Let's determine the empirical formulas for the given compounds by following a step-by-step process.
### (a) Caffeine, [tex]\( \mathbf{C_8 H_{10} N_4 O_2} \)[/tex]
1. Identify the number of each type of atom in caffeine:
- Carbon (C): 8 atoms
- Hydrogen (H): 10 atoms
- Nitrogen (N): 4 atoms
- Oxygen (O): 2 atoms
2. Find the greatest common divisor (GCD) of the quantities of each type of atom. Here the quantities are 8, 10, 4, and 2.
3. The GCD of 8, 10, 4, and 2 is 2.
4. Divide the number of each type of atom by the GCD:
- Carbon: [tex]\( \frac{8}{2} = 4 \)[/tex]
- Hydrogen: [tex]\( \frac{10}{2} = 5 \)[/tex]
- Nitrogen: [tex]\( \frac{4}{2} = 2 \)[/tex]
- Oxygen: [tex]\( \frac{2}{2} = 1 \)[/tex]
5. Write the empirical formula:
[tex]\[
\text{Empirical Formula of Caffeine} = \mathbf{C_4 H_5 N_2 O_1}
\][/tex]
### (b) Fructose, [tex]\( \mathbf{C_{12} H_{22} O_{11}} \)[/tex]
1. Identify the number of each type of atom in fructose:
- Carbon (C): 12 atoms
- Hydrogen (H): 22 atoms
- Oxygen (O): 11 atoms
2. Find the greatest common divisor (GCD) of the quantities of each type of atom. Here the quantities are 12, 22, and 11.
3. The GCD of 12, 22, and 11 is 1.
4. Divide the number of each type of atom by the GCD:
- Carbon: [tex]\( \frac{12}{1} = 12 \)[/tex]
- Hydrogen: [tex]\( \frac{22}{1} = 22 \)[/tex]
- Oxygen: [tex]\( \frac{11}{1} = 11 \)[/tex]
5. Since the GCD is 1, the empirical formula is the same as the molecular formula:
[tex]\[
\text{Empirical Formula of Fructose} = \mathbf{C_{12} H_{22} O_{11}}
\][/tex]
### (c) Hydrogen Peroxide, [tex]\( \mathbf{H_2 O_2} \)[/tex]
1. Identify the number of each type of atom in hydrogen peroxide:
- Hydrogen (H): 2 atoms
- Oxygen (O): 2 atoms
2. Find the greatest common divisor (GCD) of the quantities of each type of atom. Here the quantities are 2 and 2.
3. The GCD of 2 and 2 is 2.
4. Divide the number of each type of atom by the GCD:
- Hydrogen: [tex]\( \frac{2}{2} = 1 \)[/tex]
- Oxygen: [tex]\( \frac{2}{2} = 1 \)[/tex]
5. Write the empirical formula:
[tex]\[
\text{Empirical Formula of Hydrogen Peroxide} = \mathbf{H_1 O_1}
\][/tex]
### Summary of Empirical Formulas:
- (a) Caffeine: [tex]\( \mathbf{C_4 H_5 N_2 O_1} \)[/tex]
- (b) Fructose: [tex]\( \mathbf{C_{12} H_{22} O_{11}} \)[/tex]
- (c) Hydrogen Peroxide: [tex]\( \mathbf{H_1 O_1} \)[/tex]
