The Rankine temperature scale used in engineering is to the Fahrenheit scale as the Kelvin scale is to the Celsius scale. That is, 1 Rankine degree is the same size as 1 Fahrenheit degree, and 0 °R = absolute zero. (b) What is the value of the gas constant R on the Rankine scale in 1L ~ atm2>1°R ~ mol2? (c) Use the van der Waals equation to determine the pressure inside a 400.0-mL vessel that contains 2.50 mol of CH4 at a temperature of 525 °R. For CH4, a = 2.253 1L2 ~ atm2>mol2 and b = 0.04278 L>mol.

Verified step by step guidance

1. (b) To find the value of the gas constant R on the Rankine scale, first recall the value of R in SI units (L atm / K mol). Convert the temperature from Kelvin to Rankine by using the conversion factor (1 K = 1.8 °R). Multiply the value of R by 1.8 to adjust for the scale change from Kelvin to Rankine.
2. (c) To use the van der Waals equation to determine the pressure inside the vessel, start by writing down the van der Waals equation: \( P = \frac{nRT}{V-nb} - \frac{an^2}{V^2} \), where P is the pressure, n is the number of moles, R is the gas constant, T is the temperature, V is the volume, a is the measure of attraction between particles, and b is the volume excluded by a mole of particles.
3. Substitute the given values into the van der Waals equation. Convert the volume of the vessel from mL to L (1 mL = 0.001 L).
4. Plug in the values for n (number of moles of CH4), R (gas constant in L atm / °R mol), T (temperature in °R), V (volume in L), a, and b into the van der Waals equation.
5. Simplify the equation to solve for P, the pressure inside the vessel.

Key Concepts

Gas Constant (R)

The gas constant (R) is a fundamental constant in the ideal gas law, representing the relationship between pressure, volume, temperature, and the number of moles of a gas. It has different values depending on the units used, such as 0.0821 L·atm/(K·mol) in the metric system. In the Rankine scale, R must be converted accordingly, as it is essential for calculations involving gas behavior under varying conditions.

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 is particularly useful for real gases, providing a more accurate prediction of pressure, volume, and temperature relationships under non-ideal conditions.

Temperature Scales

Temperature scales, such as Rankine, Kelvin, Celsius, and Fahrenheit, provide a way to quantify thermal energy. The Rankine scale is an absolute temperature scale where 0 °R corresponds to absolute zero, similar to Kelvin. Understanding the relationships and conversions between these scales is crucial for solving thermodynamic problems, especially when applying equations like the van der Waals equation that require consistent temperature units.