Problem 13 - Chemistry The Molecular Nature of Matter

Question

Consider the following equilibrium. \(2 \mathrm{NaHCO}_{3}(s) \rightleftharpoons \mathrm{Na}_{2} \mathrm{CO}_{3}(s)+\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(g)\) If you were converting between \(K_{\mathrm{P}}\) and \(K_{\mathrm{c}}\), what value of \(\Delta n_{\mathrm{g}}\) would you use?

Step-by-Step Solution

Answer

\(\Delta n_{\mathrm{g}} = 2\)

Step 1: Identify the Gaseous Products and Reactants

Examine the chemical equation to identify the gaseous substances. Here, \(\mathrm{CO}_{2}\text{(g)}\) and \(\mathrm{H}_{2}\mathrm{O}\text{(g)}\) are the gaseous products and there are no gaseous reactants.

Step 2: Calculate the Change in Moles of Gas

Calculate \(\Delta n_{\mathrm{g}}\), which is the change in moles of gas, by subtracting the moles of gaseous reactants from the moles of gaseous products. In this case, \(\Delta n_{\mathrm{g}} = (1 + 1) - 0 = 2\).

Step 3: Write the Value of \(\Delta n_{\mathrm{g}}\)

The value of \(\Delta n_{\mathrm{g}}\) indicates the difference in the number of moles of gases between products and reactants. For this reaction, \(\Delta n_{\mathrm{g}} = 2\).

Key Concepts

Kp and Kc relationship

Understanding the relationship between the equilibrium constants in the gas phase, represented by \(K_p\) and \(K_c\), is crucial for grasping the concept of chemical equilibrium. The equilibrium constant \(K_c\) is based on the molar concentrations of the reactants and products at equilibrium. In contrast, \(K_p\) is based on the partial pressures of the gases involved.

These two constants are connected through the equation \(K_p = K_c(RT)^{\Delta n_g}\), where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\Delta n_g\) is the change in moles of gas during the reaction. Understanding this relationship is essential when the reaction involves gaseous substances, as it allows for conversion from concentration terms to pressure terms and vice versa.

Change in Moles of Gas (\(\Delta n_g\))

When dealing with gaseous equilibrium reactions, \(\Delta n_g\) is a term that indicates the net change in moles of gas when the reaction reaches equilibrium. It is calculated by subtracting the sum of the moles of gaseous reactants from the sum of the moles of gaseous products.

For the reaction in the exercise, \(\Delta n_g\) is calculated as the sum of the moles of \(CO_2\) and \(H_2O\), both gases, minus any gaseous reactants, which in this case there are none. Hence, \(\Delta n_g\) equals 2, representing two moles of gases produced. This value is key when converting between \(K_p\) and \(K_c\), as it determines the exponent of the temperature and gas constant product in the conversion formula.

Equilibrium Constant Calculation

Equilibrium constant calculation is fundamental for understanding the extent of a chemical reaction at equilibrium. This is represented by either \(K_c\) or \(K_p\), depending on whether the reaction involves molarity or partial pressure.

In order to calculate either of these constants, all one needs is the balanced chemical equation and the equilibrium concentrations or partial pressures of the reactants and products. The formula for \(K_c\) is the product of the concentrations of the products raised to the power of their stoichiometric coefficients, divided by the product of the concentrations of the reactants raised to the power of their stoichiometric coefficients. The process is similar for \(K_p\), but with partial pressures instead of concentrations.

Gaseous Equilibrium Reactions

Gaseous equilibrium reactions are a subset of chemical reactions that specifically involve gases and reach a state where the rate of the forward reaction equals the rate of the reverse reaction. At this point, the concentrations or partial pressures of all reactants and products remain constant over time.

In the context of the given exercise, since the reactants are solids and the products include gases, we are concerned only with the partial pressures of the gaseous CO_2 and H_2O when determining the equilibrium constant \(K_p\). It's important to note that for these reactions, changes in conditions such as temperature, pressure, or volume can shift the equilibrium position, and this is described by Le Chatelier's principle.