What is the difference between R and Ru? How are these two related?
The Correct Answer and Explanation is:
Correct Answer:
R typically refers to the universal gas constant, while Ru refers to the universal gas constant per mole (often also denoted as R̅). In practice, R and Ru are often the same and used interchangeably, but they are conceptually linked to different contexts depending on the formulation of the ideal gas law—whether it’s based on moles or mass.
Detailed Explanation (300+ words):
In thermodynamics and physical chemistry, the ideal gas law is a fundamental equation that relates pressure, volume, temperature, and amount of gas. The equation can be expressed in two different forms, depending on whether you use moles (n) or mass (m):
- Molar form of the ideal gas law: PV=nRuTPV = nRuT
- Here, P is pressure, V is volume, n is the number of moles, T is temperature in Kelvin, and Ru is the universal gas constant (also written as R̅).
- Its value is: Ru=8.314 J/mol\cdotpKRu = 8.314\ \text{J/mol·K}
- Mass-based form of the ideal gas law: PV=mRTPV = mRT
- Here, m is the mass of the gas, and R is the specific gas constant.
- The specific gas constant R is related to Ru by: R=RuMR = \frac{Ru}{M} where M is the molar mass of the specific gas (in kg/mol).
Relationship Between R and Ru:
- Ru is constant for all ideal gases.
- R varies depending on the type of gas, since it depends on the gas’s molar mass.
- For example, the specific gas constant for dry air is: R=8.314 J/mol\cdotpK0.02897 kg/mol≈287 J/kg\cdotpKR = \frac{8.314\ \text{J/mol·K}}{0.02897\ \text{kg/mol}} \approx 287\ \text{J/kg·K}
Summary:
- Ru (or R̅) is the universal gas constant per mole.
- R is the specific gas constant per kilogram, unique to each gas.
- They are related by: R = Ru /