Question

How does the gas constant R is related to the universal gas constant \(\bar{R}\) and molecular mass M?

• A. \( R = \frac{\bar{R}^2}{M} \)
• B. \( R = \frac{\bar{R}}{M^2} \)
• C. \( R = \frac{\bar{R}}{M} \)
• D. \( R = \frac{\bar{R}^2}{M^3} \)

Hint:

Remember that the "universal" constant (\(\bar{R}\)) is the same for all gases, while the "specific" gas constant (R) is different for each gas because it depends on the molar mass. The specific constant is always smaller than the universal one (since M>1).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks for the relationship between the specific gas constant (R) and the universal gas constant (\(\bar{R}\) or sometimes \(R_u\)). The ideal gas law can be written in two forms: one using moles and the universal constant, and one using mass and the specific constant.
Step 2: Key Formula or Approach:
The molar form of the ideal gas law is:
\[ PV = n\bar{R}T \]
where \(n\) is the number of moles.
The mass-based form of the ideal gas law is:
\[ PV = mRT \]
where \(m\) is the mass of the gas and R is the specific gas constant.
The number of moles \(n\) is related to mass \(m\) and molar mass \(M\) by:
\[ n = \frac{m}{M} \]
Step 3: Detailed Explanation:
We can derive the relationship by equating the two forms of the ideal gas law. Substitute \(n = m/M\) into the molar form:
\[ PV = \left(\frac{m}{M}\right)\bar{R}T \]
Compare this with the mass-based form, \(PV = mRT\):
\[ mRT = m\left(\frac{\bar{R}}{M}\right)T \]
By comparing the terms, we can see that:
\[ R = \frac{\bar{R}}{M} \]
Step 4: Final Answer:
The specific gas constant R is equal to the universal gas constant \(\bar{R}\) divided by the molar mass M.