11.18 = 2.82 \]

Find the concentration of H+ ions

Use the calculated pH values to find the concentration of H+ ions, [H+], by using the relationship \(pH = -\log[H^{+}]\). a. For \(\text{pH} = 3.30\): \[ [H^{+}] = 10^{-3.30} = 5.01 \times 10^{-4} \, M \] b. For \(\text{pH} = 3.00\): \[ [H^{+}] = 10^{-3.00} = 1.00 \times 10^{-3} \, M \] c. For \(\text{pH} = 2.82\): \[ [H^{+}] = 10^{-2.82} = 1.51 \times 10^{-3} \, M \]

Calculate the Ka value

Calculate the Ka value using the equilibrium expression for each acid and the given initial concentration of the acid. The equilibrium expression for an acid is in the form: \[ \text{Ka} = \frac{[H^{+}][A^{-}]}{[HA]} \] Since the dissociated amount of acid is equal to [H+], we can simplify the expression for Ka as: \[ \text{Ka} = \frac{[H^{+}]^{2}}{[HA]_{0} - [H^{+}]} \] a. For benzoic acid (\([C_{6}H_{5}COOH]_0 = 0.00330\, M\)): \[ \text{Ka} = \frac{(5.01 \times 10^{-4})^2}{0.00330 - 5.01 \times 10^{-4}} = 4.21 \times 10^{-5} \] b. For cyanic acid (\([HCNO]_0 = 0.100 \, M\)): \[ \text{Ka} = \frac{(1.00 \times 10^{-3})^2}{0.100 - 1.00 \times 10^{-3}} = 1.01 \times 10^{-4} \] c. For butanoic acid (\([C_{3}H_{7}COOH]_0 = 0.150 \, M\)): \[ \text{Ka} = \frac{(1.51 \times 10^{-3})^2}{0.150 - 1.51 \times 10^{-3}} = 1.56 \times 10^{-5} \] That is, the Ka values for benzoic acid, cyanic acid, and butanoic acid are approximately \(4.21 \times 10^{-5}\), \(1.01 \times 10^{-4}\), and \(1.56 \times 10^{-5}\), respectively.

Key concepts

Benzoic Acid

Benzoic acid is a simple aromatic carboxylic acid with the formula \(C_{6}H_{5}COOH\). In the context of acid dissociation constant (\(K_a\)) calculations, it's crucial to understand that benzoic acid partially dissociates in water to form benzoate (\(C_{6}H_{5}COO^{-}\)) and hydrogen ions (\(H^{+}\)). The value of \(K_a\) gives us an idea of how readily benzoic acid donates its proton to water, effectively measuring the strength of the acid. A low \(K_a\) value indicates a weak acid, which benzoic acid is considered to be. The calculation of its \(K_a\) involves the equilibrium concentrations of the acid and its dissociated ions.

Cyanic Acid

Cyanic acid (\(HCNO\)) is another interesting example when studying acid strengths and dissociation constants. Despite its simple structure, it has a relatively strong \(K_a\) compared to other weak acids. Its dissociation in water produces hydrogen ions (\(H^{+}\)) and cyanate ions (\(CNO^{-}\)). When \(K_a\) for cyanic acid is determined, it reflects how much the acid dissociates, which in turn affects the pH of the solution. The procedure for finding the \(K_a\) is similar to that of other acids, following the basic principles of chemical equilibrium and the manipulation of concentration ratios.

Butanoic Acid

Butanoic acid, with the molecular formula \(C_{3}H_{7}COOH\), is a member of the fatty acids and is known for its distinctive rancid smell. It's a carboxylic acid that dissociates in water to yield \(C_{3}H_{7}COO^{-}\) and \(H^{+}\) ions, similar to the other acids we have been discussing. The degree to which butanoic acid dissociates is quantified by its \(K_a\) value. This value is critical because it provides important information regarding the acid's behavior in various chemical processes, including those in biological systems.

pH Calculation

The pH of a solution is a logarithmic measure of the hydrogen ion concentration. Calculating the pH is fundamental in chemistry because it helps in determining the acidity or basicity of a solution. To calculate the pH, you must know the concentration of hydrogen ions in the solution. The relationship \(pH = -\log[H^{+}]\) provides a direct method for calculating pH from the hydrogen ion concentration. This calculation is a central part of determining the \(K_a\) value for an acid, as it allows us to infer the concentration of hydrogen ions in equilibrium.

pOH to pH Conversion

The pOH is another logarithmic measure that represents the hydroxide ion concentration in a solution. The pH and pOH are related to each other through the equation \(pH + pOH = 14\) at 25°C, which reflects the relationship between hydrogen and hydroxide ions in water. This intrinsic connection allows chemists to easily convert between pOH and pH values, facilitating calculations involving the acidity or basicity of a solution. One can quickly derive the pH from a given pOH, assisting in the broader process of acid-loss constant (\(K_a\)) determination.

Chemical Equilibrium

Chemical equilibrium occurs when the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time. At equilibrium, the ratio of the concentrations of products to reactants remains constant, known as the equilibrium constant (\(K\)). For acid dissociation, this is specifically termed the acid dissociation constant (\(K_a\)). Understanding equilibrium principles is essential when calculating \(K_a\), as it involves predicting the extent of acid dissociation and thus, the strength of an acid in solution. It is the underpinning concept for many calculations in chemistry, especially for those involving acid-base reactions.