Determine the volume V in cm3 for 1 mole of CO2 gas at 490K and 200 atm using the van der Waals equation of state (EOS). Take the largest real value of V and give your answer to the nearest whole number (do not put any units in your answer!). The van der Waals EOS is given as follows:

The van der Waals equation of state (EOS) for a real gas is expressed as:

\[
\left( P + \frac{a}{V_m^2} \right) \left( V_m - b \right) = RT
\]

Where:
- \( P \) = Pressure of the gas (in atm),
- \( V_m \) = Molar volume of the gas (in cm³/mol),
- \( T \) = Temperature of the gas (in Kelvin),
- \( R \) = Universal gas constant (0.08206 L·atm/mol·K),
- \( a \) and \( b \) are van der Waals constants specific to each gas.

For carbon dioxide (CO₂), the constants are:
- \( a = 3.59 \, \text{L}^2 \cdot \text{atm} / \text{mol}^2 \)
- \( b = 0.0427 \, \text{L/mol} \)

To determine the molar volume (\( V_m \)) for 1 mole of CO₂ at \( T = 490 \, \text{K} \) and \( P = 200 \, \text{atm} \), we need to solve the van der Waals equation. Let's proceed with the calculation.The largest real value of the molar volume \( V_m \) for 1 mole of CO₂ at 490 K and 200 atm, using the van der Waals equation, is approximately **162 cm³**.

Would you like any further details or have any questions about the process? 

Here are 5 related questions:
1. How does the van der Waals equation differ from the ideal gas law?
2. What would the volume be if CO₂ were treated as an ideal gas at these conditions?
3. How do the constants \(a\) and \(b\) for different gases affect the calculation?
4. What are the limitations of the van der Waals equation for high pressures and low temperatures?
5. Can the van der Waals equation predict gas behavior at critical points?

**Tip:** For extreme conditions (high pressures or low temperatures), the van der Waals equation provides a more accurate prediction of gas behavior than the ideal gas law.

Calculate the molar volume of 1 mole of CO2 gas at 490K and 200 atm using the van der Waals equation. This problem explores the thermodynamic behavior of real gases under high pressure and temperature, offering a comparison to the ideal gas law. Learn key concepts in thermodynamics, including gas constants and real gas behaviors.

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