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,
 $K_c={\displaystyle \frac{{[C]}^c{[D]}^d}{{[A]}^a{[B]}^b}}----(1)$ and
 $K_p={\displaystyle \frac{p{C}^{c}p{D}^d}{p{A}^{a}p{B}^b}}----(2)$
We know that ideal gas equation is,
 $PV=nRT$
 $\Rightarrow P={\displaystyle \frac{nRT}{V}}=CRT$ ,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,
 $K_p={\displaystyle \frac{{[C]}^c{(RT)}^c{[D]}^d{(RT)}^d}{{[A]}^a{(RT)}^a{[B]}^b{(RT)}^b}}$
 $\Rightarrow K_p={\displaystyle \frac{{[C]}^c{[D]}^d{(RT)}^{(c+d)-(a+b)}}{{[A]}^a{[B]}^b}}$
But we know that $K_c={\displaystyle \frac{{[C]}^c{[D]}^d}{{[A]}^a{[B]}^b}}$ 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,
 $\Rightarrow K_p={K_c}^{\mathrm{\Delta }ng}$
Here, $\mathrm{\Delta }ng=$ 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 $K_p={K_c}^{\mathrm{\Delta }ng}$ .

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.