Relationship between Kp and Kc: Derivation, Equilibrium Constants & Solved Examples

Relationship between Kp and Kc is given by K_p= K_c (RT)^Δn , where K_p and K_c are the equilibrium constants for an ideal gaseous mixture. K_p is the equilibrium constant used to measure equilibrium concentrations represented in atmospheric pressure.

When equilibrium concentrations are expressed in molarity, K_c is the equilibrium constant used. The equilibrium constant, denoted as k, is a number that helps to describe the relationship between the reactants and the number of products present at equilibrium for a reversible chemical reaction at a given temperature.

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Read More: Important Questions on Equilibrium

Key Terms: K_p, K_c, Equilibrium, Gas Constants, Molarity, Equilibrium Constant, Chemical Reaction, Products, Reactants

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Relationship between K_p and K_c

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Consider a generalised reaction as follows:

A + B ⇆ C + D

From the law of mass action, the rate at which A and B react is directly proportional to A and B.

• Thus the rate of forward reaction = k_f AB, where k_f is the velocity constant for the forward reaction.
• Similarly if k_b is the rate constant for backward reaction then, the rate of backward reaction = k_bCD
• At equilibrium, rate of forward reaction is equal to rate of backward reaction
• Therefore, k_fAB  = k_bCD
• Thus, \(\frac{[C][[D]}{[A][B]} = \frac{k_f}{k_b}\)
• At constant temperature, k_f abd k_b are equal, thus \(\frac{k_f}{k_b}\)= k where k is the equilibrium constant. 

Now consider another generalised reaction,

aA + bB ⇆ cC + dD

Here, a moles of reactant A reacts with b moles of reactant B to give c moles of reactant C and d moles of reactant D. This can be understood with an example:

2A (g)+ B (g) ⇆ 2 C (g)

All reactants and product are in gaseous phase. To understand the relation between K_p and K_c, let us first understand what are K_p and K_c. 

What is K_p?

K_p is the equilibrium constant determined from the partial pressure of the equation of a reaction. It's a mathematical expression to determine the relation between product and reactant pressures. Although it connects the pressures, it is a unitless number. 

We can represent K_p formula as:

K_p = \({{P^2_C}\over P^2_A P_B}\)

What is K_c?

K_c is the equilibrium constant for a reversible reaction, which depicts the ratio of the equilibrium concentrations of products over the concentrations of reactants, where each is raised to the power of their stoichiometric coefficients. The Kc formula can be represented as:

K_c = \([C]^c[D]^d\over[A]^a[B]^b\)

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Derivation of Relation between K_p and K_c

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To derive the K_p and K_c relationship, let us suppose there is a reversible reaction, 

qQ(g) + rR(g) \(\rightleftharpoons\) sS(g) +tT(g)

Here, q moles of reactant Q reacts with r moles of reactant R to produce s moles of reactant S and t moles of reactant T.

To derive the relation between K_p and K_c for a general equilibrium reaction, we get

K_c = \((S)^s(T)^t \over(Q)^q(R)^r\) -------- 1)

Now we know that K_p formula in terms of partial pressure can be written as: 

K_p = \(P(S)^sP(T)^t \over P(Q)^qP(R)^r\) ------- 2)

For an ideal gas, we have

PV = nRT

P = (n/V)RT = CRT

Here, C refers to the concentration 

Therefore,

• P_q = [Q]RT
• P_r = [R]RT
• P_s = [S]RT
• P_t = [T]RT

Now, substituting the P_Q, P_R, P_s, P_t in equation 2

We get, 

K_p = \(P(S)^sP(T)^t \over P(Q)^qP(R)^r\)

= \([S]^s(RT)^s[T]^t (RT)^t\over [Q]^q(RT)^q[R]^r(RT)^r\) 

= \([S]^s[T]^t (RT)^{t+s}\over [Q]^q[R]^r(RT)^{r+q}\)

= \({[S]^s[T]^t \over [Q]^q[R]^r.} (RT)^{{(t+s)} -{(q+r)}}\)

K_p = \({[S]^s[T]^t \over [Q]^q[R]^r.} (RT)^{\bigtriangleup ng}\)

We, know that K_c = \((S)^s(T)^t \over(Q)^q(R)^r\)

Therefore K_p and K_c relation can be denoted as, 

K_p = K_c(RT)^{\(\bigtriangleup \)n} 

Where \(\bigtriangleup\)n = no. of moles of gaseous products - no. of moles of gaseous reactants for balanced equations

R = 0.082062 L.atom.K^-1 mol^-1 

To derive the relation between K_p and K_c, when there is no change in the no. of gas molecules, n = 0

K_p = K_c

Hence, generally, the relationship between K_p and K_c can be represented as:

K_p = K_c (RT)^{\(\bigtriangleup \)n} 

K_c = K_p (RT)^{-\(\bigtriangleup \)n} 

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Factors Affecting K_p and K_c Relation

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The relation between Kp and Kc changes based on the value of number of moles of the gas molecules involved in the reaction. The three possible scenarios are listed below:

Case 1

When the change in the number of gas molecules is zero, i. e. Δn = 0, then K_p= K_c. 

Case 2

When the change in number of gas molecules is positive, i.e.  Δn > 0, then K_p > K_c. 

Case 3

If the change in the number of moles of the gas molecules is negative, i.e. Δn < 0, then K_p < K_c.

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Equilibrium Constant Units

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Consider a generalised equation as follows:

aA + bB ⇆ cC + dD

Unit of K_c

K_c for the above reaction can be expressed as:

\(K_c= \frac{CC^c.CD^d}{CA^a.CB^b}\)

\(= \frac{(Mol L^-1)^{c+d}}{(Mol L^-1)^{a+b}}\)

\(= (Mol L^-1)^{\Delta n}\)

Thus the unit of K_c= Mol/L

Unit of Kp

Kp for the above reaction can be expressed as:

\(K_p= \frac{PC^c.PD^d}{PA^a.PB^b}\)

\(= \frac{(atm)^{c+d}}{(atm)^{a+b}}\)

\(= (atm)^{(c+d)-(a+b)}\)

\(= (atm)^{\Delta n}\)

Thus the unit of Kp is \( (atm)^{\Delta n}\) or \( (bar)^{\Delta n}\)

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Gas Equilibrium Constants

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K_p and K_c are the equilibrium constants of gaseous mixtures. K_c is defined with respect to molar concentrations whereas K_p is defined with respect to the partial pressure of gases that exist inside a closed system. The ways to write gas equilibrium constants are:

• In the equilibrium equations, the arrows point both ways ( \(\rightleftharpoons\) ), left side is associated with the reactants and the right side with the products.
• The products are the ones that lie on the top of the fraction as the numerator.
• The reactants are the ones that lie on the bottom of the fraction as the denominator. 
• The concentrations of the reactants and products are always raised to the coefficient power in the balanced chemical equation.
• If any of the products or reactants are solids or liquids, their concentrations are denoted as one as they are pure substances. 
• Equilibrium constant, k, is the ratio of product of molar concentrations of products and reactants, raised to the power of its stoichiometric coefficients.

Read More: Vapor Pressure

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Things to Remember

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• K_p is the equilibrium constant in atmospheric pressure determined from the partial pressures of the equation of a reaction.
• Kc is the equilibrium constant, in molarity, which depicts the ratio of the equilibrium concentrations of products over the concentrations of reactants.
• Generally, the relation between K_p and K_c can be represented as:
  • K_p = K_c (RT)^{\(\bigtriangleup \)n} 
  • K_c = K_p(RT)-^{\(\bigtriangleup \)n}

Read More: 

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