Question

A sample of \(\mathrm{NH}_{4} \mathrm{HS}(\mathrm{s})\) is placed in a \(2.58 \mathrm{L}\) flask containing 0.100 mol \(\mathrm{NH}_{3}(\mathrm{g}) .\) What will be the total gas pressure when equilibrium is established at \(25^{\circ} \mathrm{C} ?\) $$\begin{aligned} \mathrm{NH}_{4} \mathrm{HS}(\mathrm{s}) \rightleftharpoons \mathrm{NH}_{3}(\mathrm{g})+\mathrm{H}_{2} \mathrm{S}(\mathrm{g}) & \\ K_{\mathrm{p}} &=0.108 \text { at } 25^{\circ} \mathrm{C} \end{aligned}$$

Short answer

The total gas pressure of the system at equilibrium is 0.1218 atm.

Step-by-step solution

Writing the Equation

The equilibrium equation is given as: \(\mathrm{NH}_{4} \mathrm{HS}(\mathrm{s}) \rightleftharpoons \mathrm{NH}_{3}(\mathrm{g}) + \mathrm{H}_{2} \mathrm{S}(\mathrm{g})\).

Express the Equilibrium Constant

The equilibrium constant \(K_p\) for the reaction is expressed as \( K_p = \frac{[NH_3][H_2S]}{[NH_4HS]} \). However, since \(NH_4HS\) is a solid, its concentration doesn't appear in the expression. Thus, in this case, considering pressure, \( K_p = [NH_3][H_2S] \).

Set up the ICE Table

An ICE (Initial, Change, Equilibrium) table can be utilized to visualize what's happening in the reaction.\n- Initially (I), there were 0.100 mol of \(NH_3\), while the amount of \(H_2S\) isn't provided.\n- The change (C) that occurs as the reaction approaches equilibrium can be expressed as -x for \(NH_3\) and +x for \(H_2S\).\n- At equilibrium (E), there would be (0.100 - x) mol of \(NH_3\) and (x) mol of \(H_2S\).

Solve for 'x'

For \(K_p = [NH_3][H_2S]\), substitute the equilibrium conditions into the equation, yielding: \(0.108 = (0.100 - x)x \). This equation can be rearranged into \(x^2 - 0.100x + 0.108 = 0\). Solving this quadratic equation yields two possible solutions, but only the physically meaningful one (x > 0) is considered. The solution for x is approximately 0.0609.

Calculate Total Pressure

The total pressure at equilibrium will be the sum of the partial pressures of the individual components. Therefore, the pressure of the \(NH_3\) and \(H_2S\) gases at equilibrium are equivalent to the 'x' mol/L value found earlier which is 0.0609. So, \(P_{total} = P_{NH_3} + P_{H_2S} = 2x = 2 * 0.0609 = 0.1218 atm.\)

Key concepts

Chemical Equilibrium

In the realm of chemistry, a state of balance called chemical equilibrium is reached when the rate of the forward reaction equals the rate of the reverse reaction. This does not imply that the reaction has stopped, but rather that the concentrations of the reactants and products remain constant over time.

Imagine you’re at a busy market where people buy and sell fruit. The amount of fruit doesn't change when the number of people buying fruit is the same as the number of people selling it. Similarly, in a chemical reaction at equilibrium, the amount of reactants and products remains unchanged. This state is dynamic, which means that there's continuous movement, with molecules reacting all the time, but there is no net change in concentrations.

In our sample exercise involving ammonia and hydrogen sulfide, reaching equilibrium means the rate at which ammonia turns into hydrogen sulfide equals the rate at which hydrogen sulfide reverts to ammonia.

ICE Table Method

To deal with the math part of equilibrium, chemists often use a tool called the ICE table method, which helps organize the initial concentrations, the changes that occur as the system approaches equilibrium, and the concentrations at equilibrium itself. ICE stands for Initial, Change, and Equilibrium.

The 'ICE' part helps us create a snapshot of what’s happening in a chemical reaction over time. It's like following a recipe but keeping track of how each ingredient changes as you prepare your dish. During step 3 of our sample problem, the ICE table helped in visualizing how ammonia (NH₃) gets consumed while hydrogen sulfide (H₂S) is produced up until the point where the system is in balance (equilibrium).

Partial Pressure

Now, let’s talk about gas molecules. Each gas in a mixture exerts its own pressure, which we call partial pressure. It's a bit like each person in a room contributing to the overall noise level with their voice.

In our problem, ammonia and hydrogen sulfide are gases. Their partial pressures contribute to the total pressure in the flask. If they were the only ones in a 2.58-liter ‘room’, the total pressure equals the sum of their individual pressures. The calculation is similar to adding up the noise from two separate sources to find the overall noise level.

Equilibrium Constant (Kp)

Last, let's illuminate the concept of equilibrium constant (K_p). This is a number that tells us the ratio of the products to reactants at equilibrium under a given set of conditions, specifically in terms of pressure when dealing with gases. It's like a score that tells us whether the tea party is favoring tea (products) or scones (reactants).

In the exercise, we looked at the equilibrium constant in terms of the partial pressures of the gaseous products, ammonia and hydrogen sulfide, for a solid-gas reaction. A K_p value of 0.108 means at 25°C, the product of the pressures of ammonia and hydrogen sulfide, when multiplied together, equals 0.108 atm². This K_p guides us in predicting the direction of the reaction and calculating the equilibrium concentrations or pressures of the reactants and products.