Answer :
To determine the pressure in a 6.00 L tank containing 14.4 grams of nitrogen gas at 385 K, we can use the Ideal Gas Law. The Ideal Gas Law is expressed as:
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure in atmospheres,
- [tex]\( V \)[/tex] is the volume in liters,
- [tex]\( n \)[/tex] is the number of moles of gas,
- [tex]\( R \)[/tex] is the ideal gas constant (0.0821 L atm/mol K),
- [tex]\( T \)[/tex] is the temperature in Kelvin.
### Step-by-Step Solution:
1. Identify the Given Values:
- Volume ([tex]\( V \)[/tex]) = 6.00 L
- Mass of nitrogen ([tex]\( \text{mass} \)[/tex]) = 14.4 grams
- Temperature ([tex]\( T \)[/tex]) = 385 K
- Ideal gas constant ([tex]\( R \)[/tex]) = 0.0821 L atm/mol K
2. Find the Molar Mass of Nitrogen:
- Nitrogen gas ([tex]\( \text{N}_2 \)[/tex]) has a molar mass of approximately 28.02 grams/mol.
3. Calculate the Number of Moles ([tex]\( n \)[/tex]):
- Use the formula: [tex]\( n = \frac{\text{mass}}{\text{molar mass}} \)[/tex]
- Substitute the values: [tex]\( n = \frac{14.4 \, \text{g}}{28.02 \, \text{g/mol}} \)[/tex]
- Calculate [tex]\( n \)[/tex]: [tex]\( n \approx 0.514 \, \text{moles} \)[/tex]
4. Apply the Ideal Gas Law to Find the Pressure ([tex]\( P \)[/tex]):
- Use the formula: [tex]\( P = \frac{nRT}{V} \)[/tex]
- Substitute the values:
- [tex]\( n = 0.514 \, \text{moles} \)[/tex]
- [tex]\( R = 0.0821 \, \text{L atm/mol K} \)[/tex]
- [tex]\( T = 385 \, \text{K} \)[/tex]
- [tex]\( V = 6.00 \, \text{L} \)[/tex]
- Calculate [tex]\( P \)[/tex]:
[tex]\( P = \frac{(0.514 \, \text{moles}) \times (0.0821 \, \text{L atm/mol K}) \times (385 \, \text{K})}{6.00 \, \text{L}} \)[/tex]
- The pressure [tex]\( P \approx 2.71 \, \text{atm} \)[/tex]
Thus, the pressure in the tank is approximately 2.71 atmospheres.
[tex]\[ PV = nRT \][/tex]
where:
- [tex]\( P \)[/tex] is the pressure in atmospheres,
- [tex]\( V \)[/tex] is the volume in liters,
- [tex]\( n \)[/tex] is the number of moles of gas,
- [tex]\( R \)[/tex] is the ideal gas constant (0.0821 L atm/mol K),
- [tex]\( T \)[/tex] is the temperature in Kelvin.
### Step-by-Step Solution:
1. Identify the Given Values:
- Volume ([tex]\( V \)[/tex]) = 6.00 L
- Mass of nitrogen ([tex]\( \text{mass} \)[/tex]) = 14.4 grams
- Temperature ([tex]\( T \)[/tex]) = 385 K
- Ideal gas constant ([tex]\( R \)[/tex]) = 0.0821 L atm/mol K
2. Find the Molar Mass of Nitrogen:
- Nitrogen gas ([tex]\( \text{N}_2 \)[/tex]) has a molar mass of approximately 28.02 grams/mol.
3. Calculate the Number of Moles ([tex]\( n \)[/tex]):
- Use the formula: [tex]\( n = \frac{\text{mass}}{\text{molar mass}} \)[/tex]
- Substitute the values: [tex]\( n = \frac{14.4 \, \text{g}}{28.02 \, \text{g/mol}} \)[/tex]
- Calculate [tex]\( n \)[/tex]: [tex]\( n \approx 0.514 \, \text{moles} \)[/tex]
4. Apply the Ideal Gas Law to Find the Pressure ([tex]\( P \)[/tex]):
- Use the formula: [tex]\( P = \frac{nRT}{V} \)[/tex]
- Substitute the values:
- [tex]\( n = 0.514 \, \text{moles} \)[/tex]
- [tex]\( R = 0.0821 \, \text{L atm/mol K} \)[/tex]
- [tex]\( T = 385 \, \text{K} \)[/tex]
- [tex]\( V = 6.00 \, \text{L} \)[/tex]
- Calculate [tex]\( P \)[/tex]:
[tex]\( P = \frac{(0.514 \, \text{moles}) \times (0.0821 \, \text{L atm/mol K}) \times (385 \, \text{K})}{6.00 \, \text{L}} \)[/tex]
- The pressure [tex]\( P \approx 2.71 \, \text{atm} \)[/tex]
Thus, the pressure in the tank is approximately 2.71 atmospheres.
