Universal (molar) gas constant – relation to specific gas constant The universal gas constant R_u (per mole) is related to a gas’s specific gas constant R (per unit mass) by which product?

Introduction / Context:
Two common forms of the gas constant appear in thermodynamics: R_u, the universal (molar) gas constant with units J/(mol·K), and R, the specific gas constant with units J/(kg·K). Converting between them requires the molar (molecular) mass of the gas.

Given Data / Assumptions:

• R_u ≈ 8.314 kJ/(kmol·K) or 8.314 J/(mol·K).
• R depends on the particular gas: R = R_u / M.
• M is the molar (molecular) mass in kg/kmol or kg/mol consistent with units.

Concept / Approach:
The relationship is R_u = M * R, where M is the molar mass of the gas and R is its specific gas constant on a per-mass basis. This follows directly from writing the ideal-gas equation either as p * V = n * R_u * T (molar form) or p * v = R * T (specific form) and substituting n = m/M.

Step-by-Step Solution:
Start with p * V = n * R_u * T and n = m / M.Divide by mass m to get p * v = (R_u / M) * T.Identify R = R_u / M → rearrange to R_u = M * R.Therefore, the correct product is molecular mass times specific gas constant.

Verification / Alternative check:
For air: M ≈ 28.97 kg/kmol, R ≈ 0.287 kJ/(kg·K). Product M * R ≈ 28.97 * 0.287 ≈ 8.314 kJ/(kmol·K) = R_u, confirming the relation.

Why Other Options Are Wrong:

• Atomic mass pertains to single atoms; many gases are diatomic or polyatomic; the correct parameter is molar mass of the gas species.
• Products with c_p or c_v do not produce R_u; instead, for ideal gases c_p − c_v = R (not R_u).
• “Molar mass divided by c_p” is dimensionally inconsistent.

Common Pitfalls:
Mixing per-mole and per-mass quantities. Always ensure unit consistency when converting between R and R_u.

Final Answer:
molecular mass * specific gas constant (R)