Can CP Be Less Than CV ?

Abstract

Can CP be less than CV? This is a fundamental question in physics, chemistry, chemical engineering, and mechanical engineering. This question hangs in the minds of many students, instructors, and researchers. The first instinct is to answer “Yes, for water between 0 and 4 °C” if one knows that water expands as temperature decreases in this temperature range. The same question is asked in several Physical Chemistry and Physics textbooks. Students are supposed to answer that water contracts when heated at below 4 °C in an isobaric process. Because work is done to the contracting water, less heat is required to increase the water temperature in an isobaric process than in an isochoric process. Therefore, CP is less than CV. However, this answer is fundamentally flawed because it assumes, implicitly and incorrectly, that the internal energy change of water depends solely on its temperature change. Neglecting the variation of the internal energy with volume (internal pressure) will invalidate the Clausius inequality and violate the second law of thermodynamics. Once the internal pressure is properly taken into account, it becomes clear that CP cannot be less than CV for any substance at any temperature regardless of the sign of the thermal expansion coefficient of the substance.