To determine the molecular formula of the compound with the given composition and molecular mass, follow these steps:
### Step 1: Determine the mass of phosphorus (P) and chlorine (Cl) in one mole of the compound.
- The given percentage of phosphorus (P) is 22.5%.
- The given percentage of chlorine (Cl) is 77.5%.
- The molecular mass of the compound is 137.32 g/mol.
We calculate the mass of phosphorus and chlorine in one mole based on the given percentages:
[tex]\[ \text{Mass of P} = \left(\frac{22.5}{100}\right) \times 137.32 = 30.897 \, \text{g} \][/tex]
[tex]\[ \text{Mass of Cl} = \left(\frac{77.5}{100}\right) \times 137.32 = 106.423 \, \text{g} \][/tex]
### Step 2: Determine the number of moles of phosphorus and chlorine in one mole of the compound.
The atomic mass of phosphorus (P) is 30.97 g/mol.
The atomic mass of chlorine (Cl) is 35.45 g/mol.
Using these atomic masses, we find the number of moles of each element:
[tex]\[ \text{Moles of P} = \frac{30.897 \, \text{g}}{30.97 \, \text{g/mol}} = 0.997642 \, \text{mol} \][/tex]
[tex]\[ \text{Moles of Cl} = \frac{106.423 \, \text{g}}{35.45 \, \text{g/mol}} = 3.002059 \, \text{mol} \][/tex]
### Step 3: Determine the simplest whole number ratio of moles of phosphorus to chlorine.
To find the simplest whole number ratio, divide each mole value by the smallest number of moles calculated:
[tex]\[ \text{Ratio of P} = \frac{0.997642}{0.997642} = 1.0 \][/tex]
[tex]\[ \text{Ratio of Cl} = \frac{3.002059}{0.997642} = 3.009152 \][/tex]
### Step 4: Round to the nearest whole numbers to find the empirical formula.
Rounding the ratios to the nearest whole number gives:
[tex]\[ \text{Formula ratio of P: } 1 \][/tex]
[tex]\[ \text{Formula ratio of Cl: } 3 \][/tex]
So, the empirical formula has one phosphorus (P) atom and three chlorine (Cl) atoms.
### Final Step: Determine the molecular formula.
The empirical formula is [tex]\( \text{PCl}_3 \)[/tex].
Given that the molecular mass calculated from this empirical formula (137.32 g/mol) matches the provided molecular mass of the compound, the molecular formula is:
[tex]\[ \boxed{\text{PCl}_3} \][/tex]
Therefore, the molecular formula of the compound is [tex]\( \text{PCl}_3 \)[/tex].
