The French chemists, Pierre L. Dulong and Alexis T. Petit, noted in 1819 that the molar heat capacity of many solids at ordinary temperatures is proportional to the number of atoms per formula unit of the solid. They quantified their observations in what is known as Dulong and Petit's rule, which says that the molar heat capacity,? P, of a solid can be expressed as? P=?⋅3? where? is the number of atoms per formula unit and? is the universal gas constant. The observed heat capacity per gram of a compound containing thallium and chlorine is 0.21 J·K-1·g-1. Use Dulong and Petit's rule to determine the empirical formula of the compound.

TlCl

Explanation:

The Dulong and Petit's rule is:

Cp = N*3R, where Cp is the heat capacity, N is the number of atoms in the molecule, and R is the gas constant (8.314 J/K. mol). Note that Cp has unity in gram and R in mol, so we need to transform Cp to unity in mol.

Let's suppose that the formula of the compound formed bt thallium (Tl) and chlorine (Cl) is TlCl, which has molar mass: 204.4 g/mol of Tl + 35.5 g/mol of Cl = 239.9 g/mol.

Cp observed = 0.21 J/K. g * 239.9 g/mol = 50.379 J/K. mol

For the compound, N = 2:

Cp theorical = 2*3*8.314 = 49.884 J/K. mol

For TlCl₂ the molar mass is: 204.4 + 2*35.5 = 275.4 g/mol

Cp observed = 0.21 * 275.4 = 57.834 J/K. mol

N = 3

Cp theorical = 3*3*8.314 = 74.826 J/K. mol

For Tl₂Cl the molar mass is: 2*204.4 + 35.5 = 444.3 g/mol

Cp observed = 0.21*444.3 = 93.303 J/K. mol

N = 3

Cp theorical = 3*3*8.314 = 74.826 J/K. mol

Note that the difference will be greater when there are more atoms in the compound, so the empirical formula must be TlCl which has the closest Cp observed and theoretical.