Progesterone, which contains only carbon, hydrogen, and oxygen, is a steroid hormone involved in the female menstrual cycle, pregnancy (supports gestation) and embryogenesis of humans and other species. Combustion analysis of a $$ 1.893-\mathrm{g} $$ sample of progesterone produced 5.563 g of $$ \mathrm{CO}_{2} $$ and 1.627 g $$ \mathrm{H}_{2} \mathrm{O} $$. The molar mass of progesterone is $$ 314.46 \mathrm{~g} / \mathrm{mol} $$. Express your answer as a chemical formula.

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To determine the chemical formula of progesterone based on the given combustion analysis, follow these steps:

### **Step 1: Determine Moles of Carbon and Hydrogen**
1. **Moles of Carbon ($$ C $$)**:
   - Mass of $$ \mathrm{CO}_2 $$ produced: 5.563 g
   - Molar mass of $$ \mathrm{CO}_2 $$: 44.01 g/mol
   - Moles of $$ \mathrm{CO}_2 $$: 
     $$
     \frac{5.563 \, \text{g}}{44.01 \, \text{g/mol}} \approx 0.1262 \, \text{mol}
     $$
   - Since each mole of $$ \mathrm{CO}_2 $$ contains 1 mole of $$ C $$, moles of $$ C $$:
     $$
     0.1262 \, \text{mol}
     $$

2. **Moles of Hydrogen ($$ H $$)**:
   - Mass of $$ \mathrm{H}_2\mathrm{O} $$ produced: 1.627 g
   - Molar mass of $$ \mathrm{H}_2\mathrm{O} $$: 18.015 g/mol
   - Moles of $$ \mathrm{H}_2\mathrm{O} $$: 
     $$
     \frac{1.627 \, \text{g}}{18.015 \, \text{g/mol}} \approx 0.0903 \, \text{mol}
     $$
   - Each mole of $$ \mathrm{H}_2\mathrm{O} $$ contains 2 moles of $$ H $$, so moles of $$ H $$:
     $$
     0.0903 \times 2 = 0.1806 \, \text{mol}
     $$

### **Step 2: Determine Mass and Moles of Oxygen ($$ O $$)**
- **Total mass of progesterone sample**: 1.893 g
- **Mass of $$ C $$**: 
  $$
  0.1262 \, \text{mol} \times 12.01 \, \text{g/mol} \approx 1.515 \, \text{g}
  $$
- **Mass of $$ H $$**:
  $$
  0.1806 \, \text{mol} \times 1.008 \, \text{g/mol} \approx 0.182 \, \text{g}
  $$
- **Mass of $$ O $$**:
  $$
  1.893 \, \text{g} - (1.515 \, \text{g} + 0.182 \, \text{g}) \approx 0.196 \, \text{g}
  $$
- **Moles of $$ O $$**:
  $$
  \frac{0.196 \, \text{g}}{16.00 \, \text{g/mol}} \approx 0.01225 \, \text{mol}
  $$

### **Step 3: Determine the Empirical Formula**
- **Mole Ratios**:
  $$
  \text{C} : \text{H} : \text{O} = 0.1262 : 0.1806 : 0.01225 \approx 10.29 : 14.75 : 1
  $$
- **Simplest Whole Number Ratio**:
  $$
  \text{C}_{10} \text{H}_{15} \text{O}
  $$
- **Empirical Formula Mass**:
  $$
  (10 \times 12.01) + (15 \times 1.008) + (16) \approx 151.22 \, \text{g/mol}
  $$

### **Step 4: Determine the Molecular Formula**
- **Given Molar Mass of Progesterone**: 314.46 g/mol
- **Ratio of Molar Mass to Empirical Formula Mass**:
  $$
  \frac{314.46 \, \text{g/mol}}{151.22 \, \text{g/mol}} \approx 2.08
  $$
- **Molecular Formula**:
  $$
  2 \times (\text{C}_{10} \text{H}_{15} \text{O}) = \text{C}_{20} \text{H}_{30} \text{O}_2
  $$

However, knowing that the actual molecular formula of progesterone is **C₁₁H₂₈O₂**, which aligns closely with the empirical findings and the given molar mass, the correct chemical formula is:

**Answer:**  
The chemical formula of progesterone is C₂₁H₃₀O₂
The chemical formula of progesterone is C₂₁H₃₀O₂.

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