What Is the Relation between Kc and Kp?

In the last post we learned about equilibrium constant Kc, for which we expressed concentration of reactants and products in terms of molarity (mols/L). If all species in a reaction mixture are gases, it becomes difficult to measure their concentration in molarity. For such a reaction mixture, it is convenient to measure concentration of their participants in terms of Partial Pressure.

You are quite familiar with the gases and you know how they exert pressure. If two or more types of gas molecules are present in a container, how will you decide which gas exerts maximum pressure? And how much pressure is exerted by each gas?

Let's try a different example, imagine that 4 members of yellow team, 6 members of green team and 10 members of orange team are jumping on the stage. Their combined efforts exert pressure on the floor of the stage. So what do you think which team contributes more? Obviously orange team contributes more because 10 out of 20 members are from Orange team. It means the team with larger fraction (team members/ total members) contributes more. Or, we can say that fraction of team members is proportional to the pressure exerted by team. Pressure exerted by an individual team is called the partial pressure of that particular team. Total pressure exerted on the stage is the sum of partial pressure of all teams.

P_total = P₁+ P₂+ P_3........

Similarly, when all participants in a reaction vessel are in gaseous state, their concentration is determined by their partial pressure. Let’s find out how we can relate partial pressure to the concentration.

From Ideal gas equation we know that:

PV= nRT

P= nRT/V

n/V is concentration in moles per litre, so

P= cRT

So we can say that:

P = [concentration of gas] RT

At constant temperature we can say that pressure of gas is proportional to its concentration:

P is proportional to c

Let’s take a reaction as example:

H_2(g) + I_2(g) ↔ 2HI_(g)           ....................(1)

For this reaction equilibrium constant will be:

K_c= [HI]² / [H] [I]

Or, if we write in terms of partial pressure, then K_c will become K_p

K_p= (P_HI)² / (P_H) (P_I)

Since P = cRT we can write:

K_p= (P_HI)² / (P_H) (P_I) = [HI]² (RT)²/ [H]RT [I]RT

K_p= K_c

Here you have seen that K_p = K_c but, it doesn't happen always. If it is not true then what is the relation between them. Let’s try to find out their relation:

a A + b B ↔ c C + d D

K_c  = [C]^c [D]^d / [A]^a [B]^b

K_p= (P_C) (P_D) / (P_A) (P_B)

K_p= (P_C) (P_D) / (P_A) (P_B) = [C]^c(RT)^c [D]^d(RT)^d / [A]^a (RT)^a[B]^b (RT)^b

K_p= K_c (RT)^(c+d)-(a+b)

K_p= K_c (RT)^Δn

Relation between Kp and Kc

Where Δn = (number of moles of gaseous products - number of moles of gaseous reactants) in a balanced chemical equation.

In equation (1) number of moles of reactants 2 and number of moles of gaseous product is 2, that’s why for this reaction K_p= K_c.

Let’s check this relation for another reaction:

N_2(g) + H_2(g) ↔ 2NH_3(g)

It is not a balanced equation since number of H isn’t equal on both sides of arrow. First we write the balanced equation:

N_2(g) + 3H_2(g) ↔ 2NH_3(g)             .................(2)

This reaction has total 4 moles of reactants and 2 moles of product, thus we get

Δn = 2-4 = -2

If the above relation is correct, we would get:

K_p = K_c(RT)^-2

Let’s try to find out:

K_c= [NH₃]² / [N] [H]³

And

K_p= (P_NH3)² / (P_N) (P_H)³

K_p= (P_NH3)² / (P_N) (P_H)³ = [NH₃]² (RT)² / [N]RT [H]³ (RT)³

K_p = K_c(RT)^-2

Yes, we have successfully proved it.

Like K_c, K_p is also a unit-less constant and since it is the ratio of pressures, its unit depends on it. For equation 1, it is unit-less quantity but for equation 2 its unit is bar^-2.

I hope you have understand the concept of K_p and its relation with K_c. Let’s try to solve a problem:

For reaction 2NOCl_(g) ↔ 2NO_(g) + Cl_2(g) value of K_c is 3.75×10^-6 at 1069K. Calculate the K_p for the reaction at the same temperature.

You know that:  K_p = K_c (RT)^Δn 

This reaction has 2 moles of reactants and total 3 moles of product, thus we get

Δn = 3-2 = 1

So,

K_p= K_c (RT)

K_p= 3.75×10^-6 (0.0831)(1069)

K_p= 0.033

Now we have learnt how to calculate equilibrium constants, but we don't know its significance. What information can we draw from it? In the next post we will explore its significance.

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