To determine how many molecules of vitamin A are in a sample containing [tex]\( 4.0 \times 10^{22} \)[/tex] carbon atoms, follow these steps:
1. Identify given information:
- Number of carbon atoms, [tex]\( \text{atoms}_C = 4.0 \times 10^{22} \)[/tex]
- Avogadro's number, [tex]\( N_A = 6.022 \times 10^{23} \)[/tex] atoms/mol
2. Determine the number of carbon atoms per molecule of vitamin A:
- Assume each molecule of vitamin A contains 20 carbon atoms.
3. Calculate the moles of carbon atoms:
[tex]\[
\text{moles}_C = \frac{\text{atoms}_C}{N_A} = \frac{4.0 \times 10^{22}}{6.022 \times 10^{23}}
\][/tex]
4. Convert moles of carbon atoms to moles of vitamin A molecules:
[tex]\[
\text{moles of vitamin A} = \frac{\text{moles}_C}{\text{carbon atoms per molecule of vitamin A}} = \frac{\text{moles}_C}{20}
\][/tex]
5. Convert moles of vitamin A to molecules:
[tex]\[
\text{molecules}_{\text{vitamin A}} = \text{moles of vitamin A} \times N_A
\][/tex]
6. Combine the calculations:
First, calculate the moles of carbon atoms:
[tex]\[
\text{moles}_C = \frac{4.0 \times 10^{22}}{6.022 \times 10^{23}} = \frac{4.0}{6.022} \times 10^{-1} \approx 0.6646 \times 10^{-1} = 6.646 \times 10^{-2} \text{ mol}
\][/tex]
Next, determine the moles of vitamin A:
[tex]\[
\text{moles of vitamin A} = \frac{6.646 \times 10^{-2}}{20} = 3.323 \times 10^{-3} \text{ mol}
\][/tex]
Finally, convert moles of vitamin A to molecules:
[tex]\[
\text{molecules}_{\text{vitamin A}} = 3.323 \times 10^{-3} \times 6.022 \times 10^{23} = 2.000 \times 10^{21} \text{ molecules}
\][/tex]
7. Convert the resulting molecules to the required scientific notation format [tex]\([?] \times 10^{[7]}\)[/tex]:
[tex]\[
2.000 \times 10^{21} = 200000000000000.0 \times 10^7
\][/tex]
8. Summarize the answer: In scientific notation, this is equivalent to:
- Coefficient: [tex]\(2.0 \times 10^{14}\)[/tex]
- Exponent adjustment as [tex]\(200000000000000.0 = 2.0 \times 10^{14}\)[/tex]
Therefore, the solution to the problem is:
[tex]\[ \boxed{200000000000000.0} \times 10^{\boxed{7}} \][/tex]
