To calculate the number of molecules in a breath of air, we'll follow these steps:
1. Convert temperature from Fahrenheit to Kelvin:
- The formula to convert Fahrenheit to Kelvin is:
[tex]\[ K = (F - 32) \times \frac{5}{9} + 273.15 \][/tex]
- Given the body temperature [tex]\(98.6^{\circ} F\)[/tex]:
[tex]\[ K = (98.6 - 32) \times \frac{5}{9} + 273.15 \][/tex]
Thus, the temperature in Kelvin is [tex]\(310.15 \, K\)[/tex].
2. Convert pressure from torr to atmospheres:
- The conversion factor is:
[tex]\[ 1 \, atm = 760 \, torr \][/tex]
- Given the pressure [tex]\(746 \, torr\)[/tex]:
[tex]\[ \text{Pressure in atmospheres} = \frac{746}{760} \][/tex]
Therefore, the pressure is approximately [tex]\(0.981578947 \, atm\)[/tex].
3. Use the ideal gas law to find the number of moles (n):
- The ideal gas law is given by [tex]\( PV = nRT \)[/tex], where:
- [tex]\( P \)[/tex] is the pressure in atmospheres ([tex]\(0.981578947 \, atm\)[/tex]),
- [tex]\( V \)[/tex] is the volume in liters ([tex]\(2.29 \, L\)[/tex]),
- [tex]\( R \)[/tex] is the ideal gas constant ([tex]\(0.0821 \, L \cdot atm / (K \cdot mol)\)[/tex]),
- [tex]\( T \)[/tex] is the temperature in Kelvin ([tex]\(310.15 \, K\)[/tex]).
- Rearranging the ideal gas law to solve for [tex]\( n \)[/tex] (number of moles):
[tex]\[ n = \frac{PV}{RT} \][/tex]
Plugging in the values:
[tex]\[ n = \frac{0.981578947 \times 2.29}{0.0821 \times 310.15} \][/tex]
Hence, the number of moles is [tex]\(0.088276636 \, mol\)[/tex].
4. Calculate the number of molecules using Avogadro's number:
- Avogadro's number ([tex]\(6.022 \times 10^{23}\)[/tex]) indicates the number of molecules per mole.
- The number of molecules ([tex]\(N\)[/tex]) is:
[tex]\[ N = n \times \text{Avogadro's number} \][/tex]
Using the calculated number of moles:
[tex]\[ N = 0.088276636 \times 6.022 \times 10^{23} \][/tex]
Thus, the number of molecules is approximately [tex]\(5.316 \times 10^{22}\)[/tex].
Therefore, the number of molecules in a breath of air with the given conditions is:
[tex]\[ 5.316 \times 10^{22} \, \text{molecules} \][/tex]
