This question involves selecting the correct relation between the equilibrium constants K_p and K_c for a given chemical reaction. The images provided show different mathematical relations, and we must identify which one is correct based on the general formula connecting these constants.

The fundamental relation between K_p and K_c is derived from the ideal gas law and is given by:

$K_{p} = K_{c} {(R T)}^{Δ n}$

Where:

$K_{p}$ is the equilibrium constant in terms of partial pressures.
$K_{c}$ is the equilibrium constant in terms of concentrations.
$R$ is the universal gas constant.
$T$ is the temperature in Kelvin.
$Δ n$ is the change in the number of moles of gas, calculated as (moles of gaseous products - moles of gaseous reactants).

This relation is crucial because it connects the two different ways of expressing the equilibrium constant. It shows that K_p and K_c are only equal when Δn = 0, meaning there is no change in the number of moles of gas during the reaction.

Related Topics and Formulae

Equilibrium Constants (K_p and K_c): For a general reaction: aA + bB ⇌ cC + dD

$K_{c} = \frac{{[C]}^{c} {[D]}^{d}}{{[A]}^{a} {[B]}^{b}}$

$K_{p} = \frac{{(P_{C})}^{c} {(P_{D})}^{d}}{{(P_{A})}^{a} {(P_{B})}^{b}}$

Ideal Gas Law: The partial pressure P of a gas is related to its concentration [ ] by $P = [] R T$ . Substituting this into the K_p expression leads directly to the K_p = K_c(RT)^Δn relation.

To solve the original question, you must identify which of the provided images matches this standard formula $K_{p} = K_{c} {(R T)}^{Δ n}$ .