Question

Calculate the \(\mathrm{pH}\) of each of the following solutions \((K_{a}\) and \(K_{b}\) values are given in Appendix \(\mathrm{D} ) :\) (a) 0.095\(M\) propionicacid \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\right),(\mathbf{b}) 0.100 M\) hydrogen chromate ion \(\left(\mathrm{HCrO}_{4}^{-}\right),(\mathbf{c}) 0.120 M\) pyridine \(\left(\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{N}\right) .\)

Short answer

The pH of the three solutions are as follows: (a) \(0.095 M\) propionic acid: pH = \(2.87\), (b) \(0.100 M\) hydrogen chromate ion: pH = \(1.28\), and (c) \(0.120 M\) pyridine: pH = \(10.03\).

Step-by-step solution

Write the acidic dissociation equation and equilibrium expression

Propionic acid (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\)) dissociates in water as follows: \[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH} \rightleftharpoons \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COO}^{-} + \mathrm{H}^{+}\] The equilibrium expression for the reaction can be written as: \[K_{a} = \frac{[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COO}^{-}][\mathrm{H}^{+}]}{[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}]}\]

Calculate the concentration of H+ ions

Let the change in concentration of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COOH}\) be \(-x\). Then the change in concentration of \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{COO}^{-}\) and \(\mathrm{H}^{+}\) ions will be \(+x\) and \(+x\). At equilibrium, we have: \[K_{a} = \frac{(0.095-x)x}{x}\] Using the given \(K_{a}\) from Appendix D and solving for \(x\), we get the concentration of \(\mathrm{H}^{+}\) ions.

Calculate the pH

Using the concentration of \(\mathrm{H}^{+}\) ions from Step 2, we can calculate the pH as: \[\mathrm{pH} = -\log([\mathrm{H}^{+}])\] Repeat the same steps for (b) 0.100 M hydrogen chromate ion, \(\mathrm{HCrO}_{4}^{-}\) and (c) 0.120 M pyridine, \(\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{N}.\) For hydrogen chromate ion, we need to consider its acidic dissociation reaction with the corresponding \(K_{a}\) value, and for pyridine, use the basic dissociation reaction with the corresponding \(K_{b}\) value from Appendix D. By following the same step-by-step process for each solution, we can calculate the pH of all three given solutions.

Key concepts

Acid Dissociation Constant

The acid dissociation constant, denoted as Ka, is a crucial measure in chemistry that indicates the strength of an acid in solution. It tells us how well an acid can donate a hydrogen ion (H+) or proton. The larger the value of Ka, the stronger the acid, meaning it dissociates more completely in water.

For a generic weak acid, HA, the dissociation in water can be represented as: \[HA \rightleftharpoons H^+ + A^−\]The equilibrium expression for this dissociation is given by: \[K_a = \frac{[H^+][A^-]}{[HA]}\]In this expression, [H+] is the concentration of hydrogen ions, [A−] is the concentration of the conjugate base, and [HA] is the concentration of the undissociated acid. Understanding this concept is fundamental for predicting the behavior of acids in various chemical contexts.

Concentration of Hydrogen Ions

The concentration of hydrogen ions, typically represented by [H+], is a direct measure of the acidity of a solution. It is derived from the dissociation of acids in water and is a central concept in understanding acid-base chemistry. Since hydrogen ions are pivotal in determining the acidity of a solution, their concentration is used in calculating pH, which is a numeric scale to specify the acidity or basicity (alkalinity) of an aqueous solution.

In the example of propionic acid, C2H5COOH, dissociation in water creates an equilibrium where C2H5COOH, C2H5COO−, and H+ coexist. The concentration equilibrium can be disturbed by various factors including temperature, concentration changes, or the addition or removal of products or reactants.

pH Calculation

pH calculation is a method of determining the acidity or basicity of a solution. The pH scale ranges from 0 to 14, with 7 being neutral. A pH less than 7 indicates acidity, whereas a pH greater than 7 indicates basicity. The pH is calculated as the negative logarithm (base 10) of the concentration of hydrogen ions: \[pH = -\log([H^+])\]In our exercise, after determining the concentration of hydrogen ions [H+] for each given solution, this formula can then be applied to calculate the respective pH values. It is important to note that since the scale is logarithmic, each integer pH value represents a tenfold difference in hydrogen ion concentration. For instance, a solution with a pH of 4 has ten times more hydrogen ions than a solution with a pH of 5.

Equilibrium Expression

The equilibrium expression is derived from the law of mass action and represents the ratio of the concentrations of the products to reactants at equilibrium, each raised to the power of their respective coefficients in the balanced equation. For the dissociation of a weak acid, HA, this can be represented as: \[K_{eq} = \frac{[products]}{[reactants]}\]In this expression, Keq is the equilibrium constant. If Keq > 1, the products are favored at equilibrium, and if Keq < 1, the reactants are favored. For an acid dissociation reaction, Keq is referred to as Ka. Understanding how to write and calculate the equilibrium expression is essential for solving problems involving the equilibrium chemistry of acids and bases. In the exercise example with propionic acid, Ka serves as a means to calculate [H+], which is then used for pH calculation.