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Sagot :
To solve this problem, we will systematically explore the properties of each flask to determine which statement is true:
### Data Given:
- Flask Volume (both flasks): 2.0 L
- Temperature (both flasks): 298 K
- Mass of O2 in Flask A: 16.00 grams
- Mass of N2 in Flask B: 28.0 grams
### Molecular Weights:
- Molar Mass of O2: 32.00 g/mol
- Molar Mass of N2: 28.02 g/mol
### Calculations:
1. Number of Moles:
For Flask A (O2):
[tex]\[ \text{Moles of O2} = \frac{\text{mass}}{\text{molar mass}} = \frac{16.00 \text{ g}}{32.00 \text{ g/mol}} = 0.5 \text{ mol} \][/tex]
For Flask B (N2):
[tex]\[ \text{Moles of N2} = \frac{\text{mass}}{\text{molar mass}} = \frac{28.0 \text{ g}}{28.02 \text{ g/mol}} \approx 1.0 \text{ mol} \][/tex]
2. Using the Ideal Gas Law (PV = nRT):
Where R (gas constant) = 0.0821 L·atm/(mol·K)
For Flask A:
[tex]\[ P_A = \frac{n \cdot R \cdot T}{V} = \frac{0.5 \cdot 0.0821 \cdot 298}{2.0} \approx 6.12 \text{ atm} \][/tex]
For Flask B:
[tex]\[ P_B = \frac{n \cdot R \cdot T}{V} = \frac{1.0 \cdot 0.0821 \cdot 298}{2.0} \approx 12.22 \text{ atm} \][/tex]
3. Density:
Density is calculated as:
[tex]\[ \text{Density} = \frac{\text{mass}}{\text{volume}} \][/tex]
For Flask A:
[tex]\[ \text{Density}_A = \frac{16.00 \text{ g}}{2.0 \text{ L}} = 8.0 \text{ g/L} \][/tex]
For Flask B:
[tex]\[ \text{Density}_B = \frac{28.0 \text{ g}}{2.0 \text{ L}} = 14.0 \text{ g/L} \][/tex]
4. Number of Molecules:
Using Avogadro's number ([tex]\(6.022 \times 10^{23}\)[/tex] molecules/mol):
For Flask A:
[tex]\[ \text{Number of molecules in Flask A} = 0.5 \text{ mol} \times 6.022 \times 10^{23} \approx 3.011 \times 10^{23} \text{ molecules} \][/tex]
For Flask B:
[tex]\[ \text{Number of molecules in Flask B} = 1.0 \text{ mol} \times 6.022 \times 10^{23} \approx 6.0177 \times 10^{23} \text{ molecules} \][/tex]
### Comparison of Statements:
- Statement a: Flask A contains molecules with higher average kinetic energy than Flask B.
- This is false because the average kinetic energy of gas molecules depends on temperature, which is the same for both flasks.
- Statement b: Flask A has a higher density than Flask B.
- This is false because the density of Flask A (8.0 g/L) is less than the density of Flask B (14.0 g/L).
- Statement c: Flask A contains fewer molecules than Flask B.
- This is true because Flask A contains approximately [tex]\(3.011 \times 10^{23}\)[/tex] molecules, which is less than the approximately [tex]\(6.0177 \times 10^{23}\)[/tex] molecules in Flask B.
- Statement d: Flask A has a higher pressure than Flask B.
- This is false because the pressure in Flask A (6.12 atm) is less than the pressure in Flask B (12.22 atm).
### Conclusion:
The correct statement is:
c. Flask A contains fewer molecules than Flask B.
### Data Given:
- Flask Volume (both flasks): 2.0 L
- Temperature (both flasks): 298 K
- Mass of O2 in Flask A: 16.00 grams
- Mass of N2 in Flask B: 28.0 grams
### Molecular Weights:
- Molar Mass of O2: 32.00 g/mol
- Molar Mass of N2: 28.02 g/mol
### Calculations:
1. Number of Moles:
For Flask A (O2):
[tex]\[ \text{Moles of O2} = \frac{\text{mass}}{\text{molar mass}} = \frac{16.00 \text{ g}}{32.00 \text{ g/mol}} = 0.5 \text{ mol} \][/tex]
For Flask B (N2):
[tex]\[ \text{Moles of N2} = \frac{\text{mass}}{\text{molar mass}} = \frac{28.0 \text{ g}}{28.02 \text{ g/mol}} \approx 1.0 \text{ mol} \][/tex]
2. Using the Ideal Gas Law (PV = nRT):
Where R (gas constant) = 0.0821 L·atm/(mol·K)
For Flask A:
[tex]\[ P_A = \frac{n \cdot R \cdot T}{V} = \frac{0.5 \cdot 0.0821 \cdot 298}{2.0} \approx 6.12 \text{ atm} \][/tex]
For Flask B:
[tex]\[ P_B = \frac{n \cdot R \cdot T}{V} = \frac{1.0 \cdot 0.0821 \cdot 298}{2.0} \approx 12.22 \text{ atm} \][/tex]
3. Density:
Density is calculated as:
[tex]\[ \text{Density} = \frac{\text{mass}}{\text{volume}} \][/tex]
For Flask A:
[tex]\[ \text{Density}_A = \frac{16.00 \text{ g}}{2.0 \text{ L}} = 8.0 \text{ g/L} \][/tex]
For Flask B:
[tex]\[ \text{Density}_B = \frac{28.0 \text{ g}}{2.0 \text{ L}} = 14.0 \text{ g/L} \][/tex]
4. Number of Molecules:
Using Avogadro's number ([tex]\(6.022 \times 10^{23}\)[/tex] molecules/mol):
For Flask A:
[tex]\[ \text{Number of molecules in Flask A} = 0.5 \text{ mol} \times 6.022 \times 10^{23} \approx 3.011 \times 10^{23} \text{ molecules} \][/tex]
For Flask B:
[tex]\[ \text{Number of molecules in Flask B} = 1.0 \text{ mol} \times 6.022 \times 10^{23} \approx 6.0177 \times 10^{23} \text{ molecules} \][/tex]
### Comparison of Statements:
- Statement a: Flask A contains molecules with higher average kinetic energy than Flask B.
- This is false because the average kinetic energy of gas molecules depends on temperature, which is the same for both flasks.
- Statement b: Flask A has a higher density than Flask B.
- This is false because the density of Flask A (8.0 g/L) is less than the density of Flask B (14.0 g/L).
- Statement c: Flask A contains fewer molecules than Flask B.
- This is true because Flask A contains approximately [tex]\(3.011 \times 10^{23}\)[/tex] molecules, which is less than the approximately [tex]\(6.0177 \times 10^{23}\)[/tex] molecules in Flask B.
- Statement d: Flask A has a higher pressure than Flask B.
- This is false because the pressure in Flask A (6.12 atm) is less than the pressure in Flask B (12.22 atm).
### Conclusion:
The correct statement is:
c. Flask A contains fewer molecules than Flask B.
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