Problem 98 - General Chemistry: Principles and Modern Applications

Question

In your own words, define or explain the following terms or symbols: (a) \(K_{\mathrm{p}} ;\) (b) \(Q_{\mathrm{c}} ;\) (c) \(\Delta n_{\text {gas }}\)

Step-by-Step Solution

Answer

\(K_{\mathrm{p}}\) is the equilibrium constant determined by the partial pressures of reactants and products in a gas-phase reaction. \(Q_{\mathrm{c}}\) is the reaction quotient, representing the ratio of products to reactants at any stage during the reaction. \(\Delta n_{\text {gas }}\) is the change in number of moles of gases in a chemical reaction often determined when considering the impacts of volume and pressure changes on a reaction's equilibrium.

Step 1: Defining \(K_{\mathrm{p}}\)

The symbol \(K_{\mathrm{p}}\) is used in chemistry to represent the equilibrium constant. It's a constant value which is specific for each chemical reaction and is calculated using the partial pressures of the products and reactants that are gases. The expression for it is given by the ratio of the product of partial pressures of products to the product of partial pressures of reactants, each raised to its stoichiometric coefficients.

Step 2: Defining \(Q_{\mathrm{c}}\)

\(Q_{\mathrm{c}}\) represents the reaction quotient. It is a quantity that changes as the reaction progresses. It is calculated in the same way as the equilibrium constant, but it doesn't necessarily describe the system at equilibrium. Instead, it provides the ratio of the product of concentrations of products to the product of concentrations of reactants, each raised to its stoichiometric coefficients, at any point in time during the reaction.

Step 3: Defining \(\Delta n_{\text {gas }}\)

\(\Delta n_{\text {gas }}\) stands for the change in the number of moles of gases in a chemical reaction. It is determined by subtracting the total number of moles of gaseous reactants from the total number of moles of gaseous products. It can be negative, zero, or positive, and it is important for determining how factors such as pressure and volume changes impact the direction of the reaction according to Le Chatelier's Principle.

Key Concepts

Equilibrium Constant (Kp)

The equilibrium constant, \( K_{\mathrm{p}} \), is a fundamental concept in chemical reactions involving gases. It defines the balance point for a reversible chemical reaction at a certain temperature. Specifically, \( K_{\mathrm{p}} \) is derived by calculating the ratio of the partial pressures of the gaseous products to that of the reactants, each raised to the power of their respective stoichiometric coefficients in the balanced chemical equation.

To understand it better:

• The partial pressure of a gas is a measure of its tendency to occupy space in a mixture with other gases. This means it significantly influences how we calculate \( K_{\mathrm{p}} \).
• Each equilibrium reaction has its own specific \( K_{\mathrm{p}} \) value at a given temperature. This means it's vital to know the temperature at which \( K_{\mathrm{p}} \) is evaluated.
• \( K_{\mathrm{p}} \) is dimensionless; however, it gives insight into the extent of the reaction and whether the products or reactants are favored at equilibrium.

The crux is that \( K_{\mathrm{p}} \) helps chemists understand whether the chemical species in a reaction have reached a stable state where their rates of forward and reverse reactions are equal. Thus, providing essential insight into the reaction's dynamics.

Reaction Quotient (Qc)

The reaction quotient, \( Q_{\mathrm{c}} \), is akin to the equilibrium constant, but it measures the state of a reaction at any given moment, not just at equilibrium. It is a snapshot of the reaction that helps determine which direction the reaction needs to move to reach equilibrium.

Let's break down the intricacies of \( Q_{\mathrm{c}} \):

• \( Q_{\mathrm{c}} \) is calculated just like \( K_{\mathrm{c}} \), using the concentrations of the reactants and products at any given point in time, each elevated to the power reflecting their stoichiometric coefficients in the balanced equation.
• If \( Q_{\mathrm{c}} = K_{\mathrm{c}} \), the system is at equilibrium.
• If \( Q_{\mathrm{c}} < K_{\mathrm{c}} \), the reaction shifts towards the products to reach equilibrium.
• If \( Q_{\mathrm{c}} > K_{\mathrm{c}} \), the reaction shifts towards the reactants.

Understanding \( Q_{\mathrm{c}} \) is vital for predicting the outcome of changes in the system, such as variations in concentration, pressure, or temperature, and how they might shift the position of equilibrium.

Change in Moles of Gas (Δn_gas)

The change in moles of gas, denoted as \( \Delta n_{\text{gas}} \), is an important factor in understanding reactions involving gases. This concept assesses how the number of moles of gas change from the reactants to the products.

Here's how to effectively consider \( \Delta n_{\text{gas}} \):

• Calculate \( \Delta n_{\text{gas}} \) by subtracting the sum of moles of gaseous reactants from the sum of moles of gaseous products.
• This value can be negative, zero, or positive, reflecting whether gas is produced, unchanged, or consumed respectively.
• \( \Delta n_{\text{gas}} \) is critical for understanding how changes in pressure or volume according to Le Chatelier's Principle might influence the direction of the reaction.

Understanding \( \Delta n_{\text{gas}} \) is indispensable for applications where gaseous reactions occur, as it lends insight into post-reaction conditions and how external changes impact gaseous equilibria.