To determine the empirical formula of the unknown sample, we need to use the given composition and the molar masses of the elements to find the number of moles of each element in the sample. Then, we can divide the number of moles of each element by the smallest number of moles to get the simplest, whole-number ratio of the elements in the compound.
First, we can calculate the number of moles of carbon in the sample:
mass of carbon = 0.3751 x 5.91 g = 2.216 g
moles of carbon = 2.216 g / 12.01 g/mol = 0.1845 mol
Next, we can calculate the number of moles of hydrogen and oxygen in the sample using Avogadro's number:
moles of hydrogen = 1.4830 x 10^23 atoms / 6.022 x 10^23 mol^-1 = 0.2461 mol
moles of oxygen = 1.2966 x 10^23 atoms / 6.022 x 10^23 mol^-1 = 0.2154 mol
Now, we can divide each number of moles by the smallest number of moles, which is 0.1845:
0.1845 mol C / 0.1845 mol = 1.000
0.2461 mol H / 0.1845 mol = 1.333
0.2154 mol O / 0.1845 mol = 1.167
These ratios are not whole numbers, so we need to multiply each ratio by a common factor to get the simplest, whole-number ratio. The simplest ratio is obtained by dividing each ratio by the smallest ratio:
1.000 / 1.000 = 1
1.333 / 1.000 = 4/3
1.167 / 1.000 = 7/6
We can then multiply each ratio by 6 to obtain whole numbers:
6 x 1 = 6
6 x 4/3 = 8
6 x 7/6 = 7
Therefore, the empirical formula of the unknown sample is C6H8O7.