Question

Penicillin \((\mathrm{MM}=356 \mathrm{~g} / \mathrm{mol})\), an antibiotic often used to treat bacterial infections, is a weak acid. Its \(K_{\mathrm{a}}\) is \(1.7 \times 10^{-3} .\) Calculate \(\left[\mathrm{H}^{+}\right]\) in solutions prepared by adding enough water to the following to make \(725 \mathrm{~mL}\). (a) \(0.187 \mathrm{~mol}\) (b) \(127 \mathrm{~g}\)

Short answer

Answer: The concentrations of \([\mathrm{H}^+]\) in the solutions are: (a) 0.0120 M, and (b) 0.0148 M.

Step-by-step solution

Calculate the initial concentration of Penicillin

We need to find the initial concentration of Penicillin out of the given molar quantity and mass. For Part (a), we have the moles of Penicillin: \(0.187 \mathrm{~mol}\) For Part (b), first, we need to convert the mass of Penicillin to moles using molar mass: Remember that the molecular mass (MM) of Penicillin is \(356\,g/mol\). \(\frac{127\,g}{356\,\frac{\mathrm{g}}{\mathrm{mol}}}=0.357\,\mathrm{mol}\) For both situations, the solution volume is given as \(725\,\mathrm{mL}\), so we'll convert it to Liters: \(725\,\mathrm{mL} = 0.725\,L\) Now, we will divide the moles by the volume to get the initial concentration of Penicillin in both cases: For Part (a): \(\frac{0.187\,\mathrm{mol}}{0.725\,L}=0.258\,M\) For Part (b): \(\frac{0.357\,\mathrm{mol}}{0.725\,L}=0.492\,M\)

Set up an I.C.E. table

Now, we will set up an I.C.E. (Initial, Change, Equilibrium) table for both cases to find the final equilibrium concentrations. We can represent the weak acid dissociation as: \(HA \rightleftharpoons H^{+} + A^{-}\) For Part (a): | \(HA\) | \(H^+\) | \(A^-\) Initial (M) | 0.258 | 0 | 0 Change (M) | -x | +x | +x Equilibrium (M)| 0.258-x | x | x For Part (b): | \(HA\) | \(H^+\) | \(A^-\) Initial (M) | 0.492 | 0 | 0 Change (M) | -x | +x | +x Equilibrium (M)| 0.492-x | x | x

Solve the equilibrium expression for \(\left[\mathrm{H}^+\right]\)

The equilibrium expression for the weak acid is given by \(K_a = \frac{[\mathrm{H}^+][A^-]}{[HA]}\). Now we will plug in the relevant equilibrium concentrations into this equation for both cases and solve for \(\left[\mathrm{H}^+\right]\). For Part (a): \(1.7\times10^{-3} = \frac{[H^+][A^-]}{[HA]} = \frac{x^2}{0.258-x}\) Solving for x, we get: \(x = 0.0120\,M \approx \left[\mathrm{H}^+\right]\) (in Part a) For Part (b): \(1.7\times10^{-3} = \frac{[H^+][A^-]}{[HA]} = \frac{x^2}{0.492-x}\) Solving for x, we get: \(x = 0.0148\,M \approx \left[\mathrm{H}^+\right]\) (in Part b) So we have found the concentrations of \(\left[\mathrm{H}^+\right]\) for both cases: (a) \(0.0120\,M\) (b) \(0.0148\,M\)

Key concepts

Penicillin as a Weak Acid

Many students encounter the concept of weak acids in chemistry, specifically in pharmacology when learning about drugs like penicillin. Penicillin, with a relatively high molecular mass (356 g/mol), is a classic example of a weak acid. Unlike strong acids, which dissociate completely in water, penicillin only partially separates into its constituent ions. Its acid dissociation constant, denoted as Ka, which for penicillin is reported to be 1.7 x 10^-3, provides insight into the degree to which it dissociates in aqueous solutions. This value speaks to its relative strength among weak acids; a smaller value would indicate a weaker acid. Understanding penicillin's weak acid behavior is crucial for predicting how it interacts in the body and its availability to treat infections.

Weak Acid Equilibrium Concentration

When dealing with a weak acid like penicillin in solution, it's important to understand that it does not fully ionize. Therefore, we speak of an equilibrium condition where the weak acid (HA) is in a state of dynamic balance with its ions (H+ and A-). The equilibrium concentrations of each species are of great interest in pharmacological and chemical contexts. To represent this state, we use an expression that relates these concentrations with the acid dissociation constant, or Ka. For penicillin, given its Ka value, we can predict how much of the acid will remain in the non-ionized form HA, and how much will exist as H+ and A- at equilibrium. This balance affects everything from the pH of the solution to the bioavailability of the drug in a biological system.

Calculating Hydrogen Ion Concentration

One of the key outcomes of understanding weak acid behavior is the ability to calculate the concentration of hydrogen ions, \[\mathrm{H}^+\], in solution. This is vital because the hydrogen ion concentration dictates the acidity of the solution. In pharmacology, the acidity can influence how a drug like penicillin is absorbed or metabolized. For chemists and healthcare professionals, calculating the \[\mathrm{H}^+\] provides critical information for formulating drug dosages and understanding their mechanism in the body. Using the provided Ka for penicillin and its initial concentration, we can use the equilibrium constant expression to find the \[\mathrm{H}^+\] concentration, a direct indicator of the solution's pH.

ICE Table Method

The ICE table method provides a systematic approach to visualize the changes in concentrations from the initial state to the state of equilibrium for a chemical reaction. The acronym ICE stands for Initial, Change, and Equilibrium. This table is extremely helpful when solving for unknown concentrations in weak acid or base dissociation problems. Let's demonstrate it with penicillin:

Setting Up the ICE Table

First, we list the initial concentrations (I), changes in concentrations (C), and equilibrium concentrations (E) for each species. In this particular reaction, penicillin (HA) ionizes to \[\mathrm{H}^+\] and \[A^-\]. We start with the known initial concentration of HA and zero for \[\mathrm{H}^+\] and \[A^-\]. As the reaction proceeds to reach equilibrium, the concentration of HA decreases by an amount 'x', while the concentrations of \[\mathrm{H}^+\] and \[A^-\] increase by 'x'.

Analyzing Equilibrium

Based on the Ka value and the equilibrium concentrations, we can solve for 'x' which represents the change and, thus, the concentration of \[\mathrm{H}^+\]. It's a straightforward but powerful tool for chemists studying the behavior of weak acids in solution.