Question

Papaverine hydrochloride (abbreviated papH \(^{+} \mathrm{Cl}^{-} ;\) molar mass \(=378.85 \mathrm{g} / \mathrm{mol}\) ) is a drug that belongs to a group of medicines called vasodilators, which cause blood vessels to expand, thereby increasing blood flow. This drug is the conjugate acid of the weak base papaverine (abbreviated pap; \(K_{\mathrm{b}}=\) \(8.33 \times 10^{-9}\) at \(35.0^{\circ} \mathrm{C} ) .\) Calculate the \(\mathrm{pH}\) of a \(30.0-\mathrm{mg} / \mathrm{mL}\) aqueous dose of papH \(^{+} \mathrm{Cl}^{-}\) prepared at \(35.0^{\circ} \mathrm{C} . K_{\mathrm{w}}\) at \(35.0^{\circ} \mathrm{C}\) is \(2.1 \times 10^{-14} .\)

Short answer

The pH of the aqueous solution containing \(30.0 \, \mathrm{mg / mL}\) of papaverine hydrochloride at \(35.0 ^\circ \mathrm{C}\) is approximately 1.82.

Step-by-step solution

Find the molar concentration of papH+ Cl-

First, we need to convert the concentration from mg/mL to mol/L. Using the given concentration and molar mass: \[\frac{30.0 \, \mathrm{mg}}{\mathrm{mL}} \times \frac{1 \, \mathrm{mol}}{378.85 \, \mathrm{g}} \times \frac{1 \, \mathrm{g}}{1000 \, \mathrm{mg}} = 7.93 \times 10^{-5} \, \mathrm{M}\] So, the molar concentration of papH+ Cl- is \(7.93 \times 10^{-5} \mathrm{M}\).

Calculate the concentration of the weak base (pap) and conjugate acid (papH+).

Since papH+ Cl- is a conjugate acid of the weak base pap, their concentrations in the solution will be equal. Thus, the initial concentration of both pap and papH+ Cl- is \(7.93 \times 10^{-5} \mathrm{M}\). We will assume that the concentration of Cl- does not affect the pH of the solution as Cl- is the conjugate base of a strong acid (HCl) and will not hydrolyze in the solution.

Write the equilibrium expression for the reaction between the weak base (pap) and water.

When the weak base pap reacts with water, it forms papH+ and OH-. The equilibrium expression is: \[K_{\mathrm{b}} = \frac{[\mathrm{papH}^+][\mathrm{OH}^-]}{[\mathrm{pap}]}\]

Calculate the equilibrium concentration of OH-.

Let x be the change in the concentration of pap, papH+, and OH-. At equilibrium, we have: \[[\mathrm{papH}^+] = 7.93 \times 10^{-5} - x\] \[[\mathrm{OH}^-] = x\] \[[\mathrm{pap}] = 7.93 \times 10^{-5} + x\] Plugging these values into the Kb expression, we get: \[8.33 \times 10^{-9} = \frac{x(7.93 \times 10^{-5} - x)}{7.93 \times 10^{-5} + x}\] Since the value of Kb is very small, we can assume that x is negligible compared to \(7.93 \times 10^{-5}\). This simplifies the equation: \[x = 8.33 \times 10^{-9} \times 7.93 \times 10^{-5}\] Solving for x, we get: \[x = 6.60 \times 10^{-13}\] Thus, the equilibrium concentration of OH- is \(6.60 \times 10^{-13} \mathrm{M}\).

Find the pOH.

To find the pOH, we take the negative logarithm of the hydroxide ion concentration: \[pOH = -\log(6.60 \times 10^{-13}) = 12.18\]

Find the pH.

Finally, subtract the pOH from 14 to find the pH: \[pH = 14 - 12.18 = 1.82\] So, the pH of the aqueous solution containing \(30.0 \, \mathrm{mg / mL}\) of papaverine hydrochloride at \(35.0 ^\circ \mathrm{C}\) is approximately 1.82.

Key concepts

Weak Base

A weak base is a substance that partially accepts protons (or releases hydroxide ions) in an aqueous solution. Unlike strong bases, which completely dissociate in water, weak bases only partially ionize. This means they don't increase the hydroxide ion concentration much, resulting in a smaller change in pH.
In our chemical equation, papaverine (pap) is considered a weak base. Its limited ionization is why we have a low equilibrium concentration of hydroxide ions and a relatively high pH value in the solution.
It's important to remember that the strength of a weak base is often represented by its base dissociation constant, called Kb. A smaller Kb value indicates a weaker base, as it suggests less ionization occurs. In our example, papaverine has a Kb of 8.33 x 10^-9, which categorizes it as a weak base.

Equilibrium Expression

The equilibrium expression is a mathematical relationship that uses the concentrations of reactants and products to express the ratio of their equilibrium composition. For a base, like papaverine, reacting with water, the equilibrium expression involves the concentration of the resulting conjugate acid (papH+) and hydroxide ions (OH-), over the initial concentration of the base.
In our equation, the equilibrium can be written as:

• \( K_b = \frac{[\text{papH}^+][\text{OH}^-]}{[\text{pap}]} \)

This arrangement helps us understand the extent to which the base dissociates in water. When solving for x in the equilibrium formula, we estimate the concentration changes for these ions to predict the overall pH of the solution.
This step is crucial because it shows how reactants convert to products over time, until a state of balance is achieved in the chemical reaction. Equilibrium processes are essential in understanding how weak acids and bases act in solutions.

Molar Concentration

Molar concentration, also known as molarity, measures the amount of a substance (in moles) present in a unit volume of solution (usually expressed in liters). It's a vital concept in chemistry because it quantitatively characterizes the concentration of reactants or products in a solution.
To find the molar concentration of papaverine hydrochloride (papH+ Cl-), calculate:

• The conversion from mg/mL to M (moles per liter), which includes changing mass to moles using the molar mass.
• The calculation of molarity: \(\frac{30.0\text{ mg}}{\text{mL}} \times \frac{1 \text{ mol}}{378.85 \text{ g}} \times \frac{1 \text{ g}}{1000 \text{ mg}} = 7.93 \times 10^{-5} \text{ M} \)

Understanding molar concentration is critical because it directly affects the reaction's dynamics and the solution's chemical properties. A correct molarity calculation ensures accurate predictions in chemical behaviors, including pH calculation.

Vasodilators

Vasodilators are a group of medications that open (dilate) blood vessels, improving blood flow. By reducing blood vessel resistance, they lower blood pressure and increase oxygen and nutrient delivery to tissues. This effect can help treat conditions like hypertension and heart failure.
Papaverine hydrochloride, found in our exercise, is classified as a vasodilator. Though it serves a different purpose pharmacologically, its chemical properties can be analyzed through its pH in solution as we have done here. Understanding these medications' chemical behavior, such as their interaction with pH, can help predict their efficacy and safety when transporting in the bloodstream.
When studying such compounds, it's essential to consider both their therapeutic impacts as well as their chemical properties, like how they behave in aqueous solutions, to grasp their complete role in medical treatments.