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To solve this problem, we will apply the van der Waals equation of state:

\[
P = \frac{nRT}{V - nb} - \frac{a n^2}{V^2}
\]

### Given:
- \( P = 100 \, \text{atm} \)
- \( T = 480 \, \text{K} \)
- \( n = 1 \, \text{mol} \)
- \( R = 0.08206 \, \text{L·atm/(mol·K)} \) (the ideal gas constant)
- Constants for CO₂:
  - \( a = 3.59 \, \text{L}^2 \text{·atm/mol}^2 \)
  - \( b = 0.0427 \, \text{L/mol} \)

We need to solve for the volume \( V \). For simplicity, we will treat the volume in liters first, then convert to cm³ later.

Rearranging the equation:

\[
100 = \frac{(1)(0.08206)(480)}{V - (1)(0.0427)} - \frac{(3.59)(1^2)}{V^2}
\]

This is a cubic equation in terms of \( V \). Let's solve for the largest real root of this equation. I'll proceed with the calculation.The largest real value for the volume \( V \) is approximately 346 cm³ when rounded to the nearest whole number.

Would you like further details on the calculation or have any other questions?

Here are 5 related questions you might find interesting:
1. How does the van der Waals equation differ from the ideal gas law?
2. What is the significance of the constants \( a \) and \( b \) in the van der Waals equation?
3. How would the result change if the temperature were higher?
4. What happens to the volume if the pressure increases while keeping the temperature constant?
5. Can we use the ideal gas law to approximate this problem, and how would the result compare?

**Tip:** The constants \( a \) and \( b \) in the van der Waals equation account for intermolecular forces and the finite size of molecules, respectively.

Determine the volume V in cm³ for 1 mole of CO₂ gas at 480K and 100 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.

Calculate the volume of 1 mole of CO₂ gas at 480K and 100 atm using the van der Waals equation. This problem explores cubic equations and gas laws for real gases, providing a detailed step-by-step solution. Suitable for undergraduate-level thermodynamics students.

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