General (ideal) gas equation in mass form: Select the correct form of the ideal gas equation for a fixed mass m at absolute temperature T.

Introduction / Context:The ideal gas equation is the cornerstone relation connecting pressure, volume, temperature, and amount of gas. Using the mass-based form is common in mechanical engineering calculations for compressors, nozzles, and air-standard cycles.

Given Data / Assumptions:

• Ideal gas behavior.
• Specific gas constant R is on a per-mass basis.
• p is absolute pressure, v is total volume occupied by mass m.

Concept / Approach:The molar form is pV = nR_uT, with n moles and universal constant R_u. Converting to mass m with specific gas constant R = R_u/M gives the mass form: pV = mRT. Rearranging for specific volume v_hat = V/m yields pv_hat = RT, often used in property tables and cycle work.

Step-by-Step Solution:

Verification / Alternative check:Dimensional check: [p][V] = energy; [m][R]*[T] has the same dimensions, confirming consistency.

Why Other Options Are Wrong:

• pv = RTm and pv = (RT)m: Same as option A in value but unconventional ordering; however, the standard, unambiguous expression is pv = mRT.
• pvm = C: Suggests a constant product unrelated to temperature; incorrect.

Common Pitfalls:Mixing molar and mass bases; forgetting that R is gas-specific whereas R_u is universal.

Final Answer:pv = mRT