Problem 59 - Chemistry and Chemical Reactivity

Question

A Boric acid and glycerin form a complex \(\mathrm{B}(\mathrm{OH})_{3}(\mathrm{aq})+\) glycerin \((\mathrm{aq}) \rightleftarrows \mathrm{B}(\mathrm{OH})_{3} \cdot\) glycerin \((\mathrm{aq})\) with an equilibrium constant of \(0.90 .\) If the concentration of boric acid is \(0.10 \mathrm{M}\), how much glycerin should be added, per liter, so that \(60 . \%\) of the boric acid is in the form of the complex?

Step-by-Step Solution

Answer

1.73 M of glycerin should be added.

Step 1: Define Equilibrium Expression

The equilibrium expression for the reaction is given by the equation: \[K = \frac{[\mathrm{B(OH)_3 \cdot glycerin}]}{[\mathrm{B(OH)_3}]\,[\mathrm{glycerin}]}\]where \(K = 0.90\).

Step 2: Determine Initial Concentrations

Initially, the boric acid concentration is \(0.10\, \text{M}\) and the glycerin we need to determine. Let's denote the glycerin concentration initially added as \([\mathrm{glycerin}]_0\).

Step 3: Establish Changes in Concentration

We want 60% of the boric acid to form the complex. Therefore, 60% of \(0.10\, \text{M} = 0.06\, \text{M}\) of boric acid will be converted to the complex. Thus, the change in concentration for the boric acid is \(-0.06\, \text{M}\), and the concentration of the complex will be \(0.06\, \text{M}\) at equilibrium.

Step 4: Write Equilibrium Concentration Expressions

At equilibrium, the concentration of boric acid is:\[[\mathrm{B(OH)_3}] = 0.10 - 0.06 = 0.04\, \text{M}\]Let \(x\) be the initial concentration of glycerin. At equilibrium, the glycerin concentration will be:\[[\mathrm{glycerin}] = x - 0.06\]

Step 5: Substitute into Equilibrium Expression

Substitute the equilibrium concentrations into the equilibrium expression:\[K = \frac{0.06}{0.04 \cdot (x - 0.06)} = 0.90\]

Step 6: Solve for Glycerin Concentration

Rearrange the equation to solve for \(x\):\[0.06 = 0.90 \cdot 0.04 \cdot (x - 0.06)\]\[0.06 = 0.036 \cdot (x - 0.06)\]\[\frac{0.06}{0.036} = x - 0.06\]\[1.6667 = x - 0.06\]\[x = 1.6667 + 0.06 = 1.7267\]Therefore, the initial concentration of glycerin should be \(1.73\, \text{M}\).

Key Concepts

Chemical Equilibrium

Chemical equilibrium occurs when the rates of the forward and reverse reactions are equal, resulting in a stable concentration of reactants and products. In the reaction between boric acid and glycerin, an equilibrium state is achieved when the rate of formation of the boric acid-glycerin complex is equal to the rate of its dissociation. At this point, the concentrations of the reactants and products remain constant.
To describe this state, we use the equilibrium constant, denoted by the symbol \( K \). The equilibrium constant for a given reaction at a specific temperature is an expression of the ratio of the concentrations of the products to the reactants at equilibrium. It provides insight into the extent of the reaction and helps predict the direction of the reaction depending on the initial concentrations of the reactants and products.

Concentration Calculations

Concentration calculations are crucial for determining how much of each reactant or product is present at equilibrium. Concentration is typically expressed in molarity (M), which is moles of solute per liter of solution.
In our exercise, we are tasked with figuring out how much glycerin needs to be added so that 60% of the boric acid forms the complex. We start by calculating what 60% of the initial 0.10 M boric acid concentration is, which yields 0.06 M. This amount converts into the complex at equilibrium.
Starting with these initial concentrations allows us to track changes that occur as the reaction shifts to equilibrium. We use these changes to set up an equation using the equilibrium expression, which helps us solve for the unknown initial concentration of glycerin, denoted as \( x \) in the original solution.

Reaction Kinetics

Reaction kinetics is the study of the speed at which chemical reactions occur and the factors that influence this speed. In the context of equilibrium, while reaction kinetics doesn't directly change the point of equilibrium, it informs us how quickly the system will reach that point.
Kinetics depends on a variety of factors such as temperature, concentration, and the presence of catalysts. In our specific case, although we are focused on the equilibrium state, knowing the initial rates can be important for understanding how quickly the system responds when glycerin is added.
While the equilibrium constant \( K = 0.90 \) tells us about the relative concentrations at equilibrium, kinetics would inform if the reaction is fast or slow in reaching this balance. In lab scenarios, this might help in deciding how long to wait or how to adjust conditions to reach equilibrium efficiently.