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 the superconductor YBa2Cu3O7 is approximately Y: 13.36%, Ba: 41.20%, Cu: 28.60%, and O: 16.84%.

Step-by-step solution

Find the molar mass of each element in the formula

First, we need to find the molar mass for each element: Y (yttrium), Ba (barium), Cu (copper), and O (oxygen). From the periodic table, their respective molar masses are: - Yttrium (Y): 89 g/mol - Barium (Ba): 137.3 g/mol - Copper (Cu): 63.5 g/mol - Oxygen (O): 16 g/mol

Calculate molar mass of YBa2Cu3O7

To calculate the molar mass of the compound, we multiply the molar mass of each element with its respective number of moles and then sum up the results. Molar mass of YBa2Cu3O7 = (1 × 89) + (2 × 137.3) + (3 × 63.5) + (7 × 16) = 89 + 274.6 + 190.5 + 112 = 666.1 g/mol

Calculate the mass fraction of each element

Now we can compute the mass fraction of each element in the compound. The mass fraction is the mass of an element divided by the molar mass of the compound: Mass fraction of Y = Mass of Y / Molar mass of YBa2Cu3O7 = (1 × 89) / 666.1≈ 0.1336 Mass fraction of Ba = Mass of Ba / Molar mass of YBa2Cu3O7 = (2 × 137.3) / 666.1≈ 0.4120 Mass fraction of Cu = Mass of Cu / Molar mass of YBa2Cu3O7 = (3 × 63.5) / 666.1≈ 0.2860 Mass fraction of O = Mass of O / Molar mass of YBa2Cu3O7 = (7 × 16) / 666.1≈ 0.1684

Calculate the percent composition by mass

Finally, we can express the mass fractions as percentages. Percent composition of Y = Mass fraction of Y × 100 ≈ 0.1336 × 100 = 13.36% Percent composition of Ba = Mass fraction of Ba × 100 ≈ 0.4120 × 100 = 41.20% Percent composition of Cu = Mass fraction of Cu × 100 ≈ 0.2860 × 100 = 28.60% Percent composition of O = Mass fraction of O × 100 ≈ 0.1684 × 100 = 16.84% Thus, the percent composition by mass of YBa2Cu3O7 is approximately Y: 13.36%, Ba: 41.20%, Cu: 28.60%, and O: 16.84%.

Key concepts

Molar Mass Calculation

Understanding molar mass is essential in chemistry for determining how much of each element makes up a compound. Molar mass is calculated by summing the product of the atomic masses of each element with the number of times that element appears in the chemical formula.
For example, let's determine the molar mass for the compound \( \text{YBa}_2\text{Cu}_3\text{O}_7 \). This requires us to:

• Look up the atomic masses from the periodic table. The molar masses are: yttrium (\( \text{Y} \)): 89 g/mol, barium (\( \text{Ba} \)): 137.3 g/mol, copper (\( \text{Cu} \)): 63.5 g/mol, and oxygen (\( \text{O} \)): 16 g/mol.
• Multiply the atomic masses by their respective counts in the formula: \( 1 \times 89 \) for yttrium, \( 2 \times 137.3 \) for barium, \( 3 \times 63.5 \) for copper, and \( 7 \times 16 \) for oxygen.
• Add these values together to find the total molar mass: \( 89 + 274.6 + 190.5 + 112 = 666.1 \text{ g/mol} \).

This value, 666.1 g/mol, is the molar mass of \( \text{YBa}_2\text{Cu}_3\text{O}_7 \).

Mass Fraction

The mass fraction of an element in a compound offers insight into the proportion of that element relative to the entire compound's mass. It is computed as the ratio of the element's total mass to the compound's molar mass.
To find the mass fraction of each element in \( \text{YBa}_2\text{Cu}_3\text{O}_7 \), you should:

• Start by finding the total mass of each element within the compound based on the molar mass calculation: 89 for Y, 274.6 for Ba, 190.5 for Cu, and 112 for O.
• Divide these individual masses by the compound’s total molar mass of 666.1 g/mol to get the mass fraction for each element. For example, for Y: \( \frac{89}{666.1} \approx 0.1336 \).

Now you have the mass fractions, which allow you to easily move on to finding percent compositions.

Superconductor

Superconductors are fascinating materials with zero electrical resistance when cooled below a certain temperature. This characteristic allows them to conduct electricity with perfect efficiency.
Historically, superconductors required extremely low temperatures, typically close to absolute zero. However, the compound \( \text{YBa}_2\text{Cu}_3\text{O}_7 \) marked a significant leap forward in 1987 by exhibiting superconducting properties above the temperature of liquid nitrogen (\( 77 \text{ K} \)), making it more practical for various technologies.
This milestone in superconductivity helped researchers explore new applications such as magnetic levitation, lossless power cables, and advanced medical imaging equipment, accelerating technological advancement.

YBa2Cu3O7

The chemical compound \( \text{YBa}_2\text{Cu}_3\text{O}_7 \) is known as yttrium barium copper oxide, an oxide ceramic that becomes superconducting at relatively high temperatures.
This material contains:

• One yttrium (Y) atom, which contributes to the compound's unique structural properties.
• Two barium (Ba) atoms that help stabilize the structure and contribute to the superconductive properties.
• Three copper (Cu) atoms, essential for the compound's ability to exhibit superconductivity.
• Seven oxygen (O) atoms, crucial in forming the layers necessary for superconductivity.

This combination of elements forms a crystal lattice structure instrumental in allowing superconductive properties at the higher temperature of 93 K, compared to earlier superconductors.