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(w) -r( ) =T Ꭷ-p ᎧᎢ2 ᎧᏙ T V

(w) -r( ) =T Ꭷ-p ᎧᎢ2 ᎧᏙ T V

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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**Transcription:** Show that \[ \left( \frac{\partial C_v}{\partial V} \right)_T = T \left( \frac{\partial^2 p}{\partial T^2} \right)_V, \] and evaluate this quantity for a van der Waals gas. --- **Explanation:** This equation explores the relationship between the partial derivative of the heat capacity at constant volume \((C_v)\) with respect to volume \((V)\) at constant temperature \((T)\), and another partial derivative involving the pressure \((p)\) with respect to temperature \((T)\) at constant volume. This relationship is an important concept in thermodynamics, particularly when analyzing real gases like the van der Waals gas. The equation can be used to evaluate how changes in volume affect the heat capacity at constant volume for a van der Waals gas by relating it to temperature and pressure variations.
Transcribed Image Text:**Transcription:** Show that \[ \left( \frac{\partial C_v}{\partial V} \right)_T = T \left( \frac{\partial^2 p}{\partial T^2} \right)_V, \] and evaluate this quantity for a van der Waals gas. --- **Explanation:** This equation explores the relationship between the partial derivative of the heat capacity at constant volume \((C_v)\) with respect to volume \((V)\) at constant temperature \((T)\), and another partial derivative involving the pressure \((p)\) with respect to temperature \((T)\) at constant volume. This relationship is an important concept in thermodynamics, particularly when analyzing real gases like the van der Waals gas. The equation can be used to evaluate how changes in volume affect the heat capacity at constant volume for a van der Waals gas by relating it to temperature and pressure variations.
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Step 1: Effect of volume on Cv at constant T for any substance

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