Problem 40 - Chemistry The Molecular Nature of Matter

Question

For which of the following reactions does \(K_{\mathrm{P}}=K_{\mathrm{c}}\) ? (a) \(2 \mathrm{H}_{2}(g)+\mathrm{C}_{2} \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{C}_{2} \mathrm{H}_{6}(g)\) (b) \(\mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g)\) (c) \(2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g)\)

Step-by-Step Solution

Answer

\(K_{\mathrm{P}} = K_{\mathrm{c}}\) is true only for reaction (b) \(N_{2}(g) + O_{2}(g) \rightleftharpoons 2 NO(g)\) since the change in the number of moles of gas is zero.

Step 1: Define the Equilibrium Constants

The equilibrium constant in terms of concentration is given by \(K_c\) and in terms of partial pressure is given by \(K_p\). They are related by the equation \(K_p = K_c(RT)^{\Delta n}\) where \(R\) is the gas constant, \(T\) is the temperature, and \(\Delta n\) is the change in moles of gas (moles of products minus moles of reactants).

Step 2: Analyze Reaction (a)

For reaction (a), \(2 H_{2}(g) + C_{2}H_{2}(g) \rightleftharpoons C_{2}H_{6}(g)\), the change in the number of moles of gas \(\Delta n\) is \(1 - (2 + 1) = -2\). Since \(\Delta n\) is not zero, \(K_p eq K_c\).

Step 3: Analyze Reaction (b)

For reaction (b), \(N_{2}(g) + O_{2}(g) \rightleftharpoons 2 NO(g)\), the change in the number of moles of gas \(\Delta n\) is \(2 - (1 + 1) = 0\). Since \(\Delta n = 0\), \(K_p = K_c\).

Step 4: Analyze Reaction (c)

For reaction (c), \(2 NO(g) + O_{2}(g) \rightleftharpoons 2 NO_{2}(g)\), the change in the number of moles of gas \(\Delta n\) is \(2 - (2 + 1) = -1\). Since \(\Delta n\) is not zero, \(K_p eq K_c\).

Key Concepts

Equilibrium in Chemical Reactions

Chemical equilibrium is a critical concept in chemistry, describing a state where the rate of the forward chemical reaction equals the rate of the reverse reaction. At this point, the concentration of reactants and products remains constant over time, although both reactions continue to occur. It's important to note that equilibrium does not necessarily mean that the reactants and products are present in equal amounts, but that their ratios remain steady.

Understanding equilibrium is key to predicting the behavior of chemical systems and manipulating them for desired outcomes. This state is described quantitatively by equilibrium constants, which encompass two commonly used expressions: the equilibrium constant for concentrations, represented as \(K_c\), and the equilibrium constant for partial pressures, represented as \(K_p\). These constants provide insight into the proportion of reactants and products at equilibrium in a chemical reaction under a given set of conditions.

Kp and Kc Relationship

The relationship between \(K_p\) and \(K_c\) is a fundamental concept for students studying chemical equilibria involving gases. While \(K_p\) is used when we are interested in the partial pressures of the gaseous species, \(K_c\) is related to their molar concentrations. These two constants may seem distinct, but they are related by the simple equation \(K_p = K_c(RT)^{\Delta n}\), where \(R\) is the universal gas constant, \(T\) is the absolute temperature in Kelvin, and \(\Delta n\) represents the change in moles of gas, calculated by subtracting the number of moles of gaseous reactants from the number of moles of gaseous products.

Understanding this relationship helps students realize that the position of equilibrium is influenced not only by concentrations and partial pressures but also by the temperature and the change in the number of moles of gas during the reaction.

Change in Moles of Gas (\(\textDelta n\))

The \(\Delta n\) in the equation relating \(K_p\) and \(K_c\) is a pivotal component that students must grasp. It signifies the net change in the number of moles of gases as the reaction proceeds from reactants to products. When \(\Delta n = 0\), meaning there is no change in the total moles of gas, the values for \(K_p\) and \(K_c\) are equal. This is because the \( (RT)^{\Delta n} \) term in the equation raised to the zero power becomes one.

In contrast, when \(\Delta n\) is not zero, the values of \(K_p\) and \(K_c\) differ. This typically happens when the number of gaseous molecules on the reactant side is not equal to the number of gaseous molecules on the product side of the equation. The effect of temperature on the \(K_p\) and \(K_c\) relationship also becomes significant with a nonzero \(\Delta n\), reminding students that the state of chemical equilibrium is sensitive to changes in both the composition of the system and the conditions under which the reaction occurs.