Entropy Change (Ideal Gas)
Molar entropy change of an ideal gas between two T–P states using constant specific heat.
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The engineering
For an ideal gas with constant Cp, entropy change between two states splits cleanly into a temperature term and a pressure term. Heating raises entropy; compressing (raising pressure) lowers it. This is the workhorse relation for turbines, compressors, and refrigeration cycle analysis when you need entropy without a full property table.
The constant-Cp assumption is the gotcha: real gas Cp drifts with temperature, so over a wide span use a mean Cp or the polynomial form from a data book. Check your signs — compression at constant temperature always gives negative ΔS, and an isentropic (reversible adiabatic) step means the two terms cancel to zero.
Where this math comes from
Rudolf Clausius coined "entropy" in 1865, building on Sadi Carnot's 1824 cycle analysis and the reversible-heat integral ∮dQ/T. The ideal-gas working form — separating the T and P contributions — followed once the ideal-gas law and constant-heat-capacity idealizations were combined by the late 19th-century thermodynamicists.
The compact engineering statement used here is the one drilled into every chemical engineer through Smith, Van Ness & Abbott's Introduction to Chemical Engineering Thermodynamics, first published 1949 and still the standard reference for property changes across the profession.
- 1824Sadi CarnotAnalyzes the reversible heat-engine cycle underpinning entropy.
- 1865Rudolf ClausiusNames entropy and formalizes dS = dQ_rev/T.
- 1876J. Willard GibbsCasts entropy into state-property relations for gases and mixtures.
- 1949Smith, Van Ness & AbbottPublish the ideal-gas ΔS working form as the ChemE standard.
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