Animated Solution for Chemistry - Chemical Equilibrium: The values of $\frac{K_p}{K_C}$ for the following reactions at $300\text{ K}$ are, respectively (At $300\text{ K}$, $RT=24.62{\text{ dm}}^3{\text{ atm mol}}^{-1}$) ${\text{N}}_2(g)+{\text{O}}_2(g)\rightleftharpoons 2\text{NO}(g)$ ${\text{N}}_2{\text{O}}_4(g)\rightleftharpoons 2{\text{NO}}_2(g)$ ${\text{N}}_2(g)+3{\text{H}}_2(g)\rightleftharpoons 2{\text{NH}}_3(g)$

Connecting $K_p$ and $K_c$

In the world of chemical equilibrium, we often deal with reactions involving gases. For these reactions, we can express the equilibrium constant in two ways: $K_c$ (using molar concentrations) and $K_p$ (using partial pressures). But how are they related?

The bridge between them is given by the beautiful equation:

If we rearrange this to find the ratio $\frac{K_p}{K_c}$, we get:

Here, $\Delta n_g$ is the star of the show. It represents the change in the number of moles of gaseous species during the reaction. We calculate it simply as:

Let's look at our first chemical equation:

We need to find $\Delta n_g$. On the product side, we have $2$ moles of $\text{NO}$ gas. On the reactant side, we have $1$ mole of ${\text{N}}_2$ and $1$ mole of ${\text{O}}_2$, totaling $2$ moles.

Since $\Delta n_g$ is zero, our ratio becomes:

Moving on to the decomposition of dinitrogen tetroxide:

Here, we have $2$ moles of gaseous product (${\text{NO}}_2$) and $1$ mole of gaseous reactant (${\text{N}}_2{\text{O}}_4$).

Plugging this into our master equation:

The problem generously provides the value of $RT$ at $300\text{ K}$ as $24.62{\text{ dm}}^3{\text{ atm mol}}^{-1}$. So, the ratio for this reaction is exactly $24.62{\text{ dm}}^3{\text{ atm mol}}^{-1}$.

Finally, let's examine the famous Haber process for synthesizing ammonia:

On the product side, we have $2$ moles of ${\text{NH}}_3$. On the reactant side, we have $1$ mole of ${\text{N}}_2$ and $3$ moles of ${\text{H}}_2$, giving a total of $4$ moles.

Our ratio now becomes:

Substituting the given value of $RT$:

Calculating this square and taking the reciprocal might seem daunting, but a quick estimation ($25^2=625$, and $1/625=0.0016$) leads us straight to the answer:

Gathering our results, the values for the three reactions are $1$, $24.62$, and $1.65\times 10^{-3}$ respectively. Matching this sequence with the given options, we confidently select Option (b) as the correct answer.