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.
-
Apv = mRT
-
Bpv = RTm
-
Cpvm = C
-
Dpv = (RT)m
Answer
Correct Answer: pv = mRT
Explanation
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:
Start from pV = nR_uT.Use n = m/M and R = R_u/M ⇒ pV = mRT.Therefore the correct compact form with total volume v is pv = mRT.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