:) Using the ideal gas model, show that (*), (*), (),
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e36a6f6-5c57-4c3e-b4bb-5bd0d2f1f6a7%2Fd836727c-4aa2-426b-9e4d-722568ac2e5e%2Foqx32w_processed.jpeg&w=3840&q=75)
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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