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K_P and K_c

Key Concepts

K_P is an equilibrium constant calculated using the partial pressure of each reactant and product.
K_c is an equilibrium constant calculated using the molarities (concentration) of each reactant and product.
For the gaseous reaction: kA_(g) + lB_(g) mC_(g) + nD_(g)

K_P = P^m_CPⁿ_D

P^k_AP^l_B

P_A = partial pressure of gas A (usually in atmospheres)
P_B = partial pressure of gas B (usually in atmospheres)
P_C = partial pressure of gas C (usually in atmospheres)
P_D = partial pressure of gas D (usually in atmospheres)
The relationship between K_P and K_c is
K_P = K_c(RT)^Δn
R = Ideal Gas Constant = 0.0821 (gas pressures in atm)
T = temperature in Kelvin
Δn = change in the number of moles of gas = moles of products - moles of reactants
For the reaction above, Δn = (m + n) - (k + l)

Example : Calculating Equilibrium Partial Pressure

A container at 800^oC was filled NOCl gas which decomposes to form NO gas and chlorine gas.
The equilibrium partial pressure of NOCl was 0.657atm, what was the partial pressure of NO gas?
K_P = 1.8 x 10^-2 for the reaction NOCl_(g)

NO_(g) + ½Cl_2(g)

Write the decomposition equation: NOCl_(g) NO_(g) + ½Cl_2(g)
Write the expression for K_P

K_P = P_NOP^½_Cl₂

P_NOCl
Write expressions for the equilibrium partial pressure of each species:
P_NOCl = 0.657atm
P_NO = xatm
P_Cl₂ = ½xatm
Subsitute these values into the expression for K_P

1.8 x 10^-2 = x(½x)^½

0.657
Simplify the expression for K_P and solve for x
1.8 x 10^-2 x 0.657= (½x³)^½
0.0118 = (½x³)^½
(0.0118)² = ½x³
1.4 x 10^-4 = ½x³
1.4 x 10^-4 x 2 = x³
2.8 x 10^-4 = x³
³√2.8 x 10^-4 = x
0.065 = x = P_NO
Check your answer by substituting the values back into the equation:
P_NOCl = 0.657atm
P_NO = 0.065atm
P_Cl₂ = ½ x 0.065 = 0.0325atm

K_P = P_NOP^½_Cl₂

P_NOCl

K_P = 0.065(0.0325)^½

0.657

K_P = 1.8 x 10^-2

Example : Converting K_c to K_P

For the equilibrium: 2NOCl_(g)

2NO_(g) + Cl_2(g)
K_c = 3.75 x 10^-6 at 796^oC.
Calculate K_P for this reaction at this temperature.

Write the expression to convert K_c to K_p:
K_P = K_c(RT)^Δn
Extract the relevant data from the question:
K_P = ?
K_c = 3.75 x 10^-6
R = 0.0821 (Ideal Gas Constant)
T = 796^oC = 796 + 273 = 1069K
Δn = (2 + 1) - 2 = 1
Substitute the values into the equation:
K_P = K_c(RT)^Δn
K_P = 3.75 x 10^-6(0.0821 x 1069)¹
K_P = 3.29 x 10^-4