Question

In 1987 the first substance to act as a superconductor at a temperature above that of liquid nitrogen \((77 \mathrm{~K})\) was discovered. The approximate formula of this substance is \(\mathrm{YBa}_{2} \mathrm{Cu}_{3} \mathrm{O}_{7} .\) Calculate the percent composition by mass of this material.

Short answer

The percent composition by mass of YBa\(_{2}\)Cu\(_{3}\)O\(_{7}\) is approximately: Y: \(\frac{88.91}{666.62}\) × 100% \(= 13.34 \% \) Ba: \(\frac{274.66}{666.62}\) × 100% \(= 41.18 \% \) Cu: \(\frac{190.65}{666.62}\) × 100% \(= 28.60 \% \) O: \(\frac{112.00}{666.62}\) × 100% \(= 16.80 \% \)

Step-by-step solution

Calculate the molar mass of the substance

First, we need to find the molar mass of the given substance (YBa\(_{2}\)Cu\(_{3}\)O\(_{7}\)). We can do this by using the periodic table and adding the molar masses of each element in the formula. \[ \begin{aligned} \mathrm{Molar\ Mass} &= 1(\mathrm{Molar\ Mass\ of\ Y}) + 2(\mathrm{Molar\ Mass\ of\ Ba}) + 3(\mathrm{Molar\ Mass\ of\ Cu}) + 7(\mathrm{Molar\ Mass\ of\ O}) \\ &= 1(88.91 \mathrm{~g/mol}) + 2(137.33 \mathrm{~g/mol}) + 3(63.55 \mathrm{~g/mol}) + 7(16.00 \mathrm{~g/mol}) \end{aligned} \]

Calculate the individual masses of the elements

Now, let's calculate the individual masses of the elements in the substance's formula: \[ \begin{aligned} \mathrm{Mass\ of\ Y} &= 1(88.91 \mathrm{~g/mol}) \\ \mathrm{Mass\ of\ Ba} &= 2(137.33 \mathrm{~g/mol}) \\ \mathrm{Mass\ of\ Cu} &= 3(63.55 \mathrm{~g/mol}) \\ \mathrm{Mass\ of\ O} &= 7(16.00 \mathrm{~g/mol}) \end{aligned} \]

Calculate the percent composition by mass

Finally, we can determine the percent composition by mass for each element in the substance. For this, we need to divide the individual mass of each element by the molar mass of the substance and multiply the result by 100. \[ \begin{aligned} \mathrm{Percent\ Composition\ of\ Y} & = \frac{\mathrm{Mass\ of\ Y}}{\mathrm{Molar\ Mass}} \times 100\% \\ \mathrm{Percent\ Composition\ of\ Ba} & = \frac{\mathrm{Mass\ of\ Ba}}{\mathrm{Molar\ Mass}} \times 100\% \\ \mathrm{Percent\ Composition\ of\ Cu} & = \frac{\mathrm{Mass\ of\ Cu}}{\mathrm{Molar\ Mass}} \times 100\% \\ \mathrm{Percent\ Composition\ of\ O} & = \frac{\mathrm{Mass\ of\ O}}{\mathrm{Molar\ Mass}} \times 100\% \end{aligned} \] Calculate the values for each element to find their percent compositions by mass.

Key concepts

Superconductor

A superconductor is a fascinating type of material that, when chilled to very low temperatures, completely loses its electrical resistance. This means it can conduct electricity with 100% efficiency—no energy is wasted as heat. Most materials are not superconductors. They typically need to be cooled to near absolute zero (-273.15°C or 0 Kelvin) to achieve this phenomena.
The discovery of materials that act as superconductors at higher temperatures (still quite cold, but well above absolute zero) is truly groundbreaking.

• Released in 1987, the substance with the formula ext{YBa}_2 ext{Cu}_3 ext{O}_7 is renowned for being a high-temperature superconductor.
• This compound can become superconductive at the temperature of liquid nitrogen, a relatively higher temperature compared to the initial superconductors.

Understanding superconductors is pivotal in advancing technology, potentially impacting areas like power grids, magnetic resonance imaging (MRI), and quantum computing.

Molar Mass Calculation

Calculating the molar mass of a compound involves adding up the molar masses of each element in the compound according to how many atoms of each are present. For the compound YBa ext{₂−} ext{Cu}_3 ext{O}_7, this process is crucial in finding percent composition by mass. Here's the breakdown of calculations for YBa ext{₂−} ext{Cu}_3 ext{O}_7:

• Yttrium (Y): Only one atom, molar mass = 88.91 g/mol
• Barium (Ba): Two atoms, each with a molar mass of 137.33 g/mol
• Copper (Cu): Three atoms, each with a molar mass of 63.55 g/mol
• Oxygen (O): Seven atoms, each with a molar mass of 16.00 g/mol

Adding these together gives us the total molar mass of the compound. This final figure is the essential denominator when calculating the percent composition by mass of each individual element.

Elemental Analysis

Elemental analysis is the process of breaking down a chemical compound to understand the proportion of elements within it. This is important in chemistry for identifying unknown substances, verifying purity, and more.
When conducting an elemental analysis of YBa ext{₂−} ext{Cu}_3 ext{O}_7 we'd go through a series of steps:

• Identify the Elements: Recognize which elements are present in the formula: Yttrium (Y), Barium (Ba), Copper (Cu), and Oxygen (O).
• Determine Atomic Ratios: Use the formula to see how many atoms of each element are present: 1 Y, 2 Ba, 3 Cu, and 7 O.

Using these insights, you can proceed to calculate the percent composition by mass. This is achieved by determining the total mass of each element in the entire compound and comparing it against the compound's molar mass.

YBa2Cu3O7 Compound

The YBa ext{₂−} ext{Cu}_3 ext{O}_7 compound is significant in materials science as it was the first high-temperature superconductor discovered. This compound became superconductive at much warmer temperatures compared to earlier superconductors. It changed the field by massively increasing the temperature range at which superconductivity could occur. A deeper understanding of the structural makeup of this compound can provide insights into why it can act as a superconductor at such relatively high temperatures.

• Structure: It is known for a particular lattice structure that facilitates superconductivity. The atoms are arranged in precise ways that allow electron flow with zero resistance at certain temperatures.
• Importance: Its discovery unlocked more research into development of practical superconductors potentially useful in a variety of fields, from medical technologies to energy transmission.

Understanding how and why this compound acts as a superconductor can influence future innovation and lead to new types of superconductive materials.