91Ó°ÊÓ

Calcium fluorapatite \(\left(\mathrm{Ca}_{10}\left(\mathrm{PO}_{4}\right)_{6} \mathrm{~F}_{2}, \mathrm{FM} 1008.6\right)\) laser crystals were doped with chromium to improve their efficiency. It was suspected that the chromium could be in the \(+4\) oxidation state. 1\. To measure the total oxidizing power of chromium in the material, a crystal was dissolved in \(2.9 \mathrm{M} \mathrm{HClO}_{4}\) at \(100^{\circ} \mathrm{C}\), cooled to \(20^{\circ} \mathrm{C}\), and titrated with standard \(\mathrm{Fe}^{2+}\), using \(\mathrm{Pt}\) and \(\mathrm{Ag} \mid \mathrm{AgCl}\) electrodes to find the end point. Chromium oxidized above the \(+3\) state should oxidize an equivalent amount of \(\mathrm{Fe}^{2+}\) in this step. That is, \(\mathrm{Cr}^{4+}\) would consume one \(\mathrm{Fe}^{2+}\), and each atom of \(\mathrm{Cr}^{6+}\) in \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\) would consume three \(\mathrm{Fe}^{2+}\) : $$ \begin{aligned} \mathrm{Cr}^{4+}+\mathrm{Fe}^{2+} & \rightarrow \mathrm{Cr}^{3+}+\mathrm{Fe}^{3+} \\ \frac{1}{2} \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}+3 \mathrm{Fe}^{2+} & \rightarrow \mathrm{Cr}^{3+}+3 \mathrm{Fe}^{3+} \end{aligned} $$ 2\. In a second step, the total chromium content was measured by dissolving a crystal in \(2.9 \mathrm{M} \mathrm{HClO}_{4}\) at \(100^{\circ} \mathrm{C}\) and cooling to \(20^{\circ} \mathrm{C}\). Excess \(\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\) and \(\mathrm{Ag}^{+}\)were then added to oxidize all chromium to \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\). Unreacted \(\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\) was destroyed by boiling, and the remaining solution was titrated with standard \(\mathrm{Fe}^{2+}\). In this step, each \(\mathrm{Cr}\) in the original unknown reacts with three \(\mathrm{Fe}^{2+}\). $$ \begin{aligned} \mathrm{Cr}^{x+} & \stackrel{\mathrm{S}_{2} \mathrm{O}_{8}^{2-}}{\longrightarrow} \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-} \\ \frac{1}{2} \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}+3 \mathrm{Fe}^{2+} & \rightarrow \mathrm{Cr}^{3+}+3 \mathrm{Fe}^{3+} \end{aligned} $$ In step \(1,0.4375 \mathrm{~g}\) of laser crystal required \(0.498 \mathrm{~mL}\) of \(2.786 \mathrm{mM}\) \(\mathrm{Fe}^{2+}\) (prepared by dissolving \(\mathrm{Fe}\left(\mathrm{NH}_{4}\right)_{2}\left(\mathrm{SO}_{4}\right)_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O}\) in \(2 \mathrm{M} \mathrm{HClO}_{4}\) ). In step \(2,0.1566 \mathrm{~g}\) of crystal required \(0.703 \mathrm{~mL}\) of the same \(\mathrm{Fe}^{2+}\) solution. Find the average oxidation number of \(\mathrm{Cr}\) in the crystal and find the total micrograms of \(\mathrm{Cr}\) per gram of crystal.

Step by step solution

01

Calculate the Moles of Fe in Step 1

First, determine the moles of \( \mathrm{Fe}^{2+} \) used to titrate the \( 0.4375 \) g of laser crystal in step 1.Use the concentration of \( \mathrm{Fe}^{2+} \) solution: \( 2.786 \ \mathrm{mM} = 2.786 \times 10^{-3} \ \mathrm{mol/L} \):\[ \text{Moles of Fe}^{2+} = 2.786 \times 10^{-3} \ \mathrm{mol/L} \times 0.498 \times 10^{-3} \ \mathrm{L} = 1.388 \times 10^{-6} \ \mathrm{mol} \]

02

Calculate Total Chromium Oxidation in Step 1

Each \( \mathrm{Cr}^{4+} \) consumes one \( \mathrm{Fe}^{2+} \).Assuming all chromium in the higher oxidation state is \( \mathrm{Cr}^{4+} \), the moles of \( \mathrm{Cr}^{4+} \) present are:\[ \text{Moles of } \mathrm{Cr}^{4+} = 1.388 \times 10^{-6} \]

03

Calculate the Moles of Fe in Step 2

Next, determine the moles of \( \mathrm{Fe}^{2+} \) used in step 2 for the \( 0.1566 \) g of crystal:\[ \text{Moles of Fe}^{2+} = 2.786 \times 10^{-3} \ \mathrm{mol/L} \times 0.703 \times 10^{-3} \ \mathrm{L} = 1.958 \times 10^{-6} \ \mathrm{mol} \]

04

Calculate Total Moles of Chromium

In step 2, all chromium is oxidized to \( \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-} \), each \( \mathrm{Cr} \) reacting with three \( \mathrm{Fe}^{2+} \). The total moles of \( \mathrm{Cr} \) are:\[ \text{Moles of } \mathrm{Cr} = \frac{1.958 \times 10^{-6}}{3} = 6.527 \times 10^{-7} \ \mathrm{mol} \]

05

Determine the Average Oxidation State of Chromium

To find the total moles of Cr oxidized, use the moles from step 1.Moles of \( \mathrm{Cr} \) oxidized above \(+3\) (assuming all as \( \mathrm{Cr}^{4+} \)):\[ 1.388 \times 10^{-6} \]The total moles of chromium from step 4:\[ 6.527 \times 10^{-7} \]Average oxidation state:\[ \text{Average oxidation state} = 3 + \frac{1.388 \times 10^{-6}}{6.527 \times 10^{-7}} \approx 4.13 \]

06

Calculate Micrograms of Chromium per Gram of Crystal

Convert moles of chromium to grams for step 2:Use the molar mass of chromium: \( 51.996 \ \mathrm{g/mol} \).\[ \text{Mass of Cr} = 6.527 \times 10^{-7} \ \mathrm{mol} \times 51.996 \ \mathrm{g/mol} = 3.394 \times 10^{-5} \ \mathrm{g} \]Convert to micrograms:\[ 3.394 \times 10^{-5} \ \mathrm{g} \times 10^6 = 33.94 \ \mathrm{\mu g} \ \, \text{per 0.1566 g of crystal} \]To find per gram, divide by the mass used in step 2:\[ \frac{33.94}{0.1566} = 216.7 \ \mathrm{\mu g/g}\]

07

Final Answer

The average oxidation number of chromium in the crystal is approximately 4.13, and the total micrograms of chromium per gram of the crystal is approximately 216.7 \( \mathrm{\mu g/g} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chromium Titration

Chromium titration is an important technique often used to determine the oxidation state of chromium, specifically when analyzing materials where chromium's presence in various oxidation states influences its chemical behavior. In our exercise, the titration involves the use of an iron(II) solution, which acts as a reducing agent, to measure the ability of chromium in the +4 or +6 oxidation states to oxidize the iron ions.

When conducting a chromium titration, key chemical reactions occur where chromium ions in high oxidation states are reduced by iron ions. For instance:

• In the given reaction: \[\mathrm{Cr}^{4+} + \mathrm{Fe}^{2+} \rightarrow \mathrm{Cr}^{3+} + \mathrm{Fe}^{3+}\]Chromium is reduced from +4 to +3, while iron is oxidized from +2 to +3.
• For chromium in the +6 state, as in dichromate (\(\mathrm{Cr}_2\mathrm{O}_7^{2-}\)), the reaction involves more equivalents of \(\mathrm{Fe}^{2+}\):\[\frac{1}{2} \mathrm{Cr}_2\mathrm{O}_7^{2-} + 3 \mathrm{Fe}^{2+} \rightarrow \mathrm{Cr}^{3+} + 3 \mathrm{Fe}^{3+}\]Each worth of the dichromate ion reacts with three iron ions.

Understanding these reactions allows us to measure the overall oxidizing power of chromium in a given sample which is critical in determining its oxidation state.

Chemical Stoichiometry

Chemical stoichiometry is a foundational concept in chemistry that pertains to the calculation of the reactants and products in chemical reactions. Stochiometry provides the quantitative relationships governing the reactions, ensuring that the conservation of mass principle is adhered to.

In the exercise, we navigate the stoichiometry involving chromium ions and iron ions. The core idea lies in understanding that:

• One mole of chromium in the +4 oxidation state consumes one mole of iron in the +2 state.
• When chromium exists as \(\mathrm{Cr}_2\mathrm{O}_7^{2-}\), it's crucial to note that each chromium atom in this state consumes three moles of \(\mathrm{Fe}^{2+}\). This implies larger consumption due to higher oxidation levels.

By calculating the exact moles of iron involved in the titration reaction during the given steps, stoichiometry aids in determining both the oxidation state of chromium and confirming the accuracy of the experimental procedures.

Spectrophotometric Analysis

Spectrophotometric analysis is a powerful tool used to quantify the concentration of solutes based on their interaction with light. Specifically, this analysis helps by using an instrument called a spectrophotometer to measure how much a chemical substance absorbs light as a beam of light passes through a sample solution.

While not directly involved in the specific exercise, understanding how spectrophotometric analysis could apply involves noting its role in detecting changes in concentration of reactants or products during a reaction, which can be crucial when dealing with transitions of electron states in complex ions.

Some important points regarding spectrophotometry include:

• This method can help in monitoring reactions dynamically to assess the completion of a chemical process in real-time.
• It allows for non-invasive continuous observation by simply measuring changes in light absorption without alteration to the sample.

Even though this exercise primarily focuses on titration and oxidation states, spectrophotometric analysis is a related analytical technique with a broad range of applications in chemistry.

Electrochemical Cells

Electrochemical cells are integral to titration processes, especially when measuring the potentials of redox reactions. These cells convert chemical energy into electrical energy through electron transfer processes across electrodes.

In the context of our chromium titration exercise, two types of electrodes are utilized:

• The platinum (Pt) electrode, which is often used as an inert conductor in many redox reactions.
• The silver/silver chloride (\(\mathrm{Ag}\vert \mathrm{AgCl}\)) electrode, which serves as a reference electrode providing a stable potential against which the potential of the cell can be measured.

Electrochemical cells help detect the end point of titrations, which is where the reaction between chromium ions and iron(II) ions reaches completion. Through measuring changes in potential, these cells provide critical information on when the titration should conclude.