Question

Derive the relationship between $ {K_p} $ and $ {K_c} $ for a general chemical equilibrium reaction.

Answer 1

Hint :We know that $ {K_c} $ is equal to the ratio of concentration of products to the concentration of reactants. Finding the value of $ {K_c} $ for the considered reaction and then substituting it in the ideal gas equation provides us the relation between $ {K_p} $ and $ {K_c} $ . Also, remember that $ {K_p} $ is equal to the ratio of partial pressure of products to the partial pressure of reactants.

Complete Step By Step Answer:
This question belongs to the concept of chemical equilibrium. Let us see the basic terminology used in this question.
Here we have to find the relation between $ {K_p} $ and $ {K_c} $ . But let us first get an idea of what $ {K_p} $ and $ {K_c} $ are,
 $ {K_p} $ is the equilibrium constant of an ideal gas mixture. Specifically it is the equilibrium constant which is used when the concentration of a given mixture or gas is expressed in terms of pressure. Whereas $ {K_c} $ is also the equilibrium constant for an ideal gas mixture but it is used when the concentrations of the ideal gas mixture are expressed in terms of molarity .
So, in order to find the relation between $ {K_p} $ and $ {K_c} $ we will take a gaseous reaction at equilibrium.
Let the gaseous reaction which is in a state of equilibrium is,
 $ aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g) $
Let us consider pA​, pB​, pC​ and pD​ as the partial pressure of A,B,C and D respectively.
Therefore,
 $ {K_c} = \dfrac{{{{[C]}^c}{{[D]}^d}}}{{{{[A]}^a}{{[B]}^b}}} - - - - (1) $ and
 $ {K_p} = \dfrac{{p{C^c}p{D^d}}}{{p{A^a}p{B^b}}} - - - - (2) $
We know that ideal gas equation is,
 $ PV = nRT $
 $ \Rightarrow P = \dfrac{{nRT}}{V} = CRT $ ,where C is the concentration ( $ C = n/V $ where n is number of moles and V is volume)
Now let us write some relations,
 $ pA = [A]RT $
 $ pB = [B]RT $
 $ pC = [C]RT $
 $ pD = [D]RT $
Now let us substitute these values in equation (2), therefore we will get,
 $ {K_p} = \dfrac{{{{[C]}^c}{{(RT)}^c}{{[D]}^d}{{(RT)}^d}}}{{{{[A]}^a}{{(RT)}^a}{{[B]}^b}{{(RT)}^b}}} $
 $ \Rightarrow {K_p} = \dfrac{{{{[C]}^c}{{[D]}^d}{{(RT)}^{(c + d) - (a + b)}}}}{{{{[A]}^a}{{[B]}^b}}} $
But we know that $ {K_c} = \dfrac{{{{[C]}^c}{{[D]}^d}}}{{{{[A]}^a}{{[B]}^b}}} $ from equation (1), so substituting this in this above equation we get,
  $ \Rightarrow {K_p} = {K_c}{(RT)^{(c + d) - (a + b)}} $
Or we can write it as,
 $ \Rightarrow {K_p} = {K_c}^{\Delta ng} $
Here, $ \Delta ng = $ Total number of moles of gaseous product $ - $ Total number of moles of gaseous reactant
Hence, we can conclude that the relation between $ {K_p} $ and $ {K_c} $ is $ {K_p} = {K_c}^{\Delta ng} $ .

Note :
 $ {K_p} $ and $ {K_c} $ both are equilibrium constant but expressed in different quantities. $ {K_p} $ and $ {K_c} $ are dimensionless because they are ratios of concentrations only. $ {K_p} $ and $ {K_c} $ are equal to each other in a reaction where the number of moles gaseous reactants is equal to the number of moles gaseous products.