Question

A \(24.9-\mathrm{mL}\) volume of hydrochloric acid reacts completely with \(55.0 \mathrm{~mL}\) of aqueous \(\mathrm{Na}_{2} \mathrm{CO}_{3}\). The reaction is \(2 \mathrm{HCl}(a q)+\mathrm{Na}_{2} \mathrm{CO}_{3}(a q) \longrightarrow \mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(t)+2 \mathrm{NaCl}(a q)\) The volume of \(\mathrm{CO}_{2}\) formed is \(141 \mathrm{~mL}\) at \(27^{\circ} \mathrm{C}\) and \(727 \mathrm{mmHg}\). What is the molarity of the HCl solution?

Short answer

The molarity of the HCl solution is approximately 0.445 M.

Step-by-step solution

Convert Measurements

First, convert the final measurements into more useful units. Begin by noting that the pressure of 727 mmHg is equivalent to 0.957 atm (since 1 atm = 760 mmHg). The temperature needs to be converted from Celsius to Kelvin: \( T = 27^{\circ} \mathrm{C} + 273.15 = 300.15 \mathrm{~K} \).

Apply the Ideal Gas Law for CO2

The ideal gas law equation is given by \( PV = nRT \), where \( R \) is the ideal gas constant (0.0821 L atm/mol K). Use this equation to find the number of moles \( n \) of \( \mathrm{CO}_2 \):\[ n = \frac{PV}{RT} = \frac{(0.957 \text{ atm})(0.141 \text{ L})}{(0.0821 \text{ L atm/mol K})(300.15 \text{ K})} \approx 0.00554 \text{ moles of } \mathrm{CO}_2 \]

Relate CO2 Moles to HCl Moles

According to the balanced chemical equation, 1 mole of \( \mathrm{CO}_2 \) is produced from 2 moles of \( \mathrm{HCl} \). Therefore, the number of moles of \( \mathrm{HCl} \) is twice the moles of \( \mathrm{CO}_2 \):\[ n(\mathrm{HCl}) = 2 \times n(\mathrm{CO}_2) = 2 \times 0.00554 = 0.01108 \text{ moles of } \mathrm{HCl} \]

Calculate the Molarity of the HCl Solution

Molarity (M) is defined as the number of moles of solute per liter of solution. Use the volume of \( \mathrm{HCl} \) in liters (24.9 mL = 0.0249 L) to find its molarity:\[ M(\mathrm{HCl}) = \frac{n(\mathrm{HCl})}{\text{Volume in liters}} = \frac{0.01108}{0.0249} \approx 0.445 \text{ M} \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle that relates the physical properties of gases in a mathematical equation. It is expressed as \( PV = nRT \), where:

• \( P \) stands for pressure (in atmospheres or atm),
• \( V \) is the volume (in liters),
• \( n \) represents the number of moles of the gas,
• \( R \) is the ideal gas constant (approximately 0.0821 L atm/mol K), and
• \( T \) is the temperature (in Kelvin).

To effectively use this equation, it's crucial to ensure that all units are compatible. For example, if pressure is given in mmHg, it must be converted to atm by dividing by 760. Similarly, the temperature must be converted to Kelvin by adding 273.15 to the Celsius value. In this exercise, knowing how to manipulate the Ideal Gas Law helps us calculate the moles of \( \mathrm{CO}_2 \) produced.

Chemical reaction stoichiometry

Stoichiometry is a key concept in chemistry that involves the quantitative relationship between reactants and products in a chemical reaction. This helps us determine how much of each substance is consumed and produced. In the given reaction, the balanced equation shows that 2 moles of \( \mathrm{HCl} \) react with 1 mole of \( \mathrm{Na}_2\mathrm{CO}_3 \) to produce 1 mole of \( \mathrm{CO}_2 \).

• This means that for every 1 mole of \( \mathrm{CO}_2 \) produced, 2 moles of \( \mathrm{HCl} \) are required.

By using stoichiometry, we can relate the moles of \( \mathrm{CO}_2 \) calculated from the Ideal Gas Law to find the moles of \( \mathrm{HCl} \) used in the reaction. This is achieved by multiplying the moles of \( \mathrm{CO}_2 \) by the ratio determined from the balanced equation. This step is vital when transitioning from moles of one substance to another within a reaction.

Unit conversion

Unit conversion is essential for ensuring all measurements are in compatible units when performing calculations. In chemistry problems like this one, we often encounter different units that need conversion. - **Pressure Conversion:** Pressure is typically required in atmospheres for gas law equations. If given in mmHg, divide by 760 to get atmospheres. - **Temperature Conversion:** Kelvin is the standard unit for temperature in gas law calculations. Convert from Celsius to Kelvin by adding 273.15. - **Volume Conversion:** Volumes are often given in milliliters (mL) but should be in liters (L) for molarity and ideal gas calculations. To convert, divide the mL value by 1000. These conversions ensure accuracy in calculations and are crucial when using formulas like the Ideal Gas Law or calculating molarity. It's always a good habit to double-check your unit conversions to avoid errors.

Balanced chemical equation

A balanced chemical equation is essential in chemistry as it ensures the conservation of mass and the correct stoichiometric relationships between reactants and products. In the given exercise, the reaction is:\[ 2 \mathrm{HCl}(aq) + \mathrm{Na}_2\mathrm{CO}_3(aq) \rightarrow \mathrm{CO}_2(g) + \mathrm{H}_2\mathrm{O}(t) + 2 \mathrm{NaCl}(aq) \]This shows that two moles of \( \mathrm{HCl} \) are required to react completely with one mole of \( \mathrm{Na}_2\mathrm{CO}_3 \), producing one mole of \( \mathrm{CO}_2 \), one mole of water (\( \mathrm{H}_2\mathrm{O} \)), and two moles of \( \mathrm{NaCl} \).

• Balancing ensures that the number of atoms for each element is the same on both sides of the equation.
• It allows for the accurate calculation of reactant and product amounts using stoichiometry.

A balanced equation is the foundation for understanding the precise transformation that occurs during chemical reactions and is key to solving chemistry problems efficiently.