Question

Graph A YAG, or yttrium aluminum garnet \(\left(\mathrm{Y}_{3} \mathrm{Al}_{5} \mathrm{O}_{12}\right),\) is a synthetic gemstone which has no counterpart in nature. Design a bar graph to indicate the moles of each element present in a 5.67 carat yttrium aluminum garnet. ( carat \(=0.20 \mathrm{g} )\)

Short answer

In a 5.67-carat yttrium aluminum garnet (YAG), there are 0.00573 moles of yttrium (Y), 0.00955 moles of aluminum (Al), and 0.02292 moles of oxygen (O). To represent this data in a bar graph, label the x-axis as 'Elements' and the y-axis as 'Moles'. Create three bars with heights corresponding to the moles of each element: Y (0.00573 moles), Al (0.00955 moles), and O (0.02292 moles). Label each bar with the respective element symbol and the number of moles.

Step-by-step solution

Calculate the mass of YAG in grams

First, we need to find the mass of YAG in grams. We are given that 1 carat equals 0.20 grams, and we have a 5.67-carat YAG. Mass of YAG = 5.67 carats * 0.20 g/carat = 1.134 g

Calculate the moles of each element in YAG

The molecular formula of YAG is \(\mathrm{Y}_{3} \mathrm{Al}_{5} \mathrm{O}_{12}\). To calculate the moles of each element, we need to divide the mass of the element in YAG by their respective molar mass. Here, molar mass of Y = 88.91 g/mol, molar mass of Al = 26.98 g/mol, and molar mass of O = 16.00 g/mol. First, let's find the molar mass of YAG: Molar mass of YAG = (3 * molar mass of Y) + (5 * molar mass of Al) + (12 * molar mass of O) = (3 * 88.91) + (5 * 26.98) + (12 * 16.00) = 266.73 + 134.9 + 192 = 593.63 g/mol Now, using the mass of YAG and its molar mass, we can calculate the moles of YAG: Moles of YAG = 1.134 g / 593.63 g/mol = 0.00191 moles Next, we will find the moles of Y, Al, and O in 0.00191 moles of YAG: - Moles of Y = 3 * 0.00191 = 0.00573 moles - Moles of Al = 5 * 0.00191 = 0.00955 moles - Moles of O = 12 * 0.00191 = 0.02292 moles

Design a bar graph

Now, we have the moles of each element in YAG. We can use this data to design a bar graph. Label the x-axis as 'Elements' and the y-axis as 'Moles'. Divide the x-axis into three different sections for Y, Al, and O. In each section, draw a bar with height according to the moles of each element calculated in Step 2: - The height of the bar representing Y should be 0.00573 moles. - The height of the bar representing Al should be 0.00955 moles. - The height of the bar representing O should be 0.02292 moles. Make sure to label each bar with the corresponding element symbol and their respective moles (e.g., "Y: 0.00573 moles"). By following these steps, you'll create a bar graph that represents the moles of each element present in a 5.67-carat yttrium aluminum garnet.

Key concepts

Molar Mass Calculation

Understanding the molar mass of a substance is crucial in chemistry, especially when dealing with reactions and quantifying substances. One can consider molar mass as the weight of one mole of a particular element or compound.

Molar mass is calculated by summing the atomic masses of all the atoms in a molecule's formula. For instance, water (H₂O) has two hydrogen atoms and one oxygen atom. Using the atomic mass unit (amu) from the periodic table, hydrogen has an approximate mass of 1 g/mol for each atom, and oxygen is about 16 g/mol, resulting in molar mass for water being approximately 18 g/mol (2*1 + 16).

In the case of the yttrium aluminum garnet (YAG), the molar mass calculation involves taking the sum of the molar masses of yttrium (Y), aluminum (Al), and oxygen (O) multiplied by their respective numbers in the chemical formula. It's similar to a recipe where the quantities of each ingredient must be totaled to understand the overall weight of the cake you're baking.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships, or ratios, between the reactants and products in a chemical reaction. It involves using the chemical formula and the balanced equation to predict the amount of products that form from a given amount of reactants, or conversely, the amount of reactants needed to form a desired amount of product.

In practical terms, if you are told to make sandwiches for a party and you know that one sandwich needs two slices of bread and one slice of cheese, stoichiometry would help you calculate how much bread and cheese you need depending on the number of sandwiches you want to make. Similarly, in chemistry, if you are given a certain amount of YAG and asked to find out how much of each element is present, you would use stoichiometry. In the YAG example, we use the molar ratio (3:5:12 for Y:Al:O respectively) along with the moles of YAG to determine the moles of each element.

Chemical Formulas

Chemical formulas are a way of presenting information about the chemical proportions of atoms that constitute a particular chemical compound, using the standard abbreviations for the chemical elements, and digits to indicate the number of each element present in the compound. Just as a recipe might list 'flour: 3 cups, sugar: 1 cup, baking powder: 1 tsp', a chemical formula lists the elements and the count of each atom.

For YAG, the formula \( \mathrm{Y}_{3} \mathrm{Al}_{5} \mathrm{O}_{12} \) indicates that in one formula unit of YAG, there are 3 yttrium atoms, 5 aluminum atoms, and 12 oxygen atoms. The subscript numbers are the equivalent of a chef’s recipe clauses, telling you how much of each ingredient you need. It’s also the starting block for myriad calculations in chemistry, such as determining the molar mass or converting between grams and moles of a compound, which can be particularly helpful when visualizing the composition of complex structures, such as YAG, or when balancing chemical equations.