Molar (volumetric) specific heat at constant pressure: The molar specific heat at constant pressure (per mole) can be expressed as which product?

Introduction / Context:
Specific heats may be reported per unit mass (kJ/kg·K) or per mole (kJ/kmol·K). Converting between these bases is common when using equations of state and reaction stoichiometry that are naturally molar while equipment calculations are often mass-based.

Given Data / Assumptions:

• Molar mass M (kg/kmol) of the gas is known.
• Mass-based cp has units kJ/(kg·K).
• Molar cp,m should have units kJ/(kmol·K).

Concept / Approach:
The conversion from mass-based to molar basis multiplies by the molar mass: cp,molar = M * cp (mass basis). Similarly, cv,molar = M * cv. The universal gas constant R_u relates to the mass-based specific gas constant by R_u = M * R. While cp,molar − cv,molar = R_u, cp,molar itself is not simply M * R nor M * cv; it is M multiplied by cp on a per-mass basis.

Step-by-Step Solution:

Verification / Alternative check:
Example: For air (approx. M = 28.97 kg/kmol) with cp ≈ 1.004 kJ/kg·K, cp,molar ≈ 29.1 kJ/kmol·K, matching standard tables.

Why Other Options Are Wrong:

• M * cv: Gives cv,molar, not cp,molar.
• Atomic mass * gas constant or molecular mass * gas constant: Yield R_u or related constants, not cp,molar.

Common Pitfalls:
Confusing universal R_u with specific R; forgetting to convert the basis when switching between mass and molar properties.

Final Answer:
molar mass of the gas and the mass-based specific heat at constant pressure