Here is the ranking of the samples in order of increasing number of atoms:
1) 0.5 mol BCl3 molecules: This sample contains 0.5 moles of BCl3 molecules, which corresponds to 6.022 x 10^23 atoms of boron and chlorine combined.
2) 197 g gold: Gold has an atomic weight of 197 g/mol, so this sample contains approximately 1 mole of gold atoms, or 6.022 x 10^23 gold atoms.
3) 6.0 x 10^23 CCl4 molecules: This sample contains 6.0 x 10^23 molecules of CCl4, which corresponds to 6.0 x 10^23 atoms of carbon and chlorine combined.
Therefore, the ranking from the sample with the lowest number of atoms to the highest is: 0.5 mol BCl3 molecules < 197 g gold < 6.0 x 10^23 CCl4 molecules.