To determine the molecular formula of the artificial sweetener, we need to follow these steps:
1. Calculate the empirical formula mass:
The empirical formula of the sweetener is [tex]\( C_{14}H_{18}N_{2}O_{5} \)[/tex].
To calculate the empirical formula mass, we add the atomic masses of all the atoms present in the empirical formula:
- Carbon (C): [tex]\( 14 \times 12 = 168 \, \text{g/mol} \)[/tex]
- Hydrogen (H): [tex]\( 18 \times 1 = 18 \, \text{g/mol} \)[/tex]
- Nitrogen (N): [tex]\( 2 \times 14 = 28 \, \text{g/mol} \)[/tex]
- Oxygen (O): [tex]\( 5 \times 16 = 80 \, \text{g/mol} \)[/tex]
Therefore, the empirical formula mass is:
[tex]\[
168 + 18 + 28 + 80 = 294 \, \text{g/mol}
\][/tex]
2. Determine the ratio of the molecular mass to the empirical formula mass:
We are given the molecular mass of the sweetener as [tex]\( 294.34 \, \text{g/mol} \)[/tex].
The ratio is calculated by dividing the molecular mass by the empirical formula mass:
[tex]\[
\text{Ratio} = \frac{294.34}{294} \approx 1.0012
\][/tex]
3. Determine the molecular formula:
Since the ratio is approximately 1, it suggests that the molecular formula is essentially the same as the empirical formula.
Multiplying the subscripts in the empirical formula [tex]\( C_{14}H_{18}N_{2}O_{5} \)[/tex] by 1 (because 1.0012 is very close to 1) gives us:
- Carbon (C): [tex]\( 14 \times 1 = 14 \)[/tex]
- Hydrogen (H): [tex]\( 18 \times 1 = 18 \)[/tex]
- Nitrogen (N): [tex]\( 2 \times 1 = 2 \)[/tex]
- Oxygen (O): [tex]\( 5 \times 1 = 5 \)[/tex]
Therefore, the molecular formula is [tex]\( C_{14}H_{18}N_{2}O_{5} \)[/tex].
Given the choices:
- [tex]\( C_{7}H_{9}NO_{2} \)[/tex]
- [tex]\( C_{7}H_{9}NO_{3} \)[/tex]
- [tex]\( C_{14}H_{18}N_{2}O_{5} \)[/tex]
- [tex]\( C_{28}H_{36}N_{4}O_{10} \)[/tex]
The correct molecular formula for the artificial sweetener is [tex]\( C_{14}H_{18}N_{2}O_{5} \)[/tex].
