Animated Solution for Chemistry - Chemical Equilibrium: 5.1 g ${\text{NH}}_4\text{SH}$ is introduced in 3.0 L evacuated flask at $327^{\circ }\text{C}$. 30\% of the solid ${\text{NH}}_4\text{SH}$ decomposed to ${\text{NH}}_3$ and ${\text{H}}_2\text{S}$ as gases. The $K_p$ of the reaction at $327^{\circ }\text{C}$ is ($R=0.082{\text{ atm mol}}^{-1}{\text{K}}^{-1}$, molar mass of $\text{S}=32{\text{ g mol}}^{-1}$, molar mass of $\text{N}=14{\text{ g mol}}^{-1}$)

Finding the Equilibrium Constant

Imagine you are standing in a laboratory, holding a sealed, evacuated flask with a volume of exactly $3.0\text{ L}$. Inside this flask, you place $5.1\text{ g}$ of a white crystalline solid: ammonium hydrosulfide, or ${\text{NH}}_4\text{SH}$. You then place this flask into an oven and crank the temperature up to a scorching $327^{\circ }\text{C}$.

What happens next is a beautiful microscopic dance. The solid begins to decompose, breaking apart into two invisible gases: ammonia (${\text{NH}}_3$) and hydrogen sulfide (${\text{H}}_2\text{S}$). But this isn't a one-way street. As the gases build up, they start colliding and recombining to form the solid again. Eventually, the rate of decomposition equals the rate of recombination. We have reached Chemical Equilibrium.

Our mission in this problem is to find the equilibrium constant in terms of partial pressures, known as $K_p$. I know this might sound like a daunting task with all the numbers thrown at us, but let's take a breath and break it down step by step.

The Initial State

Before we can talk about equilibrium, we need to know exactly what we started with. We have $5.1\text{ g}$ of solid ${\text{NH}}_4\text{SH}$. In chemistry, grams are rarely useful on their own; we need to speak the language of molecules, which means converting to moles.

To do this, we calculate the molar mass of ${\text{NH}}_4\text{SH}$. Using the atomic masses provided (Nitrogen = $14$, Hydrogen = $1$, Sulfur = $32$), we get:

Now, finding the initial number of moles ($n_0$) is a simple division:

So, we start our experiment with exactly $0.1\text{ mol}$ of the solid reactant.

Let's write down the chemical equation that governs this entire process:

Notice the state symbols. We have a solid decomposing into two gases. This is a heterogeneous equilibrium.

To track the moles of each substance, we use an ICE table (Initial, Change, Equilibrium). Initially ($t=0$), we have $0.1\text{ mol}$ of the solid and $0\text{ mol}$ of both gases.

The problem states a crucial piece of information: 30% of the solid decomposes. This means the change in our solid is $30\%$ of $0.1\text{ mol}$.

Because the stoichiometry of the reaction is $1:1:1$, for every $1\text{ mol}$ of solid that decomposes, $1\text{ mol}$ of ${\text{NH}}_3$ and $1\text{ mol}$ of ${\text{H}}_2\text{S}$ are formed. Therefore, at equilibrium, we have produced exactly $0.03\text{ mol}$ of ammonia and $0.03\text{ mol}$ of hydrogen sulfide.

The equilibrium constant in terms of concentration, $K_c$, is defined as the product of the equilibrium concentrations of the products divided by the reactants, each raised to the power of their stoichiometric coefficients.

But here is a massive catch, a classic trap where mistakes happen: The active mass (concentration) of a pure solid is always taken as 1. Why? Because the density of a solid remains constant regardless of how much of it is present. Therefore, the solid ${\text{NH}}_4\text{SH}$ does not appear in our $K_c$ expression!

To find these concentrations, we divide the equilibrium moles by the volume of the flask ($3.0\text{ L}$).

Now, we simply multiply them together:

Converting to $K_p$

We have $K_c$, but the question asks for $K_p$. We need the bridge that connects them:

What is $\Delta n_g$? It is the difference between the sum of the stoichiometric coefficients of the gaseous products and the gaseous reactants.

Now, we must be incredibly careful with our units. The temperature must be in Kelvin.

The universal gas constant $R$ is given as $0.082{\text{ atm L mol}}^{-1}{\text{K}}^{-1}$. Let's substitute everything into our master equation:

First, calculate the term inside the parenthesis:

Now, square it:

Finally, multiply by $K_c$:

Rounding to three significant figures, we get our final, beautiful answer:

This perfectly matches option (b).

Take a moment to appreciate what we just did. We translated a physical setup into a mathematical model, navigated the traps of heterogeneous equilibrium, and arrived at a precise constant that defines the universe inside that flask.

As a thought experiment, ask yourself: what would happen to the pressure inside the flask if we injected an inert gas at constant volume? According to Le Chatelier's principle, the partial pressures of the reacting gases wouldn't change, and neither would $K_p$! Keep exploring these "what ifs"—that is where true mastery of chemistry lies.