Regnault’s Law (Perfect Gas Approximation) – Behavior of Specific Heats According to Regnault’s law for ideal gases in an approximate range, the specific heats at constant pressure (cp) and at constant volume (cv) are taken as constants—i.e., they do not change appreciably with pressure and temperature.

Introduction / Context:
Regnault’s law is a classical engineering approximation used to simplify calculations with ideal gases: take cp and cv as constants over a moderate temperature range. Although modern data show mild variation, the assumption remains practical in many problems and underpins closed-form formulas in basic thermodynamics.

Given Data / Assumptions:

• Gas behaves ideally in the range of interest.
• Moderate temperatures and pressures where property variation is small.
• Engineering accuracy acceptable for introductory analysis.

Concept / Approach:

Perfect-gas models often assume cp and cv are constants (calorically perfect gas). This leads to useful relations like h = cpT and u = cvT and supports isentropic formulas with a constant gamma = cp/cv. Regnault’s empirical observations motivated treating specific heats as essentially constant over limited ranges, even though precision work may require temperature-dependent property data.

Step-by-Step Solution:

Verification / Alternative check:

Comparing ideal-gas predictions using constant cp to NASA polynomials shows small errors near room temperature but growing deviations at high T, confirming the approximation’s limited yet useful domain.

Why Other Options Are Wrong:

Monotonic pressure dependencies are negligible for ideal gases; “become zero” is unphysical; complex opposing trends are not part of the ideal-gas model.

Common Pitfalls:

Applying constant specific heats at very high temperatures (e.g., combustion) where vibrational modes raise cp noticeably; confusing ideal-gas assumptions with real-gas behavior at high pressure.

Final Answer:

do not change