Aqueous sulfurous acid \(\left(\mathrm{H}_{2} \mathrm{SO}_{3}\right)\) was made by dissolving \(0.200 \mathrm{~L}\) of sulfur dioxide gas at \(19^{\circ} \mathrm{C}\) and \(745 \mathrm{mmHg}\) in water to yield \(500.0 \mathrm{~mL}\) of solution. The acid solution required \(10.0 \mathrm{~mL}\) of sodium hydroxide solution to reach the titration end point. What was the molarity of the sodium hydroxide solution?

The Ideal Gas Law is a fundamental principle in chemistry and physics that relates the volume, temperature, and pressure of a gas to the number of moles present. Its formula is \( PV = nRT \). Here, P is the gas pressure in atmospheres (atm), V is the volume in liters (L), T is the temperature in Kelvin (K), R is the gas constant equal to 0.0821 atm·L/mol·K, and n is the number of moles. Understanding this law is essential for converting conditions of gases to moles, a key step in our exercise. For example, converting sulfur dioxide gas conditions involved calculating the temperature to Kelvin and pressure to atmospheres, then substituting these values into the Ideal Gas Law to find the number of moles.

Titration is a technique used to determine the concentration of a solution by reacting it with a solution of known concentration. In our exercise, we use titration to find the molarity of a sodium hydroxide (NaOH) solution. The sulfurous acid (H2SO3) reacts with the NaOH in a stoichiometric ratio. By knowing the volume of the titrant (NaOH) added to reach the end point and its relationship with the analyte (H2SO3), we can calculate the concentration of the sodium hydroxide solution. This involves precise measurement and understanding the molecular interactions at play.

Molarity (M) is a way to express the concentration of a solution. It is defined as the number of moles of solute per liter of solution, using the formula \( M = \frac{n}{V} \), where n is the number of moles and V is the volume in liters. This concept is important in our exercise for calculating the molarity of the NaOH solution from the titration data. Once we know the moles of NaOH used and the volume of the solution, applying the molarity formula provides the concentration of the NaOH in moles per liter.

Stoichiometry is the calculation of reactants and products in chemical reactions. It follows the conservation of mass, meaning the total mass of reactants equals the total mass of products. Our exercise involves the stoichiometry of the reaction between sulfurous acid and sodium hydroxide. The balanced equation \( H_2SO_3 + 2NaOH \rightarrow Na_2SO_3 + 2H_2O \) tells us that one mole of H2SO3 reacts with two moles of NaOH. This relationship is used to determine the moles of NaOH needed, based on the moles of H2SO3 present, which is crucial for the titration calculation.

Conversion of units is essential in chemistry for ensuring all values are in the correct units before performing calculations. This was crucial in our exercise when converting temperature from Celsius to Kelvin (\( T = 19 + 273.15 = 292.15 \) K) and pressure from millimeters of mercury (mmHg) to atmospheres (\( P = 745 \times \frac{1}{760} = 0.98 \) atm). Correct unit conversion ensures accuracy in calculations like those involving the Ideal Gas Law and molarity determination. Each step builds on the previous, emphasizing the interconnectedness of unit conversions in problem-solving.