Question

Base Solutions Morphine is a weak base. A 0.150 \(\mathrm{M}\) solution of morphine has a pH of \(10.5 .\) What is \(K_{\mathrm{b}}\) for morphine?

Short answer

The value of Kb for morphine is approximately \( 1.5 \times 10^{-6} \) M.

Step-by-step solution

Determine the concentration of hydroxide ions

Start by calculating the concentration of hydroxide ions (\( OH^- \) ions) using the formula that relates pH and pOH, given that pH + pOH = 14. Since the pH is 10.5, the pOH is 3.5. The concentration of hydroxide ions can then be found using the equation \( \left[OH^-\right] = 10^{-\text{pOH}} \).

Calculate the hydroxide ion concentration

Now, calculate \( \left[OH^-\right] \) using the pOH value: \( \left[OH^-\right] = 10^{-3.5} \) which gives the hydroxide ion concentration in the solution.

Write the equilibrium expression for Kb

For the base dissociation of morphine (M), the ionization can be represented as \( M + H_2O \longleftrightarrow MH^+ + OH^- \). The expression for Kb is \( K_b = \frac{[MH^+][OH^-]}{[M]} \). Since we have a weak base, we can assume \( [MH^+] = [OH^-] \) and \( [M] \) is equal to the initial concentration of the base minus the concentration of \( OH^- \) (which is very small and can be ignored at this approximation level).

Calculate the value of Kb

Now substitute the hydroxide ion concentration into the equilibrium expression for Kb to solve for the base dissociation constant. This leads to \( K_b = \frac{[OH^-]^2}{[M] - [OH^-]} \). Since \( [OH^-] \) is much smaller than \( [M] \) (0.150 M), we can approximate that \( [M] - [OH^-] \approx [M] \). Therefore, \( K_b \approx \frac{[OH^-]^2}{[M]} \).

Insert the values and calculate Kb

Insert the values of \( [OH^-] \) and the initial concentration of morphine \( [M] \) into the approximate equation for Kb to get the value of the base dissociation constant: \( K_b \approx \frac{(10^{-3.5})^2}{0.150} \) and compute Kb.

Key concepts

pH and pOH

Understanding the relationship between pH and pOH is essential for studying acid-base chemistry. The pH scale is used to determine the acidity or basicity of an aqueous solution. When we discuss a substance like morphine being a weak base, knowing its pH can help us determine its pOH, since pH and pOH are inversely related through the equation \(\text{pH} + \text{pOH} = 14\). If the pH of a morphine solution is 10.5, we can calculate the pOH by subtracting this value from 14, yielding a pOH of 3.5.

With the pOH, we can now find the concentration of hydroxide ions (\(OH^-\)) in the solution, because the pOH is a measure of how much \(OH^-\) is present. The higher the pOH, the lower the \(OH^-\) concentration, and vice versa. This relationship is mathematically represented by \( \[OH^-\] = 10^{-\text{pOH}} \) which becomes an important step in quantifying the base's dissociation constant (\(K_{\mathrm{b}}\)).

Weak Base Equilibrium

Weak bases like morphine do not completely dissociate in water. This partial dissociation sets up an equilibrium between the base and its ions in solution. The equilibrium expression for a base like morphine, represented by the variable M, reacting with water is \( M + H_2O \longleftrightarrow MH^+ + OH^- \). At equilibrium, the concentration of the products (\(MH^+\) and \(OH^-\)) multiplied together, divided by the concentration of the reactant (M), gives us the base dissociation constant, \(K_{\mathrm{b}}\).

For calculations involving weak bases, we often make an approximation. Since the base is weak, the concentration of the hydroxide ions produced is relatively small compared to the initial concentration of the base. This allows us to make simplifying assumptions that the concentration of the base does not change significantly, hence, \( [MH^+] \approx [OH^-] \) and \( [M] - [OH^-] \approx [M] \), which greatly simplifies our equilibrium calculations.

Hydroxide Ion Concentration

The hydroxide ion concentration (\([OH^-]\)) is a direct measure of the basicity of a solution. In the case of the morphine solution, this concentration can be calculated using the determined pOH. The concentration is found by the expression \( \[OH^-\] = 10^{-\text{pOH}} \). For a pOH of 3.5, the corresponding hydroxide ion concentration would be calculated as \(10^{-3.5}\).

This concentration plays a crucial role in determining how much the base has dissociated in water. By knowing \( [OH^-] \), we can infer how much of the morphine has reacted to form the \(MH^+\) and \(OH^-\) ions in the equilibrium of the weak base. These concentrations are the cornerstone when we calculate the \(K_{\mathrm{b}}\) value, which quantifies the base's strength and its tendency to dissociate in an aqueous solution.

Chemical Equilibrium Expressions

Chemical equilibrium expressions such as the base dissociation constant (\(K_{\mathrm{b}}\)) reflect the ratio of the concentration of the products over the reactants when a reaction is at equilibrium. For weak bases, the expression takes the form \( K_b = \frac{[MH^+][OH^-]}{[M]} \).

Whenever we deal with a weak base equilibrium, we're often able to simplify the expression due to the small amount of ionization. In practice, we can assume that the concentration of the base minus the hydroxide ion concentration is approximately equal to the initial concentration of the base: \( K_b \approx \frac{[OH^-]^2}{[M]} \).

By inserting the known values of \( [OH^-] \) and the initial concentration of the weak base \( [M] \), you can solve for the \(K_{\mathrm{b}}\). This constant is vital for understanding the behavior of bases in different chemical environments and is widely used in chemical calculation involving weak base reactions in aqueous solutions.