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:) Using the ideal gas model, show that (*), (*), (),

:) Using the ideal gas model, show that (*), (*), (),

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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### Ideal Gas Law - Fundamental Thermodynamic Relationship **Problem Statement:** Using the ideal gas model, show that: \[ \left( \frac{\partial P}{\partial V} \right)_T \left( \frac{\partial V}{\partial T} \right)_P \left( \frac{\partial T}{\partial P} \right)_V = -1 \] **Explanation:** This problem requires you to use the ideal gas law to derive a fundamental thermodynamic relationship. Here, \( P \) denotes pressure, \( V \) denotes volume, and \( T \) denotes temperature. The partial derivatives are taken with respect to the variables indicated by the subscripts: - \( \left( \frac{\partial P}{\partial V} \right)_T \) is the rate of change of pressure with respect to volume at constant temperature. - \( \left( \frac{\partial V}{\partial T} \right)_P \) is the rate of change of volume with respect to temperature at constant pressure. - \( \left( \frac{\partial T}{\partial P} \right)_V \) is the rate of change of temperature with respect to pressure at constant volume. The goal is to demonstrate that the product of these three partial derivatives is equal to \(-1\). Understanding the behavior of these partial derivatives in the context of the ideal gas law, \( PV = nRT \), is key to solving this problem. #### The Ideal Gas Law The ideal gas law is given by the equation: \[ PV = nRT \] where: - \( n \) is the number of moles of the gas. - \( R \) is the universal gas constant. From this equation, various relationships between \( P \), \( V \), and \( T \) can be derived for a given amount of gas \( n \). ### Steps to Solution 1. **Differential Form of the Ideal Gas Law:** Take the natural logarithm of both sides and differentiate accordingly to find the partial derivatives. 2. **Evaluate Each Term:** - Evaluate \( \left( \frac{\partial P}{\partial V} \right)_T \) using the ideal gas law while holding \( T \) constant. - Evaluate \( \left( \frac{\partial V}{\partial T} \right)_P \) while holding \( P \) constant.
Transcribed Image Text:### Ideal Gas Law - Fundamental Thermodynamic Relationship **Problem Statement:** Using the ideal gas model, show that: \[ \left( \frac{\partial P}{\partial V} \right)_T \left( \frac{\partial V}{\partial T} \right)_P \left( \frac{\partial T}{\partial P} \right)_V = -1 \] **Explanation:** This problem requires you to use the ideal gas law to derive a fundamental thermodynamic relationship. Here, \( P \) denotes pressure, \( V \) denotes volume, and \( T \) denotes temperature. The partial derivatives are taken with respect to the variables indicated by the subscripts: - \( \left( \frac{\partial P}{\partial V} \right)_T \) is the rate of change of pressure with respect to volume at constant temperature. - \( \left( \frac{\partial V}{\partial T} \right)_P \) is the rate of change of volume with respect to temperature at constant pressure. - \( \left( \frac{\partial T}{\partial P} \right)_V \) is the rate of change of temperature with respect to pressure at constant volume. The goal is to demonstrate that the product of these three partial derivatives is equal to \(-1\). Understanding the behavior of these partial derivatives in the context of the ideal gas law, \( PV = nRT \), is key to solving this problem. #### The Ideal Gas Law The ideal gas law is given by the equation: \[ PV = nRT \] where: - \( n \) is the number of moles of the gas. - \( R \) is the universal gas constant. From this equation, various relationships between \( P \), \( V \), and \( T \) can be derived for a given amount of gas \( n \). ### Steps to Solution 1. **Differential Form of the Ideal Gas Law:** Take the natural logarithm of both sides and differentiate accordingly to find the partial derivatives. 2. **Evaluate Each Term:** - Evaluate \( \left( \frac{\partial P}{\partial V} \right)_T \) using the ideal gas law while holding \( T \) constant. - Evaluate \( \left( \frac{\partial V}{\partial T} \right)_P \) while holding \( P \) constant.
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