Animated Solution for Chemistry - Chemical Equilibrium: Consider the reaction, $N_2{O}_4(g)\rightleftharpoons 2N{O}_2(g)$. The temperature at which $K_C=20.4$ and $K_p=600.1$, is ............ K. (Round off to the nearest integer). [Assume all gases are ideal and $R=0.0831{\text{ L bar K}}^{-1}{\text{ mol}}^{-1}$].

The relationship between the equilibrium constant in terms of pressure ($K_p$) and concentration ($K_C$) is one of the most fundamental and frequently tested concepts in chemical equilibrium. Let's dive into this problem and see how these two constants are intimately connected by the ideal gas law.

Decoding the Given Data

Imagine a closed vessel where a chemical tug-of-war is happening. On one side, we have the colorless dinitrogen tetroxide gas, $N_2{O}_4(g)$. On the other side, it's breaking apart into two molecules of the reddish-brown nitrogen dioxide gas, $N{O}_2(g)$.

The reaction is given as:

We are provided with the equilibrium constants for this specific state: - $K_p=600.1$ - $K_C=20.4$

We are also given the universal gas constant $R=0.0831{\text{ L bar K}}^{-1}{\text{ mol}}^{-1}$. Our mission is to find the exact temperature ($T$) at which this specific equilibrium state exists.

Bridging $K_p$ and $K_C$

How do we connect a constant based on pressure with one based on molarity? The bridge between them is derived directly from the ideal gas equation ($PV=nRT$).

The master equation is:

This elegant formula tells us that the difference between $K_p$ and $K_C$ is entirely dependent on the temperature, the gas constant, and a crucial stoichiometric factor known as $\Delta n_g$.

Finding $\Delta n_g$

Before we can plug numbers into our master equation, we need to determine $\Delta n_g$. This term represents the change in the number of moles of gaseous substances during the reaction.

Mathematically, it is:

Looking closely at our balanced chemical equation, we have $2$ moles of $N{O}_2$ gas on the product side and $1$ mole of $N_2{O}_4$ gas on the reactant side.

Therefore:

This positive value tells us that the reaction produces more gas molecules than it consumes, which is why $K_p$ is significantly larger than $K_C$.

Isolating Temperature

Now, let's bring all our known values together and substitute them into the master equation:

We have a straightforward linear equation. Let's isolate the temperature $T$. First, we divide both sides by $20.4$:

Finally, we divide by the gas constant $0.0831$ to find $T$:

The question asks us to round off to the nearest integer. Thus, the temperature at which this equilibrium exists is $354\text{ K}$.

Always remember, the key to mastering these problems is carefully calculating $\Delta n_g$ and ensuring your units for the gas constant $R$ align perfectly with the pressure units implied by $K_p$.