To find out how many molecules of the compound [tex]\( \text{C}_7\text{H}_{12}\text{N}_2\text{O}_3 \)[/tex] are in a 10.0 g sample, we'll need to follow several steps involving concepts from chemistry such as molar mass, moles, and Avogadro's number.
### Step 1: Determine the Molar Mass
First, we calculate the molar mass of [tex]\( \text{C}_7\text{H}_{12}\text{N}_2\text{O}_3 \)[/tex]. The molar mass is obtained by summing the atomic masses of all the atoms in the molecule:
- Carbon (C): 12 g/mol
- Hydrogen (H): 1 g/mol
- Nitrogen (N): 14 g/mol
- Oxygen (O): 16 g/mol
In the molecule [tex]\( \text{C}_7\text{H}_{12}\text{N}_2\text{O}_3 \)[/tex], we have:
- 7 Carbon atoms
- 12 Hydrogen atoms
- 2 Nitrogen atoms
- 3 Oxygen atoms
Thus, the molar mass of [tex]\( \text{C}_7\text{H}_{12}\text{N}_2\text{O}_3 \)[/tex] is calculated as follows:
[tex]\[ \text{Molar Mass} = (7 \times 12) + (12 \times 1) + (2 \times 14) + (3 \times 16) \][/tex]
[tex]\[ \text{Molar Mass} = 84 + 12 + 28 + 48 \][/tex]
[tex]\[ \text{Molar Mass} = 172 \text{ g/mol} \][/tex]
### Step 2: Calculate the Number of Moles
Next, we calculate the number of moles of [tex]\( \text{C}_7\text{H}_{12}\text{N}_2\text{O}_3 \)[/tex] in the 10.0 g sample using the molar mass:
[tex]\[ \text{Number of moles} = \frac{\text{mass of the sample}}{\text{molar mass}} \][/tex]
[tex]\[ \text{Number of moles} = \frac{10.0 \text{ g}}{172 \text{ g/mol}} \][/tex]
[tex]\[ \text{Number of moles} = 0.05813953488372093 \][/tex]
### Step 3: Determine the Number of Molecules
To determine the number of molecules, we use Avogadro's number, which is [tex]\( 6.022 \times 10^{23} \)[/tex] molecules per mole. The number of molecules is calculated by multiplying the number of moles by Avogadro's number:
[tex]\[ \text{Number of molecules} = \text{Number of moles} \times \text{Avogadro's number} \][/tex]
[tex]\[ \text{Number of molecules} = 0.05813953488372093 \times 6.022 \times 10^{23} \][/tex]
[tex]\[ \text{Number of molecules} = 3.5011627906976744 \times 10^{22} \][/tex]
### Summary
Therefore, a 10.0 g sample of [tex]\( \text{C}_7\text{H}_{12}\text{N}_2\text{O}_3 \)[/tex] contains approximately [tex]\( 3.501 \times 10^{22} \)[/tex] molecules.
