Problem 93 - Chemical Principles: The Quest for Insight

Question

A 30.1-g sample of NOCl is placed into a \(200-\mathrm{mL}\) reaction vessel and heated to \(500 \mathrm{~K}\). The value of \(K\) for the decomposition of NOCl at \(500 \mathrm{~K}\) in the reaction \(2 \mathrm{NOCl}(\mathrm{g})=2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g})\) is \(1.13 \times 10^{-3}\). (a) What are the equilibrium partial pressures of \(\mathrm{NOCl}, \mathrm{NO}\), and \(\mathrm{Cl}_{2}\) ? (b) What is the percentage decomposition of NOCl at this temperature?

Step-by-Step Solution

Answer

The equilibrium partial pressures of NOCl, NO, and Cl2 are calculated using the ideal gas law with equilibrium amounts. The percentage decomposition of NOCl is obtained from the ratio of the change in moles of NOCl to the initial moles of NOCl, multiplied by 100%.

Step 1: Calculate initial number of moles of NOCl

Using the molar mass of NOCl (65.5 g/mol), calculate the initial number of moles of NOCl before decomposition begins.

Step 2: Write the expression for the equilibrium constant

For the reaction \(2 \mathrm{NOCl(g)} \rightarrow 2 \mathrm{NO(g)} + \mathrm{Cl}_{2}(g)\), write the expression for the equilibrium constant \(K\) in terms of the partial pressures of the gases.

Step 3: Set up the ICE table

Create an ICE (Initial, Change, Equilibrium) table to organize the initial amounts, changes, and equilibrium amounts of NOCl, NO, and Cl2.

Step 4: Define the change in terms of the extent of reaction (x)

Express the changes in the concentration of NOCl, NO, and Cl2 in terms of \(x\), the extent of the reaction, where \(x\) is the amount of NOCl that decomposes.

Step 5: Write the equilibrium constant expression in terms of x

Substitute the equilibrium concentrations from the ICE table in terms of \(x\) into the expression for the equilibrium constant to find an equation that must be solved for \(x\).

Step 6: Solve for x

Solve the equation for \(x\) using the quadratic formula or approximations if justified.

Step 7: Calculate equilibrium partial pressures of NOCl, NO, and Cl2

Use the value of \(x\) to calculate the equilibrium partial pressures of NOCl, NO, and Cl2 using the ideal gas law \(PV = nRT\), where \(V\) is the volume of the reaction vessel, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature.

Step 8: Calculate the percentage decomposition of NOCl

Determine the proportion of the NOCl that decomposed by using the relationship \(\frac{\text{Change in moles of NOCl}}{\text{Initial moles of NOCl}} \times 100\%\).

Key Concepts

Chemical Equilibrium

Chemical equilibrium occurs when a chemical reaction and its reverse reaction proceed at the same rate, leading to an unchanging system. In the context of the exercise, the decomposition of NOCl into NO and Cl₂ reaches a point where the rate of NOCl breaking down equals the rate of NO and Cl₂ recombining to form NOCl. At equilibrium, the reaction has not necessarily stopped, but the concentrations of the reactants and products remain constant.

Understanding chemical equilibrium is fundamental to predicting how a change in conditions, such as temperature or concentration, will affect the position of equilibrium, and hence the concentration of the reactants and products. In the given problem, the equilibrium state is described by constant partial pressures for the gases involved at a fixed temperature of 500 K.

Partial Pressures

Partial pressure is a term used to describe the pressure that a single component of a gas mixture would exert if it occupied the entire volume on its own at the same temperature. For a reaction occurring in the gas phase, the concentrations of the gases can often be expressed in terms of their partial pressures. The use of partial pressures is very convenient when working with gas-phase reactions as in our exercise, where we calculate the equilibrium partial pressures of NOCl, NO, and Cl₂ after the system has reached a state of chemical equilibrium. The relationship among these pressures is determined by the ideal gas law and the stoichiometry of the reaction, and it's essential for calculating the final equilibrium state.

Reaction Quotient

The reaction quotient (Q) is a measure of the relative amounts of products and reactants present during a reaction at a particular point in time. Q is calculated in the same way as the equilibrium constant (K), but it isn't restricted to equilibrium conditions. It's a tool used to predict the direction in which a reaction will proceed to reach equilibrium. If Q < K, the forward reaction is favored and the reaction will proceed towards forming more products. Conversely, if Q > K, the reverse reaction is favored, and the reaction will produce more reactants. This concept is applied in our exercise to determine at which point the system will achieve equilibrium based on the initial amounts of reactants and products and the given K value.

ICE Table Method

The ICE table method is an organized approach to solving equilibrium problems. It stands for Initial, Change, Equilibrium. Initially, we state the concentrations or pressures of reactants and products. Upon reaching the stage of Change, we use a variable, typically denoted as x, to represent the changes in concentrations or pressures as the system approaches equilibrium. Finally, we express Equilibrium concentrations or pressures in terms of x. This method clarifies the relationship between the amounts of reactants and products and helps solve for the unknowns. In the exercise, the ICE table allows us to calculate the extent of decomposition of NOCl and the equilibrium concentrations (or partial pressures) of all gases involved.

Le Chatelier's Principle

Le Chatelier's principle states that if an external change is applied to a system at equilibrium, the system adjusts to minimize that change. External changes can include alterations in concentration, pressure, or temperature. This principle helps us understand how a shift in conditions affects the position of equilibrium. For instance, an increase in temperature for an endothermic reaction will shift the equilibrium to the product side. As a demonstration of this principle, if we were to change the temperature or volume of the vessel in the exercise scenario, the system would shift its equilibrium position to counteract that change, either by producing more NO and Cl₂ or more NOCl depending on the nature of the change.