To determine which gas possesses the largest internal energy, we can use the formula for internal energy (U) of an ideal gas, which is given by:
\[ U = N C_V T \]
where:
- \( N \) = number of moles of the gas
- \( C_V \) = molar heat capacity at constant volume
- \( T \) = absolute temperature in Kelvin
### Step-by-Step Solution:
1. **Identify the Gases and Their Properties:**
- Helium: 1 mole, \( V = 1 \, \text{m}^3 \), \( T = 300 \, \text{K} \)
- Nitrogen: 56 kg, \( P = 10^6 \, \text{N/m}^2 \), \( T = 300 \, \text{K} \)
- Oxygen: 8 g, \( P = 8 \, \text{atm} \), \( T = 300 \, \text{K} \)
- Argon: \( 6 \times 10^{26} \) molecules, \( V = 40 \, \text{m}^3 \), \( T = 900 \, \text{K} \)
2. **Calculate the Number of Moles (N) for Each Gas:**
- For Helium:
\[ N = 1 \, \text{mole} \]
- For Nitrogen:
\[ \text{Molar mass of } N_2 = 28 \, \text{g/mol} \]
\[ N = \frac{56000 \, \text{g}}{28 \, \text{g/mol}} = 2000 \, \text{moles} \]
- For Oxygen:
\[ \text{Molar mass of } O_2 = 32 \, \text{g/mol} \]
\[ N = \frac{8 \, \text{g}}{32 \, \text{g/mol}} = 0.25 \, \text{moles} \]
- For Argon:
\[ \text{Number of moles} = \frac{6 \times 10^{26}}{6.022 \times 10^{23}} \approx 1000 \, \text{moles} \]
3. **Determine the Molar Heat Capacity (C_V) for Each Gas:**
- For Helium (monatomic):
\[ C_V = \frac{3}{2} R \]
- For Nitrogen (diatomic):
\[ C_V = \frac{5}{2} R \]
- For Oxygen (diatomic):
\[ C_V = \frac{5}{2} R \]
- For Argon (monatomic):
\[ C_V = \frac{3}{2} R \]
4. **Calculate Internal Energy (U) for Each Gas:**
- For Helium:
\[ U_{He} = 1 \cdot \frac{3}{2} R \cdot 300 = 450 R \]
- For Nitrogen:
\[ U_{N_2} = 2000 \cdot \frac{5}{2} R \cdot 300 = 1500000 R \]
- For Oxygen:
\[ U_{O_2} = 0.25 \cdot \frac{5}{2} R \cdot 300 = 187.5 R \]
- For Argon:
\[ U_{Ar} = 1000 \cdot \frac{3}{2} R \cdot 900 = 1350000 R \]
5. **Compare the Internal Energies:**
- \( U_{He} = 450 R \)
- \( U_{N_2} = 1500000 R \)
- \( U_{O_2} = 187.5 R \)
- \( U_{Ar} = 1350000 R \)
From the calculations, we can see that the internal energy of Nitrogen is the largest.
### Conclusion:
The gas that possesses the largest internal energy is **Nitrogen** (56 kg at \( 10^6 \, \text{N/m}^2 \) and 300 K).
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