Question

A Chromium(III) chloride forms many compounds with ammonia. To find the formula of one of these compounds, you titrate the \(\mathrm{NH}_{3}\) in the compound with standardized acid. \(\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{\mathrm{x}} \mathrm{Cl}_{3}(\mathrm{aq})+\times \mathrm{HCl}(\mathrm{aq}) \rightarrow\) $$ x \mathrm{NH}_{4}+(\mathrm{aq})+\mathrm{Cr}^{3+}(\mathrm{aq})+(x+3) \mathrm{Cl}^{-}(\mathrm{aq}) $$ Assume that \(24.26 \mathrm{mL}\) of \(1.500 \mathrm{M} \mathrm{HCl}\) is used to tirate \(1.580 \mathrm{g}\) of \(\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}_{3} .\) What is the value of \(x ?\)

Short answer

The value of \( x \) is found by calculating the ratio of moles of NH3 to moles of HCl.

Step-by-step solution

Calculate Moles of HCl used

First, calculate the moles of HCl used in the titration. Use the formula \( M = \frac{n}{V} \), where \( M \) is the molarity, \( n \) is the number of moles, and \( V \) is the volume in liters. The volume of HCl is \( 24.26 \) mL, which is \( 0.02426 \) L. The molarity of HCl is \( 1.500 \) M. Thus, the number of moles is: \[ n = M \times V = 1.500 \, \text{mol/L} \times 0.02426 \, \text{L} = 0.03639 \, \text{mol} \]

Calculate the Molar Mass of the Ammonia Compound

Assume the molar mass of \( \mathrm{Cr} \left( \mathrm{NH}_{3} \right)_{x} \mathrm{Cl}_{3} \) is required. Chromium has a molar mass of approximately \( 51.9961 \, \text{g/mol} \), ammonia \( \mathrm{NH}_{3} \) has \( 17.03 \, \text{g/mol} \), and chloride \( \mathrm{Cl} \) has \( 35.453 \, \text{g/mol} \). The molar mass of \( \mathrm{Cr} \left( \mathrm{NH}_{3} \right)_{x} \mathrm{Cl}_{3} \) is:\[ 51.9961 + x \times 17.03 + 3 \times 35.453 \]

Using Mass to Find Moles

Determine the moles of the compound used. The mass given is \( 1.580 \, \text{g} \). Using the molar mass calculated in Step 2, set it equal to \( \frac{1.580}{\text{Molar Mass}} \) to find the moles of the compound.

Ratio of HCl to NH3

Since \( x \) moles of \( \mathrm{NH}_{3} \) react with HCl, the moles of \( \mathrm{NH}_{3} \) can be found if \( x \times \text{moles of compound} = \text{moles of HCl} \). Set up the equation \( 0.03639 = x \times \text{moles of compound} \) and solve for \( x \).

Calculate the Value of x

Use the values obtained from the calculations in Steps 1 to 4 to compute \( x \). If you're able to compute the exact molar mass from the values of \( \mathrm{Cr} \), \( \mathrm{NH}_{3} \), and \( \mathrm{Cl} \), you can determine \( x \) by solving the equation in Step 4.

Key concepts

Chromium(III) Chloride

Chromium(III) chloride is an important compound in chemistry, especially known for its ability to form complexes with ammonia. It often appears in its hydrated form, \([\text{CrCl}_3\cdot x\, \text{H}_2\text{O}]\). Chromium(III) chloride can create various compounds that contain ammonia, making it versatile in many chemical reactions.
Ammonia complexes often have a general formula such as \([\mathrm{Cr(NH}_3)_{x}\,\text{Cl}_3]\), where \([x]\) represents the number of ammonia molecules bound to each chromium atom. These complexes are crucial in understanding how metal ions like chromium interact with ligands such as ammonia. Such interactions are pivotal in fields like catalysis and material science. Understanding the unique characteristics of chromium's binding properties with ammonia is fundamental in predicting how it will behave in a given chemical reaction.

Titration

Titration is a laboratory technique used to determine the concentration of a solute in a solution. In the context of ammonia complexes, titration can help identify the number of ammonia molecules in a compound's chemical formula. It's particularly useful for understanding the stoichiometric relationships within a complex.
The process involves adding a solution of known concentration (titrant) to a solution of unknown concentration until the reaction reaches a completion point, often indicated by a color change due to an indicator or a pH shift. To accurately perform a titration, precise measurements of volume and concentration, as well as proper mixing, are important. Titration data can provide essential information for calculating components' molar ratios in ammonia complexes, such as \(\mathrm{Cr(NH}_3)_x\mathrm{Cl}_3\). This helps chemists understand the stoichiometry within reactions.

Stoichiometry

Stoichiometry is an essential concept in chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. In titrating an ammonia complex with an acid like HCl, stoichiometry helps us understand how much of each reactant is needed to complete the reaction. By knowing the stoichiometry of the chemical equation, one can determine the number of ammonia molecules (x) that bind to the chromium atom in the experiment.
The stoichiometric coefficients in a balanced chemical equation tell us the essential ratios of molecules involved in the reaction. For example, if \(x\) moles of ammonia react with one mole of HCl, the relationship must be maintained to preserve the chemical balance. By correlating the amounts of reactants used during the experiment to these stoichiometric coefficients, one can deduce the exact value of \(x\) in the ammonia complex.

Molar Mass Calculation

Molar mass calculation is a process in chemistry used to determine the mass of one mole of a substance. It is a vital step in stoichiometry for converting between grams and moles, essential for precise chemical analysis. By calculating the molar mass of a molecule, one can deduce many properties, including its composition and behavior in reactions.
The molar mass is calculated by summing up the atomic masses of all atoms in a chemical formula. For example, in the compound \([\text{Cr(NH}_3)_x\text{Cl}_3]\), the molar mass is derived by adding the atomic mass of chromium, \(x\) times the molar mass of ammonia, and three times the mass of chloride ions. This calculation provides a basis to find out how many moles of compound are in a given mass. This information, when used with titration data, enables the determination of unknowns like the value of \(x\) in chemical complexes.