Uranium hexafluoride, a molecular solid used for purification of the uranium isotope needed to fuel nuclear power plants, sublimes at 56.5 °C. Assume that you have a 22.9 L vessel that contains 512.9 g of UF6 at 70.0 °C. (a) What is the pressure in the vessel calculated using the ideal gas law? (b) What is the pressure in the vessel calculated using the van der Waals equation? For UF6, a = 15.80 L² atm/mol²; b = 0.1128 L/mol.

Verified step by step guidance

1. Step 1: Calculate the number of moles of UF6 using its molar mass. The molar mass of UF6 is the sum of the atomic masses of uranium (U) and six fluorine (F) atoms. Use the formula: \( n = \frac{\text{mass}}{\text{molar mass}} \).
2. Step 2: Use the ideal gas law to calculate the pressure. The ideal gas law is given by \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L atm/mol K), and \( T \) is the temperature in Kelvin. Convert the temperature from Celsius to Kelvin by adding 273.15.
3. Step 3: Rearrange the ideal gas law to solve for pressure \( P \): \( P = \frac{nRT}{V} \). Substitute the values for \( n \), \( R \), \( T \), and \( V \) to find the pressure using the ideal gas law.
4. Step 4: Use the van der Waals equation to calculate the pressure. The van der Waals equation is \( \left( P + \frac{a(n/V)^2} \right)(V - nb) = nRT \), where \( a \) and \( b \) are van der Waals constants specific to UF6.
5. Step 5: Rearrange the van der Waals equation to solve for pressure \( P \): \( P = \frac{nRT}{V - nb} - \frac{a(n/V)^2} \). Substitute the values for \( n \), \( R \), \( T \), \( V \), \( a \), and \( b \) to find the pressure using the van der Waals equation.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. This law assumes that gas particles do not interact and occupy no volume, making it a good approximation under many conditions but not all.

Van der Waals Equation

The Van der Waals equation is an adjustment of the Ideal Gas Law that accounts for the volume occupied by gas molecules and the attractive forces between them. It is expressed as (P + a(n/V)²)(V - nb) = nRT, where 'a' and 'b' are constants specific to each gas. This equation provides a more accurate description of real gas behavior, especially at high pressures and low temperatures, where deviations from ideality are significant.

Sublimation

Sublimation is the process by which a solid transitions directly into a gas without passing through the liquid phase. This occurs under specific temperature and pressure conditions, as seen with uranium hexafluoride, which sublimes at 56.5 °C. Understanding sublimation is crucial for interpreting the behavior of substances in different states and for calculations involving phase changes in thermodynamic contexts.