Answer :
To find the volume of oxygen (O₂) that a person inhales under given conditions, we can use the Ideal Gas Law, which is expressed by the formula:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure in atmospheres (atm)
- [tex]\( V \)[/tex] is the volume in liters (L)
- [tex]\( n \)[/tex] is the number of moles of gas
- [tex]\( R \)[/tex] is the ideal gas constant (approximately 0.0821 L·atm/(mol·K))
- [tex]\( T \)[/tex] is the temperature in Kelvin (K)
Here are the steps to solve the problem:
1. Convert the temperature from Celsius to Kelvin:
The temperature given is 36.9 °C. To convert Celsius to Kelvin, use the formula:
[tex]\[ T(K) = T(°C) + 273.15 \][/tex]
Thus, [tex]\( T = 36.9 + 273.15 = 310.05 \)[/tex] K
2. Calculate the number of moles of O₂:
Given the mass of inhaled oxygen is 82.3 g and the molar mass of oxygen (O₂) is approximately 32.00 g/mol, the number of moles of oxygen can be calculated using:
[tex]\[ n = \frac{\text{mass}}{\text{molar mass}} = \frac{82.3\, \text{g}}{32.00\, \text{g/mol}} = 2.57 \, \text{moles} \][/tex]
3. Calculate the volume using the Ideal Gas Law:
Rearrange the Ideal Gas Law to solve for volume ([tex]\( V \)[/tex]):
[tex]\[ V = \frac{nRT}{P} \][/tex]
Plugging in the values:
- [tex]\( n = 2.57 \)[/tex] moles
- [tex]\( R = 0.0821 \)[/tex] L·atm/(mol·K)
- [tex]\( T = 310.05 \)[/tex] K
- [tex]\( P = 1.03 \)[/tex] atm
[tex]\[ V = \frac{2.57 \, \text{moles} \times 0.0821 \, \text{L·atm/(mol·K)} \times 310.05 \, \text{K}}{1.03 \, \text{atm}} \][/tex]
[tex]\[ V \approx 63.6 \, \text{L} \][/tex]
The volume of oxygen inhaled is approximately 63.6 liters, ensuring the result has the appropriate number of significant figures based on the given data.
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure in atmospheres (atm)
- [tex]\( V \)[/tex] is the volume in liters (L)
- [tex]\( n \)[/tex] is the number of moles of gas
- [tex]\( R \)[/tex] is the ideal gas constant (approximately 0.0821 L·atm/(mol·K))
- [tex]\( T \)[/tex] is the temperature in Kelvin (K)
Here are the steps to solve the problem:
1. Convert the temperature from Celsius to Kelvin:
The temperature given is 36.9 °C. To convert Celsius to Kelvin, use the formula:
[tex]\[ T(K) = T(°C) + 273.15 \][/tex]
Thus, [tex]\( T = 36.9 + 273.15 = 310.05 \)[/tex] K
2. Calculate the number of moles of O₂:
Given the mass of inhaled oxygen is 82.3 g and the molar mass of oxygen (O₂) is approximately 32.00 g/mol, the number of moles of oxygen can be calculated using:
[tex]\[ n = \frac{\text{mass}}{\text{molar mass}} = \frac{82.3\, \text{g}}{32.00\, \text{g/mol}} = 2.57 \, \text{moles} \][/tex]
3. Calculate the volume using the Ideal Gas Law:
Rearrange the Ideal Gas Law to solve for volume ([tex]\( V \)[/tex]):
[tex]\[ V = \frac{nRT}{P} \][/tex]
Plugging in the values:
- [tex]\( n = 2.57 \)[/tex] moles
- [tex]\( R = 0.0821 \)[/tex] L·atm/(mol·K)
- [tex]\( T = 310.05 \)[/tex] K
- [tex]\( P = 1.03 \)[/tex] atm
[tex]\[ V = \frac{2.57 \, \text{moles} \times 0.0821 \, \text{L·atm/(mol·K)} \times 310.05 \, \text{K}}{1.03 \, \text{atm}} \][/tex]
[tex]\[ V \approx 63.6 \, \text{L} \][/tex]
The volume of oxygen inhaled is approximately 63.6 liters, ensuring the result has the appropriate number of significant figures based on the given data.
