Problem 27 - Chemistry and Chemical Reactivity

Question

Consider the isomerization of butane with an equilibrium constant of \(K=2.5 .\) (See Study Question 13.) The system is originally at equilibrium with [butane] \(=1.0 \mathrm{M}\) and [isobutane] \(=2.5 \mathrm{M}\) (a) If \(0.50 \mathrm{mol} / \mathrm{L}\) of isobutane is suddenly added and the system shifts to a new equilibrium position, what is the equilibrium concentration of each gas? (b) If \(0.50 \mathrm{mol} / \mathrm{L}\) of butane is added and the system shifts to a new equilibrium position, what is the equilibrium concentration of each gas?

Step-by-Step Solution

Answer

(a) [butane] = 1.143 M, [isobutane] = 2.857 M; (b) [butane] = 1.143 M, [isobutane] = 2.857 M.

Step 1: Understand the Problem

Initially, the system is at equilibrium with concentrations: \([\text{butane}] = 1.0 \, \text{M}\) and \([\text{isobutane}] = 2.5 \, \text{M}\). The equilibrium constant \(K\) is given as 2.5 for the reaction: \[ \text{butane} \rightleftharpoons \text{isobutane} \]. Our task is to determine the new equilibrium concentrations when either isobutane or butane is added.

Step 2: First Scenario Analysis

In this scenario, 0.50 M of isobutane is added to the system. The initial concentration of isobutane becomes 3.0 M (2.5 M + 0.5 M), while butane remains at 1.0 M. We assume a change of \(x\) M occurs as the system shifts back to equilibrium. Therefore, at new equilibrium, \([\text{butane}] = 1.0 + x\) and \([\text{isobutane}] = 3.0 - x\).

Step 3: Apply Equilibrium Constant

According to the expression of the equilibrium constant: \[K = \frac{[\text{isobutane}]}{[\text{butane}]} = 2.5 \]This implies: \[\frac{3.0-x}{1.0+x} = 2.5 \]

Step 4: Solve for x (First Scenario)

Multiply both sides by \(1.0+x\) to eliminate the fraction: \[3.0 - x = 2.5(1.0 + x) \]Simplify and solve the equation: \[3.0 - x = 2.5 + 2.5x \]\[0.5 = 3.5x \]\[x = \frac{0.5}{3.5} \]\[x \approx 0.143 \]Plug \(x\) back into our expressions: \([\text{butane}] = 1.0 + 0.143 = 1.143 \, \text{M}\) and \([\text{isobutane}] = 3.0 - 0.143 = 2.857 \, \text{M}\).

Step 5: Second Scenario Analysis

For the second scenario, 0.50 M of butane is added. Therefore, initial concentrations become: \([\text{butane}] = 1.5 \, \text{M}\) and \([\text{isobutane}] = 2.5 \, \text{M}\). Let the change be \(y\) M as the system re-equilibrates, then \([\text{butane}] = 1.5 - y\) and \([\text{isobutane}] = 2.5 + y\).

Step 6: Apply Equilibrium Constant (Second Scenario)

For the second scenario: \[K = \frac{2.5+y}{1.5-y} = 2.5 \]

Step 7: Solve for y (Second Scenario)

Eliminate the fraction by multiplying both sides by \(1.5-y\): \[2.5 + y = 2.5(1.5 - y) \]Simplify and solve the equation:\[2.5 + y = 3.75 - 2.5y \]\[3.5y = 3.75 - 2.5 \]\[y = \frac{1.25}{3.5} \]\[y \approx 0.357 \]Thus, new equilibrium concentrations are: \([\text{butane}] = 1.5 - 0.357 = 1.143 \, \text{M}\) and \([\text{isobutane}] = 2.5 + 0.357 = 2.857 \, \text{M}\).

Key Concepts

Equilibrium Constant

The equilibrium constant, often denoted by the symbol \(K\), is a crucial concept in the study of chemical reactions. It provides a measure of the extent to which reactants are converted into products at equilibrium. For a given reaction: \[ \text{A} \rightleftharpoons \text{B} \]The equilibrium constant \(K\) is expressed as a ratio of the concentrations of products to reactants, in this case:\[K = \frac{[\text{B}]}{[\text{A}]}\]where \([\text{B}]\) and \([\text{A}]\) represent the concentrations of B and A at equilibrium respectively.

Importantly, \(K\) is temperature-dependent and has no units. A larger \(K\) value signifies a greater concentration of products relative to reactants at equilibrium. This principle is applied when solving problems involving shifts in equilibrium, like in the isomerization of butane, where a known equilibrium constant helps calculate new equilibrium positions once concentration changes occur.

Isomerization

Isomerization refers to the process where a molecule transforms into another molecule with the same molecular formula, but with a different structural arrangement. In chemistry, isomerization is an important reaction that can significantly affect chemical properties and behaviors.

When considering the isomerization of butane to isobutane:\[\text{butane} \rightleftharpoons \text{isobutane}\]This reaction does not involve a change in the molecular formula, but only in how the atoms are arranged spatially. This structural change can influence the equilibrium position, as seen through the steadiness represented by the equilibrium constant. In isomerization, achieving equilibrium means that the rate of transformation from butane to isobutane equals the rate from isobutane back to butane. Thus, concentrations stabilize in proportions dictated by the equilibrium constant. Understanding this concept helps in predicting and calculating both initial and reaction-induced concentration changes.

Concentration Changes

Changes in concentration are often used to disturb an existing equilibrium, prompting the system to shift in response. This shift is described by Le Chatelier's Principle, where the system aims to counteract the change by favoring either the forward or the reverse reaction to restore equilibrium.

In the context of the butane-isobutane system, adding additional isobutane or butane changes the concentration. For instance, initially, the concentration of isobutane and butane were stable at equilibrium at 2.5 M and 1.0 M respectively. However, when more of either butane or isobutane is introduced:

• Increased isobutane concentration makes the system favor the conversion back to butane to re-establish equilibrium.
• Conversely, adding butane will favor the production of more isobutane.

Using the equilibrium constant, these concentration changes are calculated to determine new equilibrium values.

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept used to predict how a change in conditions affects a chemical equilibrium. Simply put, it states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium shifts to counteract the change, hence restoring a new state of balance. This principle is applicable to changes in concentration, temperature, and pressure.

In the problem involving butane and isobutane, the system applies Le Chatelier's Principle when either gas is added to the mixture:

• When more isobutane is added, the principle predicts that the excess isobutane will cause the reaction to shift towards forming butane. This shift reduces the added excess and leads to a new equilibrium state.
• Similarly, when butane is added, the system shifts towards forming more isobutane to reduce the additional butane concentration.

Thus, understanding Le Chatelier's Principle is vital for predicting and calculating the direction and extent of changes in equilibrium when disturbances are introduced.