Question

The volume of a sample of pure \(\mathrm{HCl}\) gas was \(189 \mathrm{~mL}\) at \(25^{\circ} \mathrm{C}\) and \(108 \mathrm{mmHg}\). It was completely dissolved in about \(60 \mathrm{~mL}\) of water and titrated with an \(\mathrm{NaOH}\) solution; \(15.7 \mathrm{~mL}\) of the \(\mathrm{NaOH}\) solution was required to neutralize the HCl. Calculate the molarity of the \(\mathrm{NaOH}\) solution.

Short answer

The molarity of the NaOH solution is approximately 0.0703 M.

Step-by-step solution

Convert Pressure to atm

First, convert the pressure from mmHg to atm. Use the conversion: \(1 \text{ atm} = 760 \text{ mmHg}\). Hence, \(108 \text{ mmHg} = \frac{108}{760} \text{ atm} = 0.1421 \text{ atm}\).

Convert Volume to Liters

The volume of the gas is given as \(189 \text{ mL}\). Convert it to liters by dividing by 1000: \(189 \text{ mL} = 0.189 \text{ L}\).

Use Ideal Gas Law to Find Moles of HCl

Use the ideal gas law \(PV = nRT\) to find the moles of HCl. Here, \(P = 0.1421 \text{ atm}\), \(V = 0.189 \text{ L}\), \(R = 0.0821 \text{ L atm K}^{-1} \text{ mol}^{-1}\), and \(T = 298 \text{ K} (\text{since it's 25}^{\circ} \text{C})\).\[n = \frac{PV}{RT} = \frac{0.1421 \times 0.189}{0.0821 \times 298} = 0.001103 \text{ moles of HCl}\].

Use Stoichiometry to Find Moles of NaOH

Since \(1 \text{ mol HCl}\) reacts with \(1 \text{ mol NaOH}\), the moles of NaOH required are equal to the moles of HCl, which is 0.001103 moles.

Calculate Molarity of NaOH Solution

The volume of the NaOH solution used is \(15.7 \text{ mL}\), which is \(0.0157 \text{ L}\). Molarity \(M\) is given by \(M = \frac{\text{moles of solute}}{\text{volume of solution in liters}}\). Thus, \[M = \frac{0.001103}{0.0157} = 0.0703 \text{ M}\].

Key concepts

Ideal Gas Law

Understanding the Ideal Gas Law is crucial when dealing with gases in chemistry. It relates the pressure, volume, temperature, and amount of a gas in moles. The formula is expressed as \(PV = nRT\). Here, \(P\) stands for pressure, \(V\) for volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

In the exercise, we used the ideal gas law to find the moles of HCl gas. By rearranging the formula to \(n = \frac{PV}{RT}\), we calculate the moles when pressure, volume, and temperature are known. This step is fundamental because it allows us to transition from gas state to understanding how much of the substance we have for subsequent chemical reactions.

Moles

Moles are a basic unit in chemistry used to express quantities of a chemical substance. A mole is \(6.022 \times 10^{23}\) entities, like atoms or molecules. In chemical calculations, knowing the number of moles allows chemists to predict the amounts of reactants and products.

In the provided exercise, moles allow us to calculate the amount of HCl present in the gaseous state before it was dissolved in water. By using the ideal gas law, the determined number of moles of HCl (=0.001103 moles) provides the necessary link to further calculate the required reactants in our titration process.

Molarity

Molarity describes the concentration of a solute in a solution. It is defined as the number of moles of solute divided by the volume of the solution in liters, expressed as \(M = \frac{\text{moles of solute}}{\text{liters of solution}}\).

In the titration scenario from our exercise, once we established the number of moles of NaOH needed to neutralize the HCl, we calculated its molarity. Knowing the moles and measuring the volume of the NaOH solution precisely allowed us to determine its concentration. The result of 0.0703 M signifies a moderately concentrated solution, often useful in analytical chemistry for quantitative analyses.

Stoichiometry

Stoichiometry is the calculation of reactants and products in chemical reactions. It is based on the balanced chemical equations that represent the law of conservation of mass. In any balanced chemical reaction, the number of atoms for each element is the same in both reactants and products.

Within the exercise, stoichiometry is applied to understand the relationship between HCl and NaOH in a reaction. Given that 1 mole of HCl reacts with 1 mole of NaOH, their mole ratio is 1:1. This information was vital because it allowed us to directly equate the moles of HCl calculated using the ideal gas law to the moles of NaOH needed for neutralization. Stoichiometry ensures precise and accurate predictions of reactant consumption and product formation in chemical processes.