Question

What is the difference between \(K_{c} K_{p}\), and \(K_{x}\) ? For a gas-phase reaction, what is the relation between them?

Short answer

Question: Explain the differences between the equilibrium constants \(K_c\), \(K_p\), and \(K_x\), and describe how they are related for a gas-phase reaction. Answer: \(K_c\), \(K_p\), and \(K_x\) are different ways to express the equilibrium constant for a chemical reaction. \(K_c\) is expressed in terms of molar concentrations (moles per liter), \(K_p\) is expressed in terms of partial pressures, and \(K_x\) is expressed in terms of mole fractions. For a gas-phase reaction, the equilibrium constants are related as follows: \(K_p = K_c (RT)^{\Delta n}\) and \(K_p = K_x (P_{total})^{\Delta n}\), where \(\Delta n\) is the change in the number of moles of gas throughout the reaction.

Step-by-step solution

Explanation of \(K_c\), \(K_p\), and \(K_x\) #

There are various ways to express the equilibrium constant for a chemical reaction. The three constants mentioned in the exercise are defined as follows: 1. \(K_c\): The equilibrium constant expressed in terms of concentrations, where all species are expressed in molar concentrations (moles per liter). 2. \(K_p\): The equilibrium constant expressed in terms of partial pressures, where all gaseous species are expressed in partial pressure units (atmospheres, or any other pressure units). 3. \(K_x\): The equilibrium constant expressed in terms of mole fractions, where the concentration of each species is expressed in terms of its mole fraction within the mixture. The main difference between them is the way the concentrations of each species are expressed.

Relation between \(K_c\), \(K_p\), and \(K_x\) for a gas-phase reaction #

For a gas-phase reaction, the Ideal Gas Law, \(PV = nRT\), can be used to relate concentrations and partial pressures in terms of molar fraction. Let's derive the relation between \(K_c\), \(K_p\), and \(K_x\) for a generalized gas-phase reaction of the form: $$aA + bB \leftrightarrow cC + dD$$ The \(K_c\) expression for this reaction is given by: $$K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b}$$ Where \([X]\) represents the molar concentration of species \(X\) in moles per liter. For each gas, we can relate its molar concentration to its partial pressure using the ideal gas law as follows: $$[X] = \frac{P_X}{RT}$$ So, the \(K_c\) expression can be rewritten in terms of partial pressures: $$K_c = \frac{\left(\frac{P_C}{RT}\right)^c \left(\frac{P_D}{RT}\right)^d}{\left(\frac{P_A}{RT}\right)^a \left(\frac{P_B}{RT}\right)^b} = \frac{P_C^c P_D^d}{P_A^a P_B^b} \cdot \frac{1}{(RT)^{c+d-a-b}}$$ Now, the \(K_p\) expression for the reaction is given by: $$K_p = \frac{P_C^c P_D^d}{P_A^a P_B^b}$$ Comparing the new \(K_c\) expression and \(K_p\) expression, we can find a relation between them: $$K_p = K_c (RT)^{\Delta n}$$ Where \(\Delta n\) is the change in the number of moles of gas throughout the reaction, given by \(\Delta n = (c+d)-(a+b)\). To compute the relation between these constants and \(K_x\), we need to know the total pressure \(P_{total}\) of the system. Using mole fractions, we can express partial pressures as: $$P_X = x_X \cdot P_{total}$$ \(x_X\) is the mole fraction of species X. Now, from the above expressions: if we replace \(P_X\) with \(x_X \cdot P_{total}\) in the \(K_p\) expression, we get: $$K_p = \frac{(x_C \cdot P_{total})^c (x_D \cdot P_{total})^d}{(x_A \cdot P_{total})^a (x_B \cdot P_{total})^b}$$ Now, the \(K_x\) expression for the reaction is given by: $$K_x = \frac{x_C^c x_D^d}{x_A^a x_B^b}$$ Comparing these two expressions, we find: $$K_p = K_x (P_{total})^{\Delta n}$$ In conclusion, for a gas-phase reaction, the equilibrium constants are related according to the following expressions: 1. \(K_p = K_c (RT)^{\Delta n}\) 2. \(K_p = K_x (P_{total})^{\Delta n}\)

Key concepts

Equilibrium Constant Expressions

In chemistry, the concept of equilibrium constants is crucial for understanding how chemical reactions reach a state of balance. When a reaction reaches equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations or pressures of the reactants and products remain unchanged. To express the state of equilibrium, various constants such as \(K_c\), \(K_p\), and \(K_x\) are used. They represent different ways of expressing the equilibrium condition depending on the nature of the reacting substances.

• \(K_c\) is the equilibrium constant in terms of molar concentrations (moles per liter). This is typically used when dealing with reactions in solutions where solutes are dissolved in a solvent.
• \(K_p\) is the equilibrium constant in terms of partial pressures. This is more applicable to gas-phase reactions where gases are involved, and pressures are easier to measure.
• \(K_x\) expresses the equilibrium constant in terms of mole fractions, which is particularly useful when dealing with calculations involving a mixture of gases.

Understanding these expressions helps predict the extent of a reaction and how changes in conditions can shift the equilibrium.

Ideal Gas Law in Chemical Reactions

The ideal gas law, represented by the equation \(PV = nRT\), is a fundamental principle connecting pressure \(P\), volume \(V\), number of moles \(n\), the ideal gas constant \(R\), and temperature \(T\). In chemical reactions, especially those involving gases, it allows us to interconvert between different forms of concentrations. This relationship is pivotal when we need to relate the molar concentrations \([X]\) to the partial pressures \(P_X\) in reactions involving gaseous mixtures.

By rearranging the ideal gas law, you can express the molar concentration \([X]\) of a gas as \([X] = \frac{P_X}{RT}\). This forms the basis for converting \(K_c\) expressions into \(K_p\) expressions since they fundamentally express the same reaction equilibrium but through different measurable quantities.

This relation is crucial in understanding how chemical reactions behave under different conditions, such as temperature and pressure, and is a foundational tool in chemical engineering and environmental science.

Relation between Kc, Kp, and Kx

The relationship between the equilibrium constants \(K_c\), \(K_p\), and \(K_x\) is a central theme in gas-phase reactions. These constants are interconnected, and their relations can be derived using the ideal gas law and basic chemistry principles.

• The link between \(K_c\) and \(K_p\) is described by the formula \(K_p = K_c (RT)^{\Delta n}\), where \(\Delta n\) is the change in the amount of gas moles \((c+d)-(a+b)\) during the reaction. This formula adjusts the concentration-based constant to a pressure-based one, accounting for the effects of temperature and changes in gas mole numbers.
• The relation between \(K_p\) and \(K_x\) is expressed as \(K_p = K_x (P_{total})^{\Delta n}\). This equation considers the total pressure in the system and is particularly valuable in systems where total pressure changes can significantly impact the behavior of equilibrium.

These relations allow chemists to switch between different forms of equilibrium constants, providing flexibility in approaching chemical problems from either concentration or pressure perspectives.

Molar Concentrations and Partial Pressures

In chemical reactions, particularly those involving gases, understanding molar concentrations and partial pressures is critical. To study reactions, chemists often rely on molar concentrations, which measure the number of moles of a substance present in a given volume (usually liters). This provides a clear picture of how much of each species is involved in a reaction.

Partial pressures, on the other hand, are used to describe how much pressure each gas in a mixture contributes to the total pressure. According to Dalton's Law, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. This is expressed by \(P_{total} = P_1 + P_2 + \cdots + P_n\).

Understanding these two concepts allows chemists to interchange and express equilibrium states more flexibly, using \(K_c\) or \(K_p\). The ability to shift between these measures is invaluable in both experimental and theoretical chemistry, where different methods of measurement may be required.