"The Relationship Between Different Equilibrium Constants: Kp, Kc, Kx, and Kn"

 

"The Relationship Between Different Equilibrium Constants: Kp, Kc, Kx, and Kn"

Introduction:

Chemical equilibrium is a fascinating topic in the world of chemistry, where the balance between reactants and products in a chemical reaction is carefully maintained. To understand and quantify this balance, chemists rely on various equilibrium constants, such as Kp, Kc, Kx, and Kn. In this article,

we'll explore the relationships between these equilibrium constants and delve into how they relate to the units of equilibrium constants, chemical equilibrium, and the fundamental principle of the Law of Mass Action.

Understanding the Equilibrium Constants:

Consider the equation for ideal gases;

Activity of the every substance is proportional to the partial pressure and to its molar concentration.

K_p = P_C^c P_D^d / P_A^a P_B^b

K_x = X_C^c X_D^d / X_A^a X_B^b

K_c = C_C^c C_D^d / C_A^a C_B^b

K_n = n_C^c n_D^d / n_A^a n_B^b

Here, K_c (Concentration-Based Equilibrium Constant) is defined in terms of molar concentrations, making it particularly useful for reactions in liquid and gaseous phases. It’s units are derived from the concentrations of the reactants and products and are often expressed in moles per liter (mol/dm^-3). The equilibrium constant expression for a chemical reaction in terms of K_c is related to the concentrations of reactants and products, following the Law of Mass Action.

Here, K_p (Pressure-Based Equilibrium Constant) is defined in terms of partial pressures and is specifically relevant for gaseous reactions. The units of K_p are derived from the partial pressures of the gases involved, typically expressed in atmospheres (atm) or pascals (Pa). The equilibrium constant expression for a reaction in terms of K_p is obtained by substituting partial pressures in place of concentrations.

K_x (Mole Fraction-Based Equilibrium Constant) is another constant used for gas-phase reactions, expressed in terms of mole fractions. The units of K_x are dimensionless since mole fractions are unitless.

K_n (Number of Particles-Based Equilibrium Constant) is used in reactions involving the number of particles, such as ionization and condensation reactions. The units of K_n are based on the number of particles involved, making it a dimensionless constant.

The Relationships Between Equilibrium Constants:

The relationships between these equilibrium constants can be summarized as follows:

K_p and K_c: The relationship between K_p and K_c is governed by the ideal gas law. For gas-phase reactions, K_p and K_c are related by the equation K_p = K_c (RT)^Δn, where Δn represents the change in the number of moles of gas between reactants and products, and R is the gas constant.

p_iV = n_iRT

p_i = n_iRT/V                    [ n_i/V=C_i ]

p_i = C_iRT

Suppose that;

p_A = n_ART/V  =   C_ART           

p_B = n_BRT/V  =   C_BRT           

p_C = n_CRT/V  =   C_CRT

p_D = n_DRT/V  =   C_DRT           

K_p = P_C^c P_D^d / P_A^a P_B^b

K_p = (C_CRT)^c (C_DRT)^d / (C_ART)^a(C_BRT)^b

K_p = C_C^c . C_D^d(RT)^c+d / C_A^a . C_B^b(RT)^a+b

K_p = K_c(RT)^c+d-(a+b)                       if    c+d = n_p ;  a+b = n_r

K_p = K_c(RT) ⁿ_p^{- n}_r

K_p = K_c (RT)^Δn

K_p and K_x:

These constants are directly related for gas-phase reactions, where K_x is determined by the mole fractions of gases involved. The relationship between K_p and K_x is represented as K_p = K_x. (P)^Δn

Suppose that,

p_i = X_iP

p_A = X_AP

p_B = X_BP

p_C = X_CP

p_D = X_DP

K_p = P_C^c P_D^d / P_A^a P_B^b

K_p = (X_CP)^c (X_DP)^d / (X_AP)^a (X_BP)^b

K_p = X_C^c . X_D(P)^c+d / X_A^a . X_B(P)^a+b

K_p = K_x(P)^c+d-(a+b)                         if    c+d = n_p ;  a+b = n_r

K_p = K_x(P) ⁿ_p^{- n}_r                         

K_p = K_x. (P)^Δn

K_p and K_n:

In some cases, K_p and K_n are related when the number of particles in a reaction remains constant. The relationship between K_p and K_n can be established by K_p = K_n. (P/N)^Δn.

Suppose the Dalton’s equation;

 p_i = X_iP

Where X_i is equal to n_i/N   

N is total number of reactants and products at equilibrium stage

n represents to no of moles of i component

So, p_i = n_iP/N

p_A = n_AP/N

p_B = n_BP/N

p_C = n_CP/N

p_D = n_DP/N

we know that;

K_p = P_C^c P_D^d / P_A^a P_B^b

K_p =( n_CP/N) ^c ( n_DP/N) ^d / ( n_AP/N) ^a ( n_BP/N) ^b

K_n = n_C^c n_D^d (P/N) ^c+d / n_A^a n_B^b (P/N) ^a+b

K_n = K_n (P/N) ^c+d-(a+b)                         if    c+d = n_p ;  a+b = n_r

K_p = K_n(P/N) ⁿ_p^{- n}_r                         

K_p = K_n. (P/N)^Δn

K_p = K_x. (P)^Δn = K_c (RT)^Δn = K_n. (P/N)^Δn

When Δn = 0 then,

K_p = K_x = K_c = K_n

Conclusion:

Understanding the relationships between different equilibrium constants, such as K_p, K_c, K_x, and K_n, is essential for accurately describing and predicting the behavior of chemical reactions in various phases. These constants, with their distinct units and applications, help chemists comprehend the complexities of chemical equilibrium, all while staying true to the fundamental principles of the Law of Mass Action. Whether you're working with gases, liquids, or solids, these equilibrium constants provide invaluable insights into the equilibrium state of chemical reactions.