Problem 31 - Chemistry: An Atoms-Focused Approach

Question

\(K_{\eta}=32\) for the following equilibrium at \(298 \mathrm{K}\) $$ \mathrm{A}(z)+\mathrm{B}(g) \rightleftharpoons \mathrm{AB}(g) $$ What is the value of \(K_{c}\) for this same equilibrium at \(298 \mathrm{K} ?\)

Step-by-Step Solution

Answer

Answer: The value of \(K_c\) for the given equilibrium reaction at \(298\, \mathrm{K}\) is \(32\).

Step 1: Understand the relationship between \(K_\eta\) and \(K_c\)

First, let's recall the relationship between the equilibrium constants \(K_\eta\) and \(K_c\) for a reaction involving gases. The relationship can be expressed as: $$ K_\eta = K_c(RT)^{\Delta n} $$ where, \(K_\eta\) is the equilibrium constant in terms of partial pressures \(K_c\) is the equilibrium constant in terms of concentrations \(R\) is the universal gas constant, which is \(0.0821\, \mathrm{L\, atm\, K^{-1}\, mol^{-1}}\) \(T\) is the temperature in Kelvin \(\Delta n\) is stoichiometric change in the number of moles of gas, i.e., \(\Delta n = n_{products} - n_{reactants}\)

Step 2: Determine \(\Delta n\) for the given reaction

For the given reaction, \(\mathrm{A}(z)+\mathrm{B}(g) \rightleftharpoons \mathrm{AB}(g)\), let's determine the stoichiometric change in the number of moles of gas: $$ \Delta n = n_{products} - n_{reactants} = 1 - 1 = 0 $$

Step 3: Use the relationship to solve for \(K_c\)

Since we know the value of \(K_\eta\), \(T\), and \(\Delta n\), we can solve for \(K_c\). Substituting the given values and the calculated value of \(\Delta n\) in the relationship equation: $$ K_\eta = K_c(RT)^{\Delta n} $$ and plugging in the values: $$ 32 = K_c \left(0.0821\times 298\right)^{0} $$ Since any non-zero number raised to the power of 0 is 1, we have: $$ 32 = K_c \times 1 $$

Step 4: Calculate the value of \(K_c\)

Now, we can simply solve for \(K_c\): $$ K_c = 32 $$ So, the value of \(K_c\) for the given equilibrium at \(298\, \mathrm{K}\) is \(32\).

Key Concepts

Partial Pressure

In chemistry, partial pressure refers to the pressure exerted by a single gas in a mixture of gases. Each gas in a mixture contributes to the overall pressure, and the partial pressure is simply the individual pressure contribution by that gas. When dealing with equilibrium in reactions where gases are present, it is essential to understand how these partial pressures play a role.

A reaction's equilibrium constant expressed in terms of partial pressures is denoted as \(K_\eta\). It represents the ratio of the partial pressures of the products to the reactants, each raised to the power of their stoichiometric coefficients.

• Partial pressures depend on the mole fraction of the gas in the mixture and the total pressure.
• In calculations for gas reactions, partial pressure helps us understand the behavior of individual gases.

Grasping partial pressures is vital in analyzing chemical equilibrium involving gases.

Concentration

Concentration measures the amount of a substance in a given volume. It is crucial in calculating equilibrium constants in terms of molarity, represented by \(K_c\).

The concentration of a substance in a solution is used to determine how reactions proceed and reach equilibrium.

• High concentrations of reactants can drive reactions forward, potentially increasing the formation of products.
• Equilibrium concentrations help define the balance point of a chemical reaction in solution.

In the context of equilibrium constants, concentration is fundamental to determining \(K_c\), where it involves expressing reactants and products in terms of their molar concentrations.

Stoichiometry

Stoichiometry refers to the calculation of reactants and products in chemical reactions. It is a vital concept that helps us understand the proportions of substances involved.

When dealing with equilibrium constants, stoichiometry allows us to calculate \(\Delta n\), the change in moles of gases.

• \(\Delta n = n_{\text{products}} - n_{\text{reactants}}\)
• This value is crucial for adjusting the relationship between \(K_\eta\) and \(K_c\).

In the given equilibrium, \(\Delta n\) results in zero, simplifying the computation of the equilibrium constants. Accurate stoichiometric calculations are indispensable for precise chemistry practices.

Gas Constant

The gas constant \(R\) is a fundamental physical constant in chemistry. It provides a link between the energy scale in thermodynamics and the molecular scale.

Typically, \(R\) has a value of \(0.0821\, \mathrm{L\, atm\, K^{-1}\, mol^{-1}}\) when used in equilibrium calculations.

• \(R\) appears in formulas that connect \(K_\eta\) and \(K_c\) by considering changes in mole numbers, \(\Delta n\).
• It helps standardize the calculations across different gas reactions.

Understanding the role of the gas constant is crucial for accurate equilibrium calculations and bridging relations between pressure, volume, and temperature.

Chemical Equilibrium

Chemical equilibrium occurs when the forward and reverse reactions in a chemical process happen at the same rate. At this point, there is no net change in the concentrations of reactants and products.

To represent this state quantitatively, we use equilibrium constants such as \(K_\eta\) and \(K_c\).

• \(K_\eta\) refers to the equilibrium constant based on partial pressures of gases.
• \(K_c\) refers to the equilibrium constant based on concentrations in solutions.

Understanding when and how chemical equilibrium is achieved helps predict the direction and extent of reactions. It is an essential principle in ensuring that reactions have reached their designated dynamic state, providing balance and stability to chemical systems.