Animated Solution for Chemistry - Chemical Equilibrium: For the reaction, ${\text{Fe}}_2\text{N}(s)+\frac{3}{2}{\text{H}}_2(g)\rightleftharpoons 2\text{Fe}(s)+{\text{NH}}_3(g)$

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 bridges the gap between measuring a system using partial pressures and measuring it using molar concentrations.

Whenever you see a problem asking to relate $K_p$ and $K_c$, your brain should immediately flash to the master equation:

Here, $R$ is the universal gas constant, $T$ is the absolute temperature, and the star of the show is $\Delta n_g$. This term represents the change in the number of moles of gaseous species during the reaction.

The most common trap students fall into is counting the moles of solids or liquids. Remember, solids and pure liquids have a constant active mass and do not exert partial pressure in the way gases do. We must strictly ignore them!

Let's look at our specific reaction:

On the reactant side, we have solid iron nitride (${\text{Fe}}_2\text{N}$) and gaseous hydrogen (${\text{H}}_2$). We only care about the gas, so our gaseous reactant moles are:

On the product side, we have solid iron ($\text{Fe}$) and gaseous ammonia (${\text{NH}}_3$). Again, ignoring the solid, our gaseous product moles are:

Now, we calculate $\Delta n_g$ by subtracting the reactant gas moles from the product gas moles:

With $\Delta n_g$ in hand, we substitute it back into our master equation:

The question asks us to find the expression for $K_c$. To isolate $K_c$, we need to move the $(RT{)}^{-1/2}$ term to the other side. Dividing by a negative exponent is mathematically identical to multiplying by the positive exponent:

And there we have it! A clean, elegant algebraic manipulation that leads us straight to the correct option. Always remember to hunt only for the gases, and the rest is just basic algebra.