The relationship between the equilibrium constants \( K_c \) and \( K_p \) is fundamental in understanding chemical equilibria, particularly in gaseous reactions. This topic is crucial for students preparing for the College Board AP Chemistry exam, as it bridges the concepts of concentration-based and pressure-based equilibrium expressions, enhancing comprehension of reaction dynamics and equilibrium shifts.

Equilibrium constants are numerical values that express the ratio of the concentrations of products to reactants at equilibrium for a reversible chemical reaction. They provide insight into the position of equilibrium and the extent to which reactants are converted into products.

\( K_c \) is the equilibrium constant expressed in terms of concentration (moles per liter). For a general reaction: $$ aA + bB \leftrightarrow cC + dD $$ the equilibrium constant \( K_c \) is given by: $$ K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} $$ where square brackets denote the molar concentrations of the species. \( K_p \) is the equilibrium constant expressed in terms of partial pressures (atmospheres or torr). Using the same reaction as above, \( K_p \) is defined as: $$ K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b} $$ where \( P \) represents the partial pressures of the gases involved.

The relationship between \( K_c \) and \( K_p \) can be derived using the ideal gas law, which relates concentration and pressure. Starting with the ideal gas equation: $$ PV = nRT \Rightarrow P = \frac{n}{V} RT = [Concentration] \times RT $$ For gases, the concentration \([C]\) can be expressed in terms of partial pressure \( P \) as: $$ [C] = \frac{P}{RT} $$ Substituting this into the expression for \( K_c \): $$ K_c = \frac{\left(\frac{P_C}{RT}\right)^c \left(\frac{P_D}{RT}\right)^d}{\left(\frac{P_A}{RT}\right)^a \left(\frac{P_B}{RT}\right)^b} $$ Simplifying, we get: $$ K_c = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b} \times (RT)^{-(c+d-a-b)} $$ Recognizing that: $$ K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b} $$ Thus, the relationship is: $$ K_p = K_c (RT)^{\Delta n} $$ where \( \Delta n = (c + d) - (a + b) \) is the change in the number of moles of gas during the reaction.

Temperature plays a pivotal role in the relationship between \( K_c \) and \( K_p \). Since \( K_p \) is directly related to \( K_c \) through the expression \( K_p = K_c (RT)^{\Delta n} \), any change in temperature affects both constants. An increase in temperature alters the value of \( RT \), thereby modifying the equilibrium position of the reaction.

Understanding the \( K_c \) and \( K_p \) relationship allows chemists to predict how changes in concentration, pressure, and temperature will shift the equilibrium. For instance, in gaseous reactions where \( \Delta n \neq 0 \), changes in pressure can significantly impact \( K_p \), thereby shifting the equilibrium to favor either the reactants or the products.

Consider the following reaction: $$ N_2(g) + 3H_2(g) \leftrightarrow 2NH_3(g) $$ Here, \( \Delta n = 2 - (1 + 3) = -2 \). Using the relationship: $$ K_p = K_c (RT)^{-2} $$ This indicates that for this reaction, an increase in temperature will decrease \( K_p \) if \( K_c \) remains constant, shifting the equilibrium towards the reactants.

While the relationship \( K_p = K_c (RT)^{\Delta n} \) is robust for ideal gases, deviations may occur in real-world scenarios where gases do not behave ideally. Additionally, this relationship is only applicable to reactions involving gaseous species; reactions in solution require different considerations.

Let’s derive \( K_p \) from \( K_c \) for the reaction: $$ CO(g) + H_2O(g) \leftrightarrow CO_2(g) + H_2(g) $$ Assume: - \( \Delta n = (1 + 1) - (1 + 1) = 0 \) - \( K_p = K_c (RT)^0 = K_c \) This illustrates that for reactions where \( \Delta n = 0 \), \( K_p \) and \( K_c \) are numerically equal, simplifying calculations and predictions.

When calculating \( K_p \) from \( K_c \), it is essential to ensure that pressure units are consistent (typically atmospheres or torr) and that temperature is in Kelvin. Additionally, care must be taken to accurately determine \( \Delta n \), as it directly influences the relationship between the two constants.

Graphing \( K_p \) versus \( K_c \) for various \( \Delta n \) values can provide visual insight into how pressure and concentration-based equilibria interrelate. Such graphs demonstrate the influence of the change in the number of moles of gas on the equilibrium position.

In more complex systems, such as those involving multiple gaseous species and varying reaction pathways, the relationship between \( K_c \) and \( K_p \) remains a foundational concept. It aids in the design of industrial processes, such as the Haber process for ammonia synthesis, where pressure and temperature optimizations are crucial for maximizing yield.

• \( K_c \) and \( K_p \) are equilibrium constants based on concentration and partial pressure, respectively.
• The relationship \( K_p = K_c (RT)^{\Delta n} \) links the two constants, where \( \Delta n \) is the change in moles of gas.
• Understanding this relationship is essential for predicting equilibrium shifts in gaseous reactions.
• Temperature and changes in pressure significantly impact the values and implications of \( K_c \) and \( K_p \).