Converting grams to number of atoms

Notice how dimensional analysis (Section 1.6) provides a straightforward route from grams to numbers of atoms. The molar mass and Avogadro s number are used as conversion factors to convert grams---> moles-----> atoms. 

First use the molar mass of the element to convert from grams to moles, and then use Avogadro s number to convert moles to number of atoms. 

How is the mass in grams of an element converted to number of atoms ... 

Since the formula of a compound indicates the numbers of atoms in the compound, we must convert the masses of the elements to numbers of atoms. The easiest way to do this is to work with 100.00 grams of the compound. In the present case, 38.67% carbon by mass means 38.67 grams of carbon per 100.00 grams of compound 16.22% hydrogen means 16.22 grams of hydrogen per 100.00 grams of compound and so on. To determine... 

Now, suppose we want to know the number of carbon atoms in the 0.58-g diamond. We first convert from grams to moles and then from moles to number of atoms. The solution map is ... 

STRATEGIZE Convert between the mass of an element in grams and the number of atoms of the element by first converting to moles (using the molar mass of the element) and then to number of atoms (using Avogadro s number). 

Our task is to estimate the volume occupied by one atom of lithium. As usual, the mole is a convenient place to begin the calculations. Visualize a piece of lithium containing one mole of atoms. The molar mass, taken from the periodic table, tells us the number of grams of Li in one mole. The density equation can be used to convert from mass to volume. Once we have the volume of one mole of lithium, we divide by the number of atoms per mole to find the volume of a single atom. 

The first step is to use the molar mass to convert grams of carbon to moles of carbon. Once the moles are obtained, Avogadro s number can be used to calculate the number of atoms. 

Skills Toolkit 3 shows that to convert from number of atoms to mass in grams, you must first convert to amount in moles. 

Let s say that you want to find an empirical formula from the percentage composition. First, convert the mass percentage of each element to grams. Second, convert from grams to moles using the molar mass of each element as a conversion factor. (Keep in mind that a formula for a compound can be read as a number of atoms or as a number of moles.) Third, as shown in Sample Problem C, compare these amounts in moles to find the simplest whole-number ratio among the elements in the compound. 

For most practical purposes we are interested in the masses of reactants and products, because those are the quantities that are directly measured. In this case, the molar masses (calculated from a table of atomic masses) are used to convert the number of moles of a substance (in moles) to its mass (in grams), as illustrated by Example 2.6. Sometimes, however, we are also interested in knowing the number of molecules in a sample. The mole allows us to convert easily from mass to numbers of molecules as follows ... 

To convert masses to moles or vice versa, we use the molar mass of the substance. Molar mass has the same numeric value as the number of atomic mass units in a formula unit, but it is expressed in units of grams per mole. For example, the molar mass of water is 18.0 g/mol because the formula mass of water is 18.0 amu/molecule. Because molar mass is a ratio, it can be used as a factor in problem solving. 

Plan We know the grams of Fe (95.8 g) and need the number of Fe atoms. We cannot convert directly from grams to atoms, so we first convert to moles by dividing grams of Fe by its molar mass. [This is the reverse of the step in part (a).] Then, we multiply number of moles by Avogadro s number to find number of atoms (see roadmap b). 

You are given the number of atoms of helium and must find the mass of the gas. First, convert the number of atoms to moles, then convert moles to grams. 

Solution Let s first calculate the number of N atoms in 1.68 x lo" g of urea. First, we must convert grams of urea to number of molecules of urea. This calculation is similar to Problem 3.26. The molecular formula of urea shows there are two N atoms in one urea molecule, which will allow us to convert to atoms of N. We need to perform three conversions ... 

Strategy We can calculate the number of atoms of sulfur if we know the number of moles—we just multiply by Avogadro s number. We are given the number of grams, however, so we must first convert 16.3 g of S to moles of S and then convert the moles to atoms using Avogadro s number ... 

Stoichiometry is the quantitative study of reactants and products in a chemical reaction. Although amounts in chemical reactions can be measured in many different units, such as grams, liters, or moles, a chemical equation can only tell us the relative numbers of atoms (or moles) that are produced and consumed in a reaction. Therefore, no matter in what units the initial amounts are given, they must be converted to moles before the chemical equation can be used to determine the quantitative relationships between reactants and products. For example, the combustion of carbon monoxide gas in air produces carbon dioxide according to the chemical equation... 

Amount-Mass-Number Conversions Involving Elements We begin with amount-mass-number relationships of elements. As Figure 3.2 shows, convert mass (in grams) or number of entities (atoms or molecules) to amount (mol) first. For molecular elements, Avogadro s number gives molecules per mole. 

So from the periodic table, we can find the mass of one mole of atoms for any element. Now, how do we use that quantity, in combination with the number of atoms in a mole, to determine the number of atoms in sample of a given mass Suppose we want to know the number of aluminum atoms in 12.5 grams of aluminum. We first convert from the mass to the number of moles, and then from the number of moles to the number of atoms ... 

Avogadro s number can be used to find the number of atoms of an element from the amount in moles or to find the amount of an element in moles from the number of atoms. While these types of problems are less common in chemistry than converting between amoimt in moles and mass in grams, they are usefiil in demonstrating the meaning of Avogadro s number. Note that in these calculations, Avogadro s number is expressed in units of atoms per mole. 

As indicated in Figure 3.7, the given number of atoms must first be converted to amount in moles by dividing by Avogadro s number. Amount in moles is then multiplied by molar mass to yield mass in grams. 

From the initial mass of and the known half-life of 1 determine how many grams of 1 remain after 30 days. Convert this mass to a number of atoms using the molar mass of and Avogadro s number. Finally, the product of the number of atoms and the decay constant yields the activity in disintegrations per second, which can be converted to curies. 

This is one source of acid rain, a serious environmental problem. The sulfur dioxide content of an air sample can be determined. A sample of air is bubbled through an aqueous solution of hydrogen peroxide to convert all of the SO2 to H2 SO4. H2 O2 + SO2 H2 SO4 Titration of the resulting solution completes the analysis (both H atoms of H2 SO4 are titrated). In one such case, the analysis of 1.55 X 10 Lof Los Angeles air gave a solution that required 5.70 mL of 5.96 X 10 M NaOH to complete the titration. Determine the number of grams of SO2 present in the air sample. 

The relationship above gives a way of converting from grams to moles to particles, and vice versa. If you have any one of the three quantities, you can calculate the other two. This becomes extremely useful in working with chemical equations, as we will see later, because the coefficients in the balanced chemical equation are not only the number of individual atoms or molecules at the microscopic level, but also the number of moles at the macroscopic level. 

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