Animated Solution for Chemistry - Chemical Equilibrium: For the reaction, $CO(g)+C{l}_2(g)\rightleftharpoons COC{l}_2(g)$, the $K_p/K_c$ is equal to

The relationship between the equilibrium constants $K_p$ and $K_c$ is one of the most fundamental and frequently tested concepts in chemical equilibrium. It beautifully connects the macroscopic world of partial pressures with the microscopic world of molar concentrations.

When dealing with gas-phase reactions, we can express the equilibrium constant in terms of partial pressures ($K_p$) or molar concentrations ($K_c$). These two are not independent; they are linked by the ideal gas law, $PV=nRT$.

By substituting concentration $C=\frac{n}{V}=\frac{P}{RT}$ into the equilibrium expression, we arrive at the master equation:

Here, $R$ is the universal gas constant, $T$ is the absolute temperature, and $\Delta n_g$ is the star of the show—the change in the number of moles of gas.

To find $\Delta n_g$, we must look exclusively at the gaseous species in our balanced chemical equation. Solids and liquids do not expand to fill the container in the same way gases do, so their concentrations remain effectively constant and are incorporated into the equilibrium constant itself.

Let's analyze our specific reaction:

First, we count the moles of gaseous products ($n_P$). We have exactly $1$ mole of phosgene gas ($COC{l}_2$).

Next, we count the moles of gaseous reactants ($n_R$). We have $1$ mole of carbon monoxide ($CO$) and $1$ mole of chlorine gas ($C{l}_2$), giving us a total of $2$ moles.

Now, we calculate the difference:

With $\Delta n_g$ in hand, the rest is straightforward algebra. We substitute $\Delta n_g=-1$ back into our master equation:

The question asks for the ratio of $K_p$ to $K_c$. By simply dividing both sides by $K_c$, we get:

Using the property of negative exponents, $(RT{)}^{-1}$ is mathematically identical to $\frac{1}{RT}$. Therefore, our final, elegant result is:

This perfectly matches option (a). The key takeaway here is to always be meticulous when counting moles—ensure the equation is balanced and strictly ignore any non-gaseous species!