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.