Question

A sample of the superconducting oxide \(\mathrm{HgBa}_{2} \mathrm{Ca}_{2} \mathrm{Cu}_{3} \mathrm{O}_{8+x}\) is found to contain \(14.99 \%\) oxygen by mass. (a) Assuming that all other elements are present in the ratios represented by the formula, what is the value of \(x\) in the formula? (b) Which of the metallicelements in the compound is (are) most likely to have noninteger average charges? Explain your answer. (c) Which of the metallic ions in the substance is likely to have the largest ionic radius? Which will have the smallest?

Short answer

(a) The value of \(x\) in the formula \(\mathrm{HgBa}_{2} \mathrm{Ca}_{2}\mathrm{Cu}_{3} \mathrm{O}_{8+x}\) is approximately 1.22. (b) Copper (Cu) is most likely to have a non-integer average oxidation state to balance the charges in the compound. (c) Among the metallic ions, Hg(+2) has the largest ionic radius, while Cu has the smallest ionic radius.

Step-by-step solution

Determine the total mass of elements represented in the formula without the extra oxygen atoms

We first calculate the mass of the elements, excluding the extra \(x\) oxygen atoms in the given compound. Recall the atomic masses of the relevant elements: - Mercury (Hg) = 200.59 amu - Barium (Ba) = 137.33 amu - Calcium (Ca) = 40.08 amu - Copper (Cu) = 63.55 amu - Oxygen (O) = 16.00 amu Now, calculate the mass of the elements using the atomic masses above: \(\mathrm{HgBa}_{2} \mathrm{Ca}_{2}\mathrm{Cu}_{3} \mathrm{O}_{8}\) Mass = (1)(200.59) + (2)(137.33) + (2)(40.08) + (3)(63.55) + (8)(16.00) \(Mass = 1130.58\,\text{amu}\)

Find the mass of 8+x oxygen atoms

Given that the sample contains \(14.99 \%\) oxygen by mass, we can find the total mass of the oxygen atoms using the calculated mass of elements without extra oxygen atoms: Mass of oxygen atoms (8+x) = \(0.1499 \times 1130.58\) Mass of oxygen atoms (8+x) = \(169.55\,\text{amu}\)

Calculate the value of x

We can now calculate the value of \(x\) by dividing the difference between the mass of oxygen atoms in the sample and the mass of 8 oxygen atoms by the molar mass of oxygen: x = \(\frac{169.55 - (8)(16.00)}{16.00}\) \(x \approx 1.22\)

Identify the oxidation states of the metallic elements

We know that the oxidation states of the metallic elements are as follows: - Mercury (Hg): +2 - Barium (Ba): +2 - Calcium (Ca): +2 Considering the fact that the compound is neutral and the total positive charge must equal the total negative charge, copper (Cu) is the most likely element to have a non-integer average oxidation state in order to balance the charges.

Compare the ionic radii of the metallic elements

When comparing the coefficients of the ionic radii of the metallic elements, we see that, - Hg(+2) has the largest ionic radius - Cu(+1), Cu(+2), or Cu(+3) has the smallest ionic radius In summary, (a) the value of \(x\) is approximately 1.22, (b) copper is the most likely metallic element that has the noninteger average charge due to charge balance requirements, and (c) Hg(+2) has the largest ionic radius, while Cu has the smallest ionic radius.

Key concepts

Oxidation States of Elements

An oxidation state is the total number of electrons that an atom gains or loses to form a chemical bond with another atom. In the analysis of a compound like \( \mathrm{HgBa}_{2} \mathrm{Ca}_{2}\mathrm{Cu}_{3} \mathrm{O}_{8+x} \), each element has a typical oxidation state that helps maintain a balance of the electrical charge within the compound.

• Mercury (Hg), Barium (Ba), and Calcium (Ca) most often adopt an oxidation state of +2. This means they each lose two electrons to become positively charged ions.
• Oxygen typically has an oxidation state of -2, as it gains two electrons when forming compounds.
• Copper (Cu) is more complex, often appearing in oxidation states +1 or +2, but in superconducting oxides like this, it may range up to +3. This variability often results in a "non-integer" average oxidation state for copper.

The non-integer oxidation state arises because copper helps maintain charge neutrality in the compound by contributing different numbers of charges, depending on its oxidation state.

Ionic Radii and Their Influences

Ionic radii refer to the size of ions and can significantly affect the structure and properties of compounds, such as superconductors. These radii influence how tightly or loosely ions pack in a lattice, which can affect conductivity.

• In the ionic compounds, the charge and size of an ion are closely related. More positive charges typically mean smaller ionic radii because the nucleus pulls the electrons more closely.
• Mercury having a +2 charge generally has the largest ionic radius among the listed metals – Hg(+2) ions have comparatively fewer positive charges versus the number of electron shells it has, leading to a larger size.
• Copper, given its multiple typical charges (Cu(+1), Cu(+2), Cu(+3)), is usually among the smallest due to the increased nuclear charge pulling its electrons in closer, especially in the +3 state.

The comparison of ionic radii is essential for understanding how elements are arranged within a material and, therefore, how the material functions.

Elemental Mass Calculation Processes

Calculating elemental mass in a compound involves using the atomic masses of the constituent elements to determine the total mass; it's vital for stoichiometric calculations and verifying the composition of complex materials.

• First, use the atomic mass of each element from the periodic table to calculate the mass contribution of each element in the compound. For example, Mercury (Hg) is 200.59 amu, while Barium (Ba) is 137.33 amu.
• Multiply the atomic mass by the number of respective atoms in the molecular formula. For HgBa₂Ca₂Cu₃O₈, that means calculating and summing [(1)(200.59) + (2)(137.33) + (2)(40.08) + (3)(63.55) + (8)(16.00)].
• Determine the percentage by mass of each element or group of elements, such as the percentage of oxygen in the compound, which can influence the determination of additional variables like the unknown \(x\) in \(O_{8+x}\).

This method allows chemists to dissect a compound's composition quantitatively, ensuring precise elemental analyzation in research and application.