Problem 110 - Chemistry The Central Science

Question

A sample of \(7.5 \mathrm{~L}\) of \(\mathrm{NH}_{3}\) gas at \(22{ }^{\circ} \mathrm{C}\) and 735 torr is bubbled into a 0.50 - \(\mathrm{L}\) solution of \(0.40 \mathrm{M} \mathrm{HCl}\). Assuming that all the \(\mathrm{NH}_{3}\) dissolves and that the volume of the solution remains \(0.50 \mathrm{~L},\) calculate the \(\mathrm{pH}\) of the resulting solution.

Step-by-Step Solution

Answer

The resulting solution after bubbling NH3 gas into the HCl solution will be acidic with a pH less than 7. This is because there is no remaining NH3 to act as a base competing with H+ ions or any other pH buffering component in the solution.

Step 1: Calculate the moles of ammonia

First, let's use the Ideal Gas Law (PV=nRT) to find the moles of ammonia. We are given the volume (V), temperature (T), and pressure (P) of the ammonia gas. R (gas constant) = 62.36 L·Torr/mol·K (use this value since the pressure is given in Torr) Rearrange the Ideal Gas Law to solve for n (moles of ammonia): n = PV / RT n = (735 Torr × 7.5 L) / (62.36 L·Torr/mol·K × (22 °C + 273.15)) n = 0.30093 moles of NH3

Step 2: Calculate the moles of hydrochloric acid

Now let's find the moles of HCl in the solution. We are given the volume and molarity of the HCl solution. moles of HCl = volume × molarity moles of HCl = 0.50 L × 0.40 mol/L moles of HCl = 0.20 moles of HCl

Step 3: Determine the balanced chemical equation

The reaction between ammonia and hydrochloric acid produces ammonium chloride (NH4Cl). Let's write the balanced chemical equation for the reaction: NH3 + HCl -> NH4+ + Cl-

Step 4: Perform stoichiometric calculations

Now let's perform stoichiometric calculations to find the amounts of NH4+ ions and leftover HCl in the solution after the reaction. Since ammonia is the limiting reactant, we will calculate the moles of NH4+ produced and subtract the moles of ammonia that reacted from the moles of hydrochloric acid to find the leftover moles of HCl. moles of NH4+ = 0.30093 moles (as it is 1:1 stoichiometry) moles of leftover HCl = moles of initial HCl - moles of reacted NH3 moles of leftover HCl = 0.20 moles - 0.30093 moles moles of leftover HCl = - 0.10093 moles (Note: Since we have a negative value, this means that there is no leftover HCl and we have 0 leftover moles of HCl)

Step 5: Calculate the concentration of ions in the solution

Now that we have the moles of NH4+ ions and leftover HCl in the solution, let's calculate their concentrations. Since the volume of the solution remains 0.50 L, we will divide the moles by 0.50 L to find the concentration of the ions. concentration of NH4+ = moles of NH4+ / volume of solution concentration of NH4+ = (0.30093 moles) / 0.50 L concentration of NH4+ = 0.60186 M

Step 6: Calculate the pH of the resulting solution

Now that we have the concentration of NH4+ ions, we can calculate the pH of the solution using the following equations: \(K_a = \frac{[H^+][\mathrm{A}^-]}{ [\mathrm{HA}] }\) \(pK_w = pK_a + pK_b\) \(K_a\) for NH4+ is \(5.6\times{10^{-10}}\), given that: \(K_b\) for NH3 is \(1.8\times{10^{-5}}\) \(10^{-14} = pK_a + pK_b\) \(10^{-14} = K_a + 1.8\times 10^{-5}\) \(K_a = 5.6\times{10^{-10}} = \frac{[H^+][\mathrm{NH_3}]}{ [\mathrm{NH_4^+}] }\) Now we can solve for the concentration of H+ ions in the solution: \([H^+] = \frac{K_a [\mathrm{NH_4^+}]}{ [\mathrm{NH_3}] }\) As there is no HCl left, all NH3 reacted, thus, the remaining concentration of NH3 is 0: \([H^+] = \frac{(5.6\times{10^{-10}}) (0.60186 M)}{ 0 }\) Since the division by zero yields an undefined result, we can assume the resulting solution is acidic with a pH less than 7, as there is no remaining NH3 to act as a base competing with H+ ions or any other pH buffering component in the solution.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is an essential tool in chemistry for determining the state of a gas under specific conditions. The formula for the Ideal Gas Law is given by \(PV = nRT\). In this equation:

• \(P\) represents pressure.
• \(V\) stands for volume.
• \(n\) signifies the number of moles of gas.
• \(R\) is the ideal gas constant.
• \(T\) corresponds to temperature measured in Kelvin.

This equation can be rearranged to solve for any variable when the appropriate values are known. It is crucial to use consistent units, where pressure is usually given in atmospheres or Torr, and temperature in Kelvin.
In this exercise, we used the Ideal Gas Law to find the moles of ammonia gas involved in the reaction. By rearranging \(n = \frac{PV}{RT}\), we are able to calculate how many moles of the gas contribute to the reaction. This number is pivotal for stoichiometric calculations. Understanding this law helps in comprehending how gases behave under different conditions, making it easier to predict how changes in one variable affect others.

Stoichiometry

Stoichiometry is the area of chemistry that involves calculating the quantities of reactants and products in a chemical reaction. It is based on the balanced chemical equation for the reaction, which gives the ratio in which the reactants combine and the products form.
In this scenario, we used stoichiometry to determine how much ammonium chloride is formed and whether any excess of hydrochloric acid remains after the reaction with ammonia. The balanced equation \( ext{NH}_3 + ext{HCl} \rightarrow ext{NH}_4^+ + ext{Cl}^-\) shows a 1:1 stoichiometric ratio, meaning one mole of \( ext{NH}_3\) reacts with one mole of \( ext{HCl}\) to produce one mole of \( ext{NH}_4^+\).
With stoichiometry, you can calculate how much of each substance is used up and how much is formed. It's crucial for determining the limiting reactant, which in this case was ammonia, as it was completely consumed. Stoichiometry ensures that reactions are understood quantitatively, helping to predict theoretical yields and evaluate reaction efficiency.

Acid-Base Reaction

Acid-base reactions are fundamental chemical processes where an acid reacts with a base to produce water and a salt, often leading to changes in pH. In our problem, ammonia \( ext{NH}_3\), a base, reacts with hydrochloric acid \( ext{HCl}\), a strong acid.
This reaction results in the formation of ammonium chloride \( ext{NH}_4 ext{Cl}\), which consists of ammonium ions \( ext{NH}_4^+\) and chloride ions \( ext{Cl}^-\). The reaction between ammonia and hydrochloric acid can be described with the equation:

• \( ext{NH}_3 + ext{HCl} \rightarrow ext{NH}_4^+ + ext{Cl}^-\)

In this acid-base reaction, all of the \( ext{NH}_3\) is converted into \( ext{NH}_4^+\). Since \( ext{NH}_3\) neutralizes \( ext{HCl}\), the solution initially becomes less acidic compared to any remaining \( ext{HCl}\) being present. However, as we calculate further, we realize there's no leftover \( ext{HCl}\), so only \( ext{NH}_4^+\) ions affect the solution's pH.
Understanding acid-base reactions helps illustrate how different substances can alter the acidity or basicity of a solution, which is essential for predicting the pH changes.

Chemical Equilibrium

Chemical equilibrium refers to the state in a chemical reaction where the forward and backward reactions occur at the same rate, and the concentrations of reactants and products remain constant. It is a dynamic process described by equilibrium constants.
In our context, although not reaching true equilibrium due to complete reaction of ammonia, the concept is crucial when considering the dissociation of formed ammonium ions (\( ext{NH}_4^+\)). These ions seek equilibrium with their dissociated forms. Such equilibrium is depicted by the equilibrium constant \(K_a\) of \( ext{NH}_4^+\), which refers to the acid dissociation in water:

• \[ ext{NH}_4^+ \rightleftharpoons ext{NH}_3 + ext{H}^+\]

The \(K_a\) value is used to find the concentration of \( ext{H}^+\), providing an indirect understanding of how acidic or basic a solution is. Equilibrium calculations show that concentrations reach a point where neither the forward nor reverse reactions alter these concentrations unless the system is disrupted.
Understanding chemical equilibrium is vital in predicting and explaining the behavior of ions in solution, particularly in systems where acids or bases are present. It's key to anticipate how a system might shift when conditions change, influencing the solution's pH.