Problem 82 - General Chemistry: Principles and Modern Applications

Question

A piece of marble (assume it is pure \(\mathrm{CaCO}_{3}\) ) reacts with \(2.00 \mathrm{L}\) of \(2.52 \mathrm{M} \mathrm{HCl}\). After dissolution of the marble, a \(10.00 \mathrm{mL}\) sample of the resulting solution is withdrawn, added to some water, and titrated with 24.87 mL of 0.9987 M NaOH. What must have been the mass of the piece of marble? Comment on the precision of this method; that is, how many significant figures are justified in the result?

Step-by-Step Solution

Answer

The mass of the marble piece was approximately 251 g. This calculation justifies three significant figures.

Step 1: Calculate moles of \(HCl\) used in reaction

Given that volume of \(HCl\) used in the reaction is 2.00 L and concentration is 2.52 M, we can calculate moles of \(HCl\) using formula moles = Volume × Molarity which gives \(2.00 L × 2.52 mol/L = 5.04 moles of \(HCl\)

Step 2: Determine moles of \(CaCO_3\)

The reaction between \(CaCO_3\) and \(HCl\) is balanced as follows: \(CaCO_3 + 2HCl -> CaCl_2 + CO_2 + H_2O\). According to this stoichiometry, 1 mole of \(CaCO_3\) reacts with 2 moles of \(HCl\). Therefore, the moles of \(CaCO_3\) are \(5.04 moles/2 = 2.52 moles\)

Step 3: Calculate moles of \(NaOH\) used in the titration

Given that the volume of \(NaOH\) used in the titration is 24.87 mL or 0.02487 L and its molarity is 0.9987 M, the number of moles of \(NaOH\) can be calculated as \(0.02487 L × 0.9987 mol/L = 0.02483 moles\)

Step 4: Convert moles of \(NaOH\) to moles of \(HCl\) and then to moles of \(CaCO_3\)

For every mole of \(NaOH\) reacted there is one mole of \(HCl\) reacted and for every two moles of \(HCl\) reacted, there's a mole of \(CaCO_3\) reacted, thus the moles of \(CaCO_3\) reacted in titration are \(0.02483 moles/2 = 0.01242 moles\)

Step 5: Calculate total moles of \(CaCO_3\) and find the mass of marble

The initial total moles of \(CaCO_3\) are 2.52 moles. If 0.01242 moles were used in titration, then 2.52 moles - 0.01242 moles = 2.50758 moles were in the original sample. Since the molar mass of \(CaCO_3\) is about 100.09 g/mol, the mass of the marble piece would be \(2.50758 moles × 100.09 g/mol = 251 g\)

Step 6: Comment on the precision of the method

Every measurement used in this calculation comes with three or more significant figures. The result has been rounded to three significant figures, in line with the least precise measurement, the molarity of \(HCl\). So, the result justifies three significant figures.

Key Concepts

Understanding Molarity

Molarity is all about concentration and is a key player in stoichiometry problems. It helps us understand how many moles of a chemical are present in a given volume of solution. - **Defining Molarity**: Molarity (M) is defined as the number of moles of solute (the substance being dissolved) divided by the volume of the solution in liters. The formula is: \[\text{Molarity (M)} = \frac{\text{moles of solute}}{\text{liters of solution}}\]- **Application in Reactions**: In chemical reactions, knowing the molarity allows you to calculate exactly how much of each reactant is needed to complete the reaction. For instance, in this problem, by knowing the molarity and the volume of the hydrochloric acid, you can easily calculate the moles of \[\mathrm{HCl}\] used by multiplying these two values together. Remember that molarity provides a clear picture of the solution's concentration and is particularly useful in reactions happening in solutions.

Exploring Chemical Reactions

Chemical reactions are transformations where reactants are converted into products. Understanding the balanced equation is crucial in stoichiometry because it tells us the precise ratios in which elements and compounds react.- **Balanced Equations**: The balanced chemical equation for the reaction between calcium carbonate \( (\mathrm{CaCO_3}) \) and hydrochloric acid \( (\mathrm{HCl}) \) is: \[\mathrm{CaCO_3} + 2\mathrm{HCl} \rightarrow \mathrm{CaCl_2} + \mathrm{CO_2} + \mathrm{H_2O}\] This equation sets the foundation for calculating the amount of reactants needed and products formed.- **Stoichiometric Ratios**: Here, 1 mol of \( \mathrm{CaCO_3} \) reacts with 2 mols of \( \mathrm{HCl} \). By using this ratio, once you know the number of moles of \( \mathrm{HCl} \), you can determine the moles of \( \mathrm{CaCO_3} \) and subsequently the mass of marble involved.Chemical reactions require precise balancing and ratio understanding, which is why stoichiometry is a central concept in chemistry.

Significant Figures in Calculations

Significant figures are important in scientific calculations because they indicate the precision of the measurements and results. They show which digits are reliable and which are estimates.- **Defining Significant Figures**: These are the digits in a number that carry meaningful information about its precision. The number of significant figures in a measurement is determined by the precision of the measuring device.- **Applying Significant Figures**: In this exercise, the molarity of \( \mathrm{HCl} \) is stated with three significant figures. Consequently, the final answer for the mass of the marble piece should also be reported with three significant figures: \( 251 \text{ g} \).- **Rounding Rules**: When performing calculations, the number of significant figures is determined by the least precise measurement. This means carrying over the least number of significant figures throughout your calculation.Following significant figures in your calculations ensures that your results are as accurate and as reliable as the data you started with. Proper use helps in maintaining the integrity of your results in scientific inquiries.