Answer :
To solve this problem, we will use the ideal gas law, which is expressed as:
[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 the gas,
- [tex]\( R \)[/tex] is the ideal gas constant (0.0821 L·atm/(mol·K)),
- [tex]\( T \)[/tex] is the temperature in Kelvin (K).
Step 1: Convert the given pressure from mmHg to atm.
The problem states that the pressure is 505 mmHg. To convert mmHg to atm, use the conversion factor:
[tex]\[ 1 \, \text{atm} = 760 \, \text{mmHg} \][/tex]
[tex]\[ \text{Pressure in atm} = \frac{505 \, \text{mmHg}}{760 \, \text{mmHg/atm}} \approx 0.6645 \, \text{atm} \][/tex]
Step 2: Calculate the number of moles of oxygen.
The given mass of oxygen is 2.00 grams, and the molar mass of oxygen ([tex]\( O_2 \)[/tex]) is 32.00 g/mol.
[tex]\[ \text{Moles of O}_2 = \frac{\text{mass}}{\text{molar mass}} = \frac{2.00 \, \text{g}}{32.00 \, \text{g/mol}} = 0.0625 \, \text{mol} \][/tex]
Step 3: Use the ideal gas law to find the temperature in Kelvin.
Rearrange the ideal gas law to solve for temperature [tex]\( T \)[/tex]:
[tex]\[ T = \frac{PV}{nR} \][/tex]
Substitute the known values:
[tex]\[ T = \frac{(0.6645 \, \text{atm})(2.00 \, \text{L})}{(0.0625 \, \text{mol})(0.0821 \, \text{L·atm/(mol·K)})} \][/tex]
[tex]\[ T \approx 258.99 \, \text{K} \][/tex]
Step 4: Convert the temperature from Kelvin to degrees Celsius.
[tex]\[ \text{Temperature in Celsius} = T - 273.15 \][/tex]
[tex]\[ \text{Temperature in Celsius} = 258.99 \, \text{K} - 273.15 \][/tex]
[tex]\[ \text{Temperature in Celsius} \approx -14.16^\circ \text{C} \][/tex]
Therefore, the temperature inside the flask is approximately [tex]\(-14.16^\circ\)[/tex] Celsius.
[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 the gas,
- [tex]\( R \)[/tex] is the ideal gas constant (0.0821 L·atm/(mol·K)),
- [tex]\( T \)[/tex] is the temperature in Kelvin (K).
Step 1: Convert the given pressure from mmHg to atm.
The problem states that the pressure is 505 mmHg. To convert mmHg to atm, use the conversion factor:
[tex]\[ 1 \, \text{atm} = 760 \, \text{mmHg} \][/tex]
[tex]\[ \text{Pressure in atm} = \frac{505 \, \text{mmHg}}{760 \, \text{mmHg/atm}} \approx 0.6645 \, \text{atm} \][/tex]
Step 2: Calculate the number of moles of oxygen.
The given mass of oxygen is 2.00 grams, and the molar mass of oxygen ([tex]\( O_2 \)[/tex]) is 32.00 g/mol.
[tex]\[ \text{Moles of O}_2 = \frac{\text{mass}}{\text{molar mass}} = \frac{2.00 \, \text{g}}{32.00 \, \text{g/mol}} = 0.0625 \, \text{mol} \][/tex]
Step 3: Use the ideal gas law to find the temperature in Kelvin.
Rearrange the ideal gas law to solve for temperature [tex]\( T \)[/tex]:
[tex]\[ T = \frac{PV}{nR} \][/tex]
Substitute the known values:
[tex]\[ T = \frac{(0.6645 \, \text{atm})(2.00 \, \text{L})}{(0.0625 \, \text{mol})(0.0821 \, \text{L·atm/(mol·K)})} \][/tex]
[tex]\[ T \approx 258.99 \, \text{K} \][/tex]
Step 4: Convert the temperature from Kelvin to degrees Celsius.
[tex]\[ \text{Temperature in Celsius} = T - 273.15 \][/tex]
[tex]\[ \text{Temperature in Celsius} = 258.99 \, \text{K} - 273.15 \][/tex]
[tex]\[ \text{Temperature in Celsius} \approx -14.16^\circ \text{C} \][/tex]
Therefore, the temperature inside the flask is approximately [tex]\(-14.16^\circ\)[/tex] Celsius.
