2.13 Relative atomic masses and empirical formulae (ticket #3497,  (Page 5/5)

Therefore, the atomic masses of non-gaseous elements can be determined by mass and volume measurements ongaseous compounds containing these elements. This procedure is fairly general, and most atomic masses can be determined in thisway.

Moles, molecular formulae and stoichiometric calculations

We began with a circular dilemma: we could determine molecular formulae provided that we knew atomic masses,but that we could only determine atomic masses from a knowledge of molecular formulae. Since we now have a method for determining allatomic masses, we have resolved this dilemma and we can determine the molecular formula for any compound for which we have percentcomposition by mass.

As a simple example, we consider a compound which is found to be 40.0% carbon, 53.3% oxygen, and 6.7% hydrogenby mass. Recall from the Law of Definite Proportions that these mass ratios areindependent of the sample, so we can take any convenient sample to do our analysis. Assuming that we have 100.0g of the compound, wemust have 40.0g of carbon, 53.3g of oxygen, and 6.7g of hydrogen. If we could count or otherwise determine the number of atoms ofeach element represented by these masses, we would have the molecular formula. However, this would not only be extremelydifficult to do but also unnecessary.

From our determination of atomic masses, we can note that 1 atom of carbon has a mass which is 12.0 times themass of a hydrogen atom. Therefore, the mass of $N$ atoms of carbon is also 12.0 times the mass of $N$ atoms of hydrogen atoms, no matter what $N$ is. If we consider this carefully, we discover that 12.0g of carbon contains exactly the same number of atoms as does 1.0g ofhydrogen. Similarly, we note that 1 atom of oxygen has a mass which is $\frac{16.0}{12.0}$ times the mass of a carbon atom. Therefore, the mass of $N$ atoms of oxygen is $\frac{16.0}{12.0}$ times the mass of $N$ atoms of carbon. Again, we can conclude that 16.0g of oxygen containsexactly the same number of atoms as 12.0g of carbon, which in turn is the same number of atoms as 1.0g of hydrogen. Without knowing (or necessarily even caring) what the number is, we can say that itis the same number for all three elements.

For convenience, then, we define the number of atoms in 12.0g of carbon to be 1 mole of atoms. Note that 1 mole is a specific number ofparticles, just like 1 dozen is a specific number, independent of what objects we are counting. The advantage to defining themole in this way is that it is easy to determine the number of moles of a substance we have, and knowing the number ofmoles is equivalent to counting the number of atoms (or molecules) in a sample. For example, 24.0g of carbon contains 2.0 moles ofatoms, 30.0g of carbon contains 2.5 moles of atoms, and in general, $x$ grams of carbon contains $\frac{x}{12.0}$ moles of atoms. Also, we recall that 16.0g of oxygen contains exactly as many atoms as does 12.0gof carbon, and therefore 16.0g of oxygen contains exactly 1.0 mole of oxygen atoms. Thus, 32.0g of oxygen contains 2.0 moles of oxygenatoms, 40.0g of oxygen contains 2.5 moles, and $x$ grams of oxygen contains $\frac{x}{16.0}$ moles of oxygen atoms. Even more generally, then, if we have $m$ grams of an element whose atomic mass is $M$ , the number of moles of atoms, $n$ , is

Now we can determine the relative numbers of atoms of carbon, oxygen, and hydrogen in our unknown compoundabove. In a 100.0g sample, we have 40.0g of carbon, 53.3g of oxygen, and 6.7g of hydrogen. The number of moles of atoms in eachelement is thus

We note that the numbers of moles of atoms of the elements are in the simple ratio $n_C:n_O:n_H=1:1:2$ . Since the number of particles in 1 mole is the same for all elements, then itmust also be true that the number of atoms of the elements are in the simple ratio 1 : 1 : 2. Therefore, the molecular formula of thecompound must be $CO{H}_2$ .

Or is it? On further reflection, we must realize that the simple ratio 1 : 1 : 2 need not represent the exactnumbers of atoms of each type in a molecule of the compound, since it is indeed only a ratio. Thus the molecular formula could just aseasily be $C_2{O}_2{H}_4$ or $C_3{O}_3{H}_6$ . Since the formula $CO{H}_2$ is based on empirical mass ratio data, we refer to this as the empirical formula of the compound. To determine the molecular formula , we need to determine the relative mass of a molecule of the compound,i.e. the molecular mass. One way to do so is based on the Law of Combining Volumes,Avogadro's Hypothesis, and the Ideal Gas Law . To illustrate, however, if we were to find that the relative mass of one molecule of the compound is60.0, we could conclude that the molecular formula is $C_2{O}_2{H}_4$ .

Review and discussion questions