Question

Which of the following statements are true and which are false? (a) For the reaction \(2 \mathrm{A}(g)+\mathrm{B}(g) \rightleftharpoons \mathrm{A}_{2} \mathrm{B}(g) K_{c}\) and \(K_{p}\) are numerically the same. (b) It is possible to distinguish \(K_{c}\) from \(K_{p}\) by comparing the units used to express the equilibrium constant. (c) For the equilibrium in (a), the value of \(K_{c}\) increases with increasing pressure.

Short answer

(a) False: Kc and Kp are numerically related, but they are not always the same. (b) True: Units can differentiate between Kc and Kp. (c) True: For the given equilibrium, the value of Kc increases with increasing pressure.

Step-by-step solution

(a) Analyzing whether Kc and Kp are numerically the same

For a general reaction: \[aA + bB \rightleftharpoons cC + dD\] \(K_C\) is the equilibrium constant in terms of concentrations, and it relates the concentrations of reactants and products at equilibrium: \[ K_C = \frac{[C]^c[D]^d}{[A]^a[B]^b} \] On the other hand, \(K_P\) is the equilibrium constant in terms of partial pressures and relates the partial pressures of reactants and products at equilibrium: \[ K_P = \frac{(P_C)^c(P_D)^d}{(P_A)^a(P_B)^b} \] The relationship between Kc and Kp is given by: \[ K_P = K_C \times (\frac{RT}{P})^{\Delta n} \] Where: - R: Ideal gas constant - T: Temperature in Kelvin - P: Total pressure - \(\Delta n\): Change in the number of moles of gas during the reaction (\(n_\text{final} - n_\text{initial}\)) For the given reaction: \[2A(g) + B(g) \rightleftharpoons A_{2}B(g)\] The change in the number of gas moles is: \[ \Delta n = 1 - (2 + 1) = -2 \] Thus: \[ K_p = K_c \times (\frac{RT}{P})^{-2} \] In general, this equation indicates that Kc and Kp have a numerical relationship, but they are not always exactly the same.

(b) Analyzing if units can differentiate between Kc and Kp

The equilibrium constant Kc is expressed in terms of concentrations, which are typically measured in mol/L. It is dimensionless when the given reaction has balanced stoichiometry, but in general, it can have units derived from the stoichiometric coefficients (\(a\), \(b\), \(c\), and \(d\)). Therefore, Kc may or may not have units. On the other hand, Kp is expressed in terms of partial pressures, typically measured in atm or bar. Thus, Kp will always have units derived from the pressure units (\(a\), \(b\), \(c\), and \(d\)). So, it is possible to differentiate Kc from Kp by comparing the units used to express the equilibrium constant.

(c) Analyzing the effect of pressure on Kc for the equilibrium in (a)

When we increase the pressure for a reaction at constant temperature, the reaction tends to shift in the direction that reduces the pressure (Le Chatelier's principle). This can be achieved by moving towards the side with fewer gas moles. For the given reaction: \[2A(g) + B(g) \rightleftharpoons A_{2}B(g)\] The relationship between Kp and Kc is given by: \[ K_p = K_c \times (\frac{RT}{P})^{\Delta n} \] In this case, since \(\Delta n\) is negative, increasing the pressure will make the \((\frac{RT}{P})^{\Delta n}\) term larger, causing Kp to increase. Since Kc and Kp are related, Kc will also increase as the total pressure of the system is increased. #Conclusion# (a) False: Kc and Kp are numerically related, but they are not always the same. (b) True: Units can differentiate between Kc and Kp. (c) True: For the given equilibrium, the value of Kc increases with increasing pressure.

Key concepts

Kc vs Kp

Understanding the distinction between the equilibrium constants Kc and Kp is essential for students studying chemical equilibria. The equilibrium constant Kc is calculated using the molarity, which is the concentration of substances in moles per liter, of reactants and products. On the other hand, Kp is based on partial pressures, commonly measured in atmospheres or bars, for gases involved in the reaction.

In mathematical terms, Kc and Kp are related through the equation \[ K_p = K_c \times (\frac{RT}{P})^{\Delta n} \.\], where \(R\) is the gas constant, \(T\) is the temperature in Kelvin, \(P\) is the total pressure, and \(\Delta n\) represents the change in moles of gas—reflecting the difference between the moles of gaseous products and reactants. An important takeaway is that Kc and Kp are not always numerically equal; they are related through the reaction conditions and the change in gas moles. This discrepancy arises due to the differing units used to measure concentration (for Kc) and pressure (for Kp).

Le Chatelier's Principle

Le Chatelier's principle, named after the French chemist Henri Louis Le Chatelier, is a fundamental concept that predicts how a system at equilibrium reacts to disturbances or changes in conditions such as concentration, pressure, and temperature. Simply put, if a dynamic equilibrium is disrupted, the system will adjust to minimize the change and restore a new balance.

This principle can be applied to changes in pressure for gas-phase reactions. When pressure is increased, the system shifts to reduce that pressure, favoring the side with fewer gas molecules. Conversely, decreasing pressure shifts the reaction toward producing more gas molecules. This shift changes the equilibrium concentrations or partial pressures of the reactants and products, affecting the calculated values of Kc and Kp.

Reaction Quotient

The reaction quotient, often denoted as \(Q\), plays a pivotal role in determining the direction of a reaction's shift when it is not at equilibrium. It is calculated with the same formula as the equilibrium constant, but with the initial concentrations or partial pressures instead of the equilibrium values.

By comparing \(Q\) to the equilibrium constant \(K\), we can predict the direction of the reaction: if \(Q < K\), the reaction will proceed forward to reach equilibrium; if \(Q > K\), the reaction will shift to the left, or reverse, to return to equilibrium. This comparison informs chemists whether the system has yet to reach equilibrium or if a shift needs to occur to restore balance.

Gas-Phase Equilibrium

Gas-phase equilibrium concerns reactions where all, or some, reactants and products are in the gas phase. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, leading to constant concentrations or partial pressures of the substances involved.

An important consideration in gas-phase equilibria is the effect of pressure changes on the equilibrium state. As dictated by Le Chatelier's principle, the position of the equilibrium will shift depending on the addition or removal of gaseous substances or changes in pressure. This aspect emphasizes the dynamic nature of equilibrium in gas systems, where changes in conditions lead to a shift in balance but do not affect the value of the equilibrium constant unless the temperature is changed.