Question

Tris(hydroxymethyl)aminomethane, known as TRIS or THAM, is a water-soluble base used in synthesizing surfactants and pharmaceuticals, as an emulsifying agent in cosmetics, and in cleaning mixtures for textiles and leather. In biomedical research, solutions of TRIS are used to maintain nearly constant pH for the study of enzymes and other cellular components. Given that the \(\mathrm{pK}_{\mathrm{b}}\) is \(5.91,\) calculate the \(\mathrm{pH}\) of \(0.075 \mathrm{M}\) TRIS.

Short answer

pH = 9.3

Step-by-step solution

Determine the given values

We know the following: - Concentration of TRIS solution, \(C = 0.075 \, \text{M}\) - \(\mathrm{pK}_{\mathrm{b}} = 5.91\)

Calculate \(\mathrm{K}_{\mathrm{b}}\)

Use the relationship \(\mathrm{K}_{\mathrm{b}} = 10^{-\mathrm{pK}_{\mathrm{b}}}\). Substitute the given \(\mathrm{pK}_{\mathrm{b}}\) value: \[\mathrm{K}_{\mathrm{b}} = 10^{-5.91} \]

Setup the equilibrium expression for the dissociation of TRIS in water

The reaction is \[ \text{TRIS} + \text{H}_2\text{O} \leftrightarrow \text{TRIS}^+ + \text{OH}^- \] The equilibrium expression in terms of \(\mathrm{K}_{\mathrm{b}}\) is \[\mathrm{K}_{\mathrm{b}} = \frac{[\text{TRIS}^+][\text{OH}^-]}{[\text{TRIS}]} \]

Define the variables for the equilibrium concentrations

Let \(x\) be the concentration of \[ \text{TRIS}^+ \] and \[ \text{OH}^- \] at equilibrium. The initial concentration of TRIS is \(0.075 \, \text{M}\) and it changes by \(x\). At equilibrium: \[ [\text{TRIS}] = 0.075 - x \]Thus, \[ \mathrm{K}_{\mathrm{b}} = \frac{x^2}{0.075 - x} \]

Solve the quadratic equation for \(x\)

Since \(\mathrm{K}_{\mathrm{b}} \) is very small, we can assume \(0.075 - x \approx 0.075\). This simplifies our equation: \[\mathrm{K}_{\mathrm{b}} = \frac{x^2}{0.075} \] Solving for \(x\): \[x^2 = \mathrm{K}_{\mathrm{b}} \cdot 0.075 \] \[x = \sqrt{(10^{-5.91})(0.075)} \]

Calculate \[ \text{OH}^- \] concentration

Calculate the concentration of \[ \text{OH}^- \] using the equation from the previous step: \[ \text{OH}^- = \sqrt{(10^{-5.91})(0.075)} \]

Calculate the \[ \text{pOH} \]

Use the formula \[ \text{pOH} = -\log_{10}[\text{OH}^-] \]

Calculate the \[ \text{pH} \]

Finally, use the relationship between \[ \text{pH} \] and \[ \text{pOH} \]: \[ \text{pH} + \text{pOH} = 14 \] \[ \text{pH} = 14 - \text{pOH} \]

Key concepts

pH

To begin with, the term pH is a measure of how acidic or basic a solution is. It is an essential concept in chemistry, specifically in studies involving buffer solutions, equilibrium expressions, and weak bases.
pH is calculated using the formula: \text{pH} = -\text{log }[ \text{H}^+ ], where \([\text{H}^+]\) is the concentration of hydrogen ions in the solution.
The pH scale ranges from 0 to 14:

• A pH less than 7 means the solution is acidic,
• a pH of 7 is neutral,
• while a pH greater than 7 indicates basic or alkaline nature.

In the problem, we calculated the pH of a TRIS solution, which involves understanding its equilibrium and the resulting concentrations of different ions.

Buffer Solutions

Buffer solutions are vital in maintaining stable pH levels in various chemical and biological processes.
A buffer solution typically contains a weak acid and its conjugate base or a weak base and its conjugate acid. In our exercise:

• TRIS acts as a weak base.
• It reacts with water to form its conjugate acid and hydroxide ions (OH-).

Buffer solutions resist changes in pH upon the addition of small amounts of acid or base thanks to the dynamic balance between the weak acid and weak base components.
This property is crucial in biochemical experiments where constant pH is required for enzyme activity and other cellular functions.

Equilibrium Expressions

Equilibrium expressions describe the balance between reactants and products in a reversible chemical reaction.
For our TRIS solution, the equilibrium expression is given by the equation: \[ \text{TRIS} + \text{H}_2\text{O} \rightleftharpoons \text{TRIS}^+ + \text{OH}^- \]
To express this equilibrium in terms of the equilibrium constant (\(\text{K}_b\)), we use: \[ \text{K}_b = \frac{[\text{TRIS}^+][\text{OH}^-]}{[\text{TRIS}]} \]
By defining the variables and setting up the expression with initial concentrations and changes due to dissociation, we can solve for the unknown concentration of ions at equilibrium. Simplifying assumptions such as ignoring minor changes in concentration help make calculations easier.

Weak Bases

Weak bases partially ionize in solution, unlike strong bases which dissociate completely.
TRIS is a weak base with a given \(\text{pK}_b\) value. The lower the value of \(\text{K}_b\), the weaker the base. In our example:

• We use the relationship \( \text{K}_b = 10^{-\text{pK}_b} \) to find the base dissociation constant.
• This helps in setting up the equilibrium expression and solving for ion concentrations at equilibrium.

Understanding weak bases involves recognizing that only a small fraction of the base molecules ionize to produce \( \text{OH}^- \) ions.
This partial ionization is key to many buffer systems, as it allows for a balanced reaction that can stabilize the pH of the solution effectively.