Relation Between Kp and Kc

The relation between K_p and K_c is expressed by the equation 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 that is used when equilibrium concentrations are expressed in atmospheric pressure and K_c is the equilibrium constant that is used when equilibrium concentrations are expressed in molarity.

The main difference between K_p & K_c is that K_c is the equilibrium constant given by the terms of concentration and K_p is the equilibrium constant given by the terms of pressure. More details about the relation between K_p and K_c are covered in the article.

Relation Between K_p and K_c

The basic relation between K_p and K_c is that they are the equilibrium constants of an ideal gaseous mixture in a reversible reaction.

K_p = K_c(RT)^Δn

or K_c= K_p(RT)^-Δn

K_p is the equilibrium constant of pressure and K_c is the equilibrium constant of concentration. R is the universal gas constant (8.314 Jmol^-1K^-1), T is the temperature in degree Kelvin (°K) and Δn is the change in the number of moles of gas molecules i.e. the difference between the total moles of gas products and the total moles of gas reactants.

Derivation of Relation Between K_p and K_c

Consider a reversible reaction to derive the relation between K_p and K_c.

qQ(g) + rR(g) ⇌ 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 denote the relation between K_p and K_c for a general equilibrium reaction, we have the equation:

K_c= (S)^s(T)^t / (Q)^q(R)^r

The K_p formula in terms of partial pressure can be written as:

K_p= P(S)^sP(T)^t / P(Q)^qP(R)^r

For an ideal gas, 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, and P_t we get:

K_p= P(S)^sP(T)^t / P(Q)^qP(R)^r

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

= [S]^s[T]^t(RT)^t+s / [Q]^q[R]^r(RT)^r+q

= [S]^s[T]^t / [Q]^q[R]^r * (RT)^(t+s)-(r+q)

K_p= [S]^s[T]^t / [Q]^q[R]^r * (RT)^Δng

We already know that K_c= [S]^s[T]^t / [Q]^q[R]^r

So the relation between K_p and K_c can be denoted as:

K_p= K_c (RT)^Δn

Where Δn = no. of moles of gaseous products - no. of moles of gaseous reactants for balanced equations and R = 0.082062 L.atom.K^-1 mol^-1

When there is no change in the number of gas molecules i.e. n = 0;

K_p=K_c

Hence, the general relation between K_p and K_c can be represented as:

K_p = K_c(RT)^Δn

or K_c= K_p(RT)^-Δn

Quick Insights On K_p & K_c

K_p is the equilibrium constant expressed as a ratio between the pressure of products and reactants whereas K_c is the equilibrium constant expressed as a ratio between concentrations of products and reactants in a reaction mixture. More details about K_p and K_c are shown below:

What is K_p?

K_p is the equilibrium constant given as a ratio between the pressure of products and reactants in a reaction mixture. This equilibrium constant is only applicable to gaseous reaction mixtures. K_p depends on the partial pressure of gaseous components in the reaction mixture.

pP + qQ ↔ rR + sS

The equilibrium constant for the above reaction can be expressed as:

K_p= p_R^r.p_S^s/ p_P^p.p_Q^q

Here, “p” indicates the partial pressure. p_P, p_Q, p_R & p_S are the partial pressures of the gas components P, Q, R and S. The exponents “p’, “q”, “r” & “s” are the stoichiometric coefficients of each product and reactant in the chemical equation.

What is K_c?

K_c is the equilibrium constant expressed as a ratio between concentrations of products and reactants in a reaction mixture. The molar concentrations of the components are utilized for the expression of K_c.

aA + bB ↔ cC + dD

The equilibrium constant for the aforementioned reaction can be written as:

K_c = [C]^c[D]^d / [A]^a[B]^b

Here, [A], [B], [C] and [D] are the concentrations of reactants A and B and products C and D. The exponents “a’, “b”, “c” & “d” are the stoichiometric coefficients of each product and reactant in the chemical equation. In the expression of K_c, the concentrations of the products and reactants are raised to the powers equal to their stoichiometric coefficients.

Example:

Q: Find the equilibrium constant between silver and copper ions.

Ans: Cu(s) + 2Ag⁺ ⇆ Cu²⁺(aq) + 2Ag(s)

This equation can be written as:

K_c = [Cu²⁺] / [Ag⁺]²

In this equation, the solid silver and copper are eliminated. Also, the coefficient for the copper ions is converted into an exponent after the equilibrium calculation is done.

Factors Affecting the Relation Between K_p and K_c

The relation between K_p and K_c changes on the basis of the value of the number of moles of the gas molecules that are involved in the reaction. The three possible scenarios that affect the relation between K_p and K_c are shown below:

Scenario 1

If the change in the number of gas molecules is 0, i. e. If Δn = 0, then the relation between K_p and K_c is K_p= K_c.

Scenario 2

If the change in the number of gas molecules is positive, i.e. if Δn > 0, then the relation between K_p and K_c is K_p > K_c.

Scenario 3

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

Difference Between K_p & K_c

The equilibrium constant of a reaction mixture demonstrates the ratio between products and reactants present in that reaction mixture in terms of either concentration (known as K_c) or partial pressure (known as K_p). After knowing the relation between K_p and K_c, find out the basic difference between K_p & K_c from the table below:

Difference Between K_p & K_c

Kp	Kc
Kp is the equilibrium constant defined as a ratio between the pressure of the products and reactants in a reaction mixture.	Kc is the equilibrium constant defined as a ratio between the concentrations of the products and reactants in a reaction mixture.
Kp can be used only for gaseous reaction mixtures.	Kc can be used for both gaseous or liquid reaction mixtures.
Kp is given by units of pressure.	Kc is given by units of concentration.

FAQs

Q1: What is the relation between K_p and K_c?

Ans: Both K_p and K_c are the equilibrium constants of the ideal gaseous mixture in a reversible reaction. They are directly proportional to each other expressed by the equation K_p = K_c(RT)^Δn.

Q2: What is the difference between K_p and K_c?

Ans: The key difference between K_p and K_c is that K_p is the equilibrium constant represented by the terms of pressure whereas K_c is the equilibrium constant represented by the terms of concentration.

Q3: What are the basic properties of gases?

Ans: Gases have three fundamental properties.

• Gases are very easy to compress.
• They occupy more space than solids/liquids.
• Gases typically expand to fill the spaces around them.

Q4: What is the ideal gas equation?

Ans: PV = nRT

Q5: What is pressure?

Ans: Pressure is defined as the force per unit area expressed as P= F/A

Q6: What is temperature?

Ans: Temperature is the measurement of heat.

Q7: What is an ideal gas?

Ans: An ideal gas is a hypothetical gas made up of a group of randomly-propagating point particles that intermix only through elastic collisions.